Cohomology
of
Number Fields
by
Jürgen Neukirch
Alexander Schmidt
Kay Wingberg
Second Edition
corrected version 2.2, July 2015
Electronic Edition
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Vorwort
Als unser Freund und Lehrer Jürgen Neukirch Anfang 1997 starb, hinterließ er den Entwurf zu einem Buch über die Kohomologie der Zahlkörper,
welches als zweiter Band zu seiner Monographie Algebraische Zahlentheorie
gedacht war. Für die Kohomologie proendlicher Gruppen, sowie für Teile der
Kohomologie lokaler und globaler Körper lag bereits eine Rohfassung vor,
die schon zu einer regen Korrespondenz zwischen Jürgen Neukirch und uns
geführt hatte.
In den letzten zwei Jahren ist, ausgehend von seinem Entwurf, das hier
vorliegende Buch entstanden. Allerdings wussten wir nur teilweise, was Jürgen
Neukirch geplant hatte. So mag es sein, dass wir Themen ausgelassen haben,
welche er berücksichtigen wollte, und anderes, nicht Geplantes, aufgenommen
haben.
Jürgen Neukirchs inspirierte und pointierte Art, Mathematik auf hohem
sprachlichen Niveau darzustellen, ist für uns stets Vorbild gewesen. Leider
erreichen wir nicht seine Meisterschaft, aber wir haben uns alle Mühe gegeben
und hoffen, ein Buch in seinem Sinne und nicht zuletzt auch zum Nutzen seiner
Leser fertig gestellt zu haben.
Heidelberg, im September 1999
Alexander Schmidt
Kay Wingberg
J.Neukirch, A.Schmidt, K.Wingberg: www.mathi.uni-heidelberg.de/∼schmidt/NSW/
Electronic Edition. Free for private, not for commercial use.
J.Neukirch, A.Schmidt, K.Wingberg: www.mathi.uni-heidelberg.de/∼schmidt/NSW/
Electronic Edition. Free for private, not for commercial use.
Introduction
Number theory, one of the most beautiful and fascinating areas of mathematics, has made major progress over the last decades, and is still developing
rapidly. In the beginning of the foreword to his book Algebraic Number Theory,
J. Neukirch wrote
" Die Zahlentheorie nimmt unter den mathematischen Disziplinen
eine ähnlich idealisierte Stellung ein wie die Mathematik selbst
unter den anderen Wissenschaften." ∗)
Although the joint authors of the present book wish to reiterate this statement,
we wish to stress also that number theory owes much of its current strong
development to its interaction with almost all other mathematical fields. In
particular, the geometric (and consequent functorial) point of view of arithmetic
geometry uses techniques from, and is inspired by, analysis, geometry, group
theory and algebraic topology. This interaction had already started in the
1950s with the introduction of group cohomology to local and global class
field theory, which led to a substantial simplification and unification of this
area.
The aim of the present volume is to provide a textbook for students, as well
as a reference book for the working mathematician on cohomological topics
in number theory. Its main subject is Galois modules over local and global
fields, objects which are typically associated to arithmetic schemes. In view
of the enormous quantity of material, we were forced to restrict the subject
matter in some way. In order to keep the book at a reasonable length, we
have therefore decided to restrict attention to the case of dimension less than
or equal to one, i.e. to the global fields themselves, and the various subrings
contained in them. Central and frequently used theorems such as the global
duality theorem of G. POITOU and J. TATE, as well as results such as the theorem
of I. R. ŠAFAREVIČ on the realization of solvable groups as Galois groups over
global fields, had been part of algebraic number theory for a long time. But the
proofs of statements like these were spread over many original articles, some
of which contained serious mistakes, and some even remained unpublished. It
was the initial motivation of the authors to fill these gaps and we hope that the
result of our efforts will be useful for the reader.
In the course of the years since the 1950s, the point of view of class field
theory has slightly changed. The classical approach describes the Galois groups
∗) “Number theory, among the mathematical disciplines, occupies a similar idealized position
to that held by mathematics itself among the sciences.”
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viii
Introduction
of finite extensions using arithmetic invariants of the local or global ground
field. An essential feature of the modern point of view is to consider infinite
Galois groups instead, i.e. one investigates the set of all finite extensions of the
field k at once, via the absolute Galois group Gk . These groups intrinsically
come equipped with a topology, the Krull topology, under which they are
Hausdorff, compact and totally disconnected topological groups. It proves to
be useful to ignore, for the moment, their number theoretical motivation and to
investigate topological groups of this type, the profinite groups, as objects of
interest in their own right. For this reason, an extensive “algebra of profinite
groups” has been developed by number theorists, not as an end in itself, but
always with concrete number theoretical applications in mind. Nevertheless,
many results can be formulated solely in terms of profinite groups and their
modules, without reference to the number theoretical background.
The first part of this book deals with this “profinite algebra”, while the
arithmetic applications are contained in the second part. This division should
not be seen as strict; sometimes, however, it is useful to get an idea of how
much algebra and how much number theory is contained in a given result.
A significant feature of the arithmetic applications is that classical reciprocity
laws are reflected in duality properties of the associated infinite Galois groups.
For example, the reciprocity law for local fields corresponds to Tate’s duality
theorem for local cohomology. This duality property is in fact so strong that it
becomes possible to describe, for an arbitrary prime p, the Galois groups of the
maximal p-extensions of local fields. These are either free groups or groups
with a very special structure, which are now known as Demuškin groups. This
result then became the basis for the description of the full absolute Galois
group of a p-adic local field by U. JANNSEN and the third author.
The global case is rather different. As was already noticed by J. TATE,
the absolute Galois group of a global field is not a duality group. It is the
geometric point of view, which offers an explanation of this phenomenon: the
duality comes from the curve rather than from its generic point. It is therefore
natural to consider the étale fundamental groups π1et (Spec(Ok,S )), where S is
a finite set of places of k. Translated to the language of Galois groups, the
fundamental group of Spec(Ok,S ) is a quotient of the full group Gk , namely,
the Galois group Gk,S of the maximal extension of k which is unramified
outside S. If S contains all places that divide the order of the torsion of a
module M , the central Poitou-Tate duality theorem provides a duality between
the localization kernels in dimensions one and two. In conjunction with Tate
local duality, this can also be expressed in the form of a long 9-term sequence.
The duality theorem of Poitou-Tate remains true for infinite sets of places S
and, using topologically restricted products of local cohomology groups, the
long exact sequence can be generalized to this case. The question of whether
J.Neukirch, A.Schmidt, K.Wingberg: www.mathi.uni-heidelberg.de/∼schmidt/NSW/
Electronic Edition. Free for private, not for commercial use.
Introduction
ix
the group Gk,S is a duality group when S is finite was positively answered by
the second author.
As might already be clear from the above considerations, the basic technique
used in this book is Galois cohomology, which is essential for class field theory. For a more geometric point of view, it would have been desirable to have
also formulated the results throughout in the language of étale cohomology.
However, we decided to leave this to the reader. Firstly, the technique of sheaf
cohomology associated to a Grothendieck topos is sufficiently covered in the
literature (see [5], [139], [228]) and, in any case, it is an easy exercise (at least
in dimension ≤ 1) to translate between the Galois and the étale languages. A
further reason is that results which involve infinite sets of places (necessary
when using Dirichlet density arguments) or infinite extension fields, can be
much better expressed in terms of Galois cohomology than of étale cohomology of pro-schemes. When the geometric point of view seemed to bring a
better insight or intuition, however, we have added corresponding remarks or
footnotes. A more serious gap, due to the absence of Grothendieck topologies,
is that we cannot use flat cohomology and the global flat duality theorem of
Artin-Mazur. In chapter VIII, we therefore use an ad hoc construction, the
group BS , which measures the size of the localization kernel for the first flat
cohomology group with the roots of unity as coefficients.
Let us now examine the contents of the individual chapters more closely.
The first part covers the algebraic background for the number theoretical applications. Chapter I contains well-known basic definitions and results, which
may be found in several monographs. This is only partly true for chapter II: the
explicit description of the edge morphisms of the Hochschild-Serre spectral
sequence in §2 is certainly well-known to specialists, but is not to be found in
the literature. In addition, the material of §3 is well-known, but contained only
in original articles.
Chapter III considers abstract duality properties of profinite groups. Among
the existing monographs which also cover large parts of the material, we should
mention the famous Cohomologie Galoisienne by J.-P. SERRE and H. KOCH’s
book Galoissche Theorie der p-Erweiterungen. Many details, however, have
been available until now only in the original articles.
In chapter IV, free products of profinite groups are considered. These are
important for a possible non-abelian decomposition of global Galois groups
into local ones. This happens only in rather rare, degenerate situations for
Galois groups of global fields, but it is quite a frequent phenomenon for
subgroups of infinite index. In order to formulate such statements (like the
arithmetic form of Riemann’s existence theorem in chapter X), we develop the
concept of the free product of a bundle of profinite groups in §3.
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x
Introduction
Chapter V deals with the algebraic foundations of Iwasawa theory. We will
not prove the structure theorem for Iwasawa modules in the usual way by using
matrix calculations (even though it may be more acceptable to some mathematicians, as it is more concrete), but we will follow mostly the presentation
found in Bourbaki, Commutative Algebra, with a view to more general situations. Moreover, we present results concerning the structure of these modules
up to isomorphism, which are obtained using the homotopy theory of modules
over group rings, as presented by U. JANNSEN.
The central technical result of the arithmetic part is the famous global duality
theorem of Poitou-Tate. We start, in chapter VI, with general facts about Galois
cohomology. Chapter VII deals with local fields. Its first three sections largely
follow the presentation of J.-P. SERRE in Cohomologie Galoisienne. The
next two sections are devoted to the explicit determination of the structure
of local Galois groups. In chapter VIII, the central chapter of this book, we
give a complete proof of the Poitou-Tate theorem, including its generalization
to finitely generated modules. We begin by collecting basic results on the
topological structure, universal norms and the cohomology of the S-idèle class
group, before moving on to the proof itself, given in sections 4 and 6. In
the proof, we apply the group theoretical theorems of Nakayama-Tate and of
Poitou, proven already in chapter III.
In chapter IX, we reap the rewards of our efforts in the previous chapters. We
prove several classical number theoretical results, such as the Hasse principle
and the Grunwald-Wang theorem. In §5, we consider embedding problems
and we present the theorem of K. IWASAWA to the effect that the maximal
prosolvable factor of the absolute Galois group of Qab is free. In §6, we give
a complete proof of Šafarevič’s theorem on the realization of finite solvable
groups as Galois groups over global fields.
The main concern of chapter X is to consider restricted ramification. Geometrically speaking, we are considering the curves Spec(Ok,S ), in contrast
to chapter IX, where our main interest was in the point Spec(k). Needless to
say, things now become much harder. Invariants like the S-ideal class group
or the p-adic regulator enter the game and establish new arithmetic obstructions. Our investigations are guided by the analogy between number fields
and function fields. We know a lot about the latter from algebraic geometry,
and we try to establish analogous results for number fields. For example,
using the group theoretical techniques of chapter IV, we can prove the number
theoretical analogue of Riemann’s existence theorem. The fundamental group
of Spec(Ok ), i.e. the Galois group of the maximal unramified extension of the
number field k, was the subject of the long-standing class field tower problem
in number theory, which was finally answered negatively by E. S. GOLOD and
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Electronic Edition. Free for private, not for commercial use.
Introduction
xi
I. R. ŠAFAREVIČ. We present their proof, which demonstrates the power of the
group theoretical and cohomological methods, in §8.
Chapter XI deals with Iwasawa theory, which is the consequent conceptual
continuation of the analogy between number fields and function fields. We
concentrate on the algebraic aspects of Iwasawa theory of p-adic local fields
and of number fields, first presenting the classical statements which one can
usually find in the standard literature. Then we prove more far-reaching results
on the structure of certain Iwasawa modules attached to p-adic local fields and
to number fields, using the homotopy theory of Iwasawa modules. The analytic
aspects of Iwasawa theory will merely be described, since this topic is covered by several monographs, for example, the book [246] of L. WASHINGTON.
Finally, the Main Conjecture of Iwasawa theory will be formulated and discussed; for a proof, we refer the reader to the original work of B. MAZUR and
A. WILES ([134], [249]).
In the last chapter, we give a survey of so-called anabelian geometry, a
program initiated by A. GROTHENDIECK. Perhaps the first result of this theory,
obtained even before this program existed, is a theorem of J. NEUKIRCH and
K. UCHIDA which asserts that the absolute Galois group of a global field, as a
profinite group, characterizes the field up to isomorphism. We give a proof of
this theorem for number fields in the first two sections. The final section gives
an overview of the conjectures and their current status.
The reader will recognize very quickly that this book is not a basic textbook
in the sense that it is completely self-contained. We use freely basic algebraic,
topological and arithmetic facts which are commonly known and contained in
the standard textbooks. In particular, the reader should be familiar with basic
number theory. While assuming a certain minimal level of knowledge, we
have tried to be as complete and as self-contained as possible at the next stage.
We give full proofs of almost all of the main results, and we have tried not
to use references which are only available in original papers. This makes it
possible for the interested student to use this book as a textbook and to find
large parts of the theory coherently ordered and gently accessible in one place.
On the other hand, this book is intended for the working mathematician as a
reference on cohomology of local and global fields.
Finally, a remark on the exercises at the end of the sections. A few of them
are not so much exercises as additional remarks which did not fit well into the
main text. Most of them, however, are intended to be solved by the interested
reader. However, there might be occasional mistakes in the way they are posed.
If such a case arises, it is an additional task for the reader to give the correct
formulation.
J.Neukirch, A.Schmidt, K.Wingberg: www.mathi.uni-heidelberg.de/∼schmidt/NSW/
Electronic Edition. Free for private, not for commercial use.
xii
Preface to the Second Edition
We would like to thank many friends and colleagues for their mathematical examination of parts of this book, and particularly, ANTON DEITMAR,
TORSTEN FIMMEL, DAN HARAN, UWE JANNSEN, HIROAKI NAKAMURA and
OTMAR VENJAKOB. We are indebted to Mrs. INGE MEIER who TEXed a large
part of the manuscript, and EVA-MARIA STROBEL receives our special gratitude
for her careful proofreading. Hearty thanks go to FRAZER JARVIS for going
through the entire manuscript, correcting our English.
Heidelberg, September 1999
Alexander Schmidt
Kay Wingberg
Preface to the Second Edition
The present second edition is a corrected and extended version of the first.
We have tried to improve the exposition and reorganize the content to some
extent; furthermore, we have included some new material. As an unfortunate
result, the numbering of the first edition is not compatible with the second.
In the algebraic part you will find new sections on filtered cochain complexes, on the degeneration of spectral sequences and on Tate cohomology of
profinite groups. Amongst other topics, the arithmetic part contains a new
section on duality theorems for unramified and tamely ramified extensions, a
careful analysis of 2-extensions of real number fields and a complete proof of
Neukirch’s theorem on solvable Galois groups with given local conditions.
Since the publication of the first edition, many people have sent us lists
of corrections and suggestions or have contributed in other ways to this edition. We would like to thank them all. In particular, we would like to thank
JAKOB STIX and DENIS VOGEL for their comments on the new parts of this
second edition and FRAZER JARVIS, who again did a great job correcting our
English.
Regensburg and Heidelberg, November 2007
Alexander Schmidt
Kay Wingberg
J.Neukirch, A.Schmidt, K.Wingberg: www.mathi.uni-heidelberg.de/∼schmidt/NSW/
Electronic Edition. Free for private, not for commercial use.
Contents
Algebraic Theory
Chapter I: Cohomology of Profinite Groups
§1. Profinite Spaces and Profinite Groups .
§2. Definition of the Cohomology Groups .
§3. The Exact Cohomology Sequence . . .
§4. The Cup-Product . . . . . . . . . . . .
§5. Change of the Group G . . . . . . . . .
§6. Basic Properties . . . . . . . . . . . .
§7. Cohomology of Cyclic Groups . . . . .
§8. Cohomological Triviality . . . . . . . .
§9. Tate Cohomology of Profinite Groups .
Chapter II: Some Homological Algebra
§1. Spectral Sequences . . . . . . . . . . .
§2. Filtered Cochain Complexes . . . . . .
§3. Degeneration of Spectral Sequences . .
§4. The Hochschild-Serre Spectral Sequence
§5. The Tate Spectral Sequence . . . . . . .
§6. Derived Functors . . . . . . . . . . . .
§7. Continuous Cochain Cohomology . . .
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Chapter III: Duality Properties of Profinite Groups
§1. Duality for Class Formations . . . . . . . . . . . . . . . . . .
§2. An Alternative Description of the Reciprocity Homomorphism
§3. Cohomological Dimension . . . . . . . . . . . . . . . . . . .
§4. Dualizing Modules . . . . . . . . . . . . . . . . . . . . . . .
§5. Projective pro-c-groups . . . . . . . . . . . . . . . . . . . . .
§6. Profinite Groups of scd G = 2 . . . . . . . . . . . . . . . . .
§7. Poincaré Groups . . . . . . . . . . . . . . . . . . . . . . . .
§8. Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
§9. Generators and Relations . . . . . . . . . . . . . . . . . . . .
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Electronic Edition. Free for private, not for commercial use.
xiv
Contents
Chapter IV: Free Products of Profinite Groups
§1. Free Products . . . . . . . . . . . . . . . . . . . . . . . . . . .
§2. Subgroups of Free Products . . . . . . . . . . . . . . . . . . . .
§3. Generalized Free Products . . . . . . . . . . . . . . . . . . . .
245
245
252
256
Chapter V: Iwasawa Modules
§1. Modules up to Pseudo-Isomorphism . . .
§2. Complete Group Rings . . . . . . . . . .
§3. Iwasawa Modules . . . . . . . . . . . . .
§4. Homotopy of Modules . . . . . . . . . .
§5. Homotopy Invariants of Iwasawa Modules
§6. Differential Modules and Presentations . .
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Arithmetic Theory
Chapter VI: Galois Cohomology
§1. Cohomology of the Additive Group
§2. Hilbert’s Satz 90 . . . . . . . . . .
§3. The Brauer Group . . . . . . . . .
§4. The Milnor K-Groups . . . . . . .
§5. Dimension of Fields . . . . . . . .
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Chapter VII: Cohomology of Local Fields
§1. Cohomology of the Multiplicative Group . . . . . .
§2. The Local Duality Theorem . . . . . . . . . . . .
§3. The Local Euler-Poincaré Characteristic . . . . . .
§4. Galois Module Structure of the Multiplicative Group
§5. Explicit Determination of Local Galois Groups . . .
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Chapter VIII: Cohomology of Global Fields
§1. Cohomology of the Idèle Class Group . . . . . . . . . .
§2. The Connected Component of Ck . . . . . . . . . . . . .
§3. Restricted Ramification . . . . . . . . . . . . . . . . . .
§4. The Global Duality Theorem . . . . . . . . . . . . . . .
§5. Local Cohomology of Global Galois Modules . . . . . .
§6. Poitou-Tate Duality . . . . . . . . . . . . . . . . . . . .
§7. The Global Euler-Poincaré Characteristic . . . . . . . .
§8. Duality for Unramified and Tamely Ramified Extensions .
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J.Neukirch, A.Schmidt, K.Wingberg: www.mathi.uni-heidelberg.de/∼schmidt/NSW/
Electronic Edition. Free for private, not for commercial use.
xv
Contents
Chapter IX: The Absolute Galois Group of a Global Field
§1. The Hasse Principle . . . . . . . . . . . . . . . . . .
§2. The Theorem of Grunwald-Wang . . . . . . . . . . .
§3. Construction of Cohomology Classes . . . . . . . . .
§4. Local Galois Groups in a Global Group . . . . . . .
§5. Solvable Groups as Galois Groups . . . . . . . . . .
§6. Šafarevič’s Theorem . . . . . . . . . . . . . . . . .
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Chapter XII: Anabelian Geometry
§1. Subgroups of Gk . . . . . . . . . . . . . . . . . . . . . . . . .
§2. The Neukirch-Uchida Theorem . . . . . . . . . . . . . . . . . .
§3. Anabelian Conjectures . . . . . . . . . . . . . . . . . . . . . .
785
785
791
798
Literature
805
Index
821
Chapter X: Restricted Ramification
§1. The Function Field Case . . . . . . . . . . .
§2. First Observations on the Number Field Case .
§3. Leopoldt’s Conjecture . . . . . . . . . . . .
§4. Cohomology of Large Number Fields . . . .
§5. Riemann’s Existence Theorem . . . . . . . .
§6. The Relation between 2 and ∞ . . . . . . . .
§7. Dimension of H i (GST , ZZ/pZZ) . . . . . . . .
§8. The Theorem of Kuz’min . . . . . . . . . . .
§9. Free Product Decomposition of GS (p) . . . .
§10. Class Field Towers . . . . . . . . . . . . . .
§11. The Profinite Group GS . . . . . . . . . . . .
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Chapter XI: Iwasawa Theory of Number Fields
§1. The Maximal Abelian Unramified p-Extension of k∞ . . . .
§2. Iwasawa Theory for p-adic Local Fields . . . . . . . . . . .
§3. The Maximal Abelian p-Extension of k∞ Unramified Outside S
§4. Iwasawa Theory for Totally Real Fields and CM-Fields . . .
§5. Positively Ramified Extensions . . . . . . . . . . . . . . . .
§6. The Main Conjecture . . . . . . . . . . . . . . . . . . . . .
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Electronic Edition. Free for private, not for commercial use.
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Algebraic Theory
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Chapter I
Cohomology of Profinite Groups
Profinite groups are topological groups which naturally occur in algebraic
number theory as Galois groups of infinite field extensions or more generally as
étale fundamental groups of schemes. Their cohomology groups often contain
important arithmetic information.
In the first chapter we will study profinite groups as objects of interest in
themselves, independently of arithmetic applications, which will be treated in
the second part of this book.
§1. Profinite Spaces and Profinite Groups
The underlying topological spaces of profinite groups are of a very specific
type, which will be described now. To do this, we make use of the concept
of inverse (or projective) limits. We refer the reader to the standard literature
(e.g. [160], [79], [139]) for the definition and basic properties of limits. All
index sets will be assumed to be filtered.
(1.1.1) Lemma. For a Hausdorff topological space T the following conditions
are equivalent.
(i) T is the (topological) inverse limit of finite discrete spaces.
(ii) T is compact and every point of T has a basis of neighbourhoods consisting of subsets which are both closed and open.
(iii) T is compact and totally disconnected.
Proof: In order to show the implication (i) ⇒ (ii), we first recall that the
inverse limit of compact spaces is compact (see [15] chap.I, §9, no.6, prop.8).
Therefore T is compact. By the definition of the inverse limit topology and
by (i), every point of T has a basis of neighbourhoods consisting of sets of the
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4
Chapter I. Cohomology of Profinite Groups
form f −1 (W ), where W is a subset of a finite discrete space V and f : T → V
is a continuous map. These sets are both open and closed.
For the implication (ii) ⇒ (iii) we have to show that the connected component
Ct of every point t ∈ T equals {t}. Since T is compact, Ct is the intersection
of all closed and open subsets containing t (see [15] chap.II, §4, no.4, prop.6).
Since T is Hausdorff, we obtain Ct = {t}.
It remains to show the implication (iii) ⇒ (i). Let I be the set of equivalence
relations R ⊆ T × T on T , such that the quotient space T /R is finite and
discrete in the quotient topology. The set I is partially ordered by inclusion
and is directed, because R1 ∩ R2 is in I if R1 and R2 are. We claim that the
canonical map φ : T → lim
T /R is a homeomorphism.
←− R∈I
First we see that the map φ is surjective, because for an element {tR }R∈I ∈
lim
T /R, the sets (pR ◦ φ)−1 (tR ) are nonempty and compact. Since I is
←− R∈I
directed, finite intersections of these sets are also nonempty and compactness
T
then implies that φ−1 ({tR }R∈I ) = R∈I (pR ◦ φ)−1 (tR ) is nonempty.
For the injectivity it suffices to show that for t, s ∈ T , t =/ s, there exists an
R ∈ I such that (t, s) ∈/ R. But since s is not in the connected component of t,
there exists a closed and open subset U ⊆ T with t ∈ U , s ∈/ U (see [15] chap.II,
§4, no.4, prop.6). Then the equivalence relation R defined by "(x, y) ∈ R if
x and y are both in U or both not in U " is of the required type. The proof is
completed by the remark that a continuous bijection between compact spaces
is a homeomorphism.
2
In fact one immediately verifies that we could have chosen the inverse system
in (i) in such a way that all transition maps are surjective.
(1.1.2) Definition. A space T is called a profinite space if it satisfies the
equivalent conditions of lemma (1.1.1).
A compactness argument shows that a subset V ⊆ lim
Xi of a profinite space
←−
is both closed and open if and only if V is the pre-image under the canonical
projection pi : X → Xi of a (necessarily closed and open) subset in Xi for
some i. Every continuous map between profinite spaces can be realized as
a projective limit of maps between finite discrete spaces. Without giving an
exact definition, we want to note that the category of profinite spaces with
continuous maps is the pro-category of the category of finite discrete spaces.
Recall that a topological group is a group G endowed with the structure of
a topological space, such that the group operations G → G, g 7→ g −1 , and
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§1. Profinite Spaces and Profinite Groups
G × G → G, (g, h) 7→ gh, are continuous. The reader will immediately
verify that the inverse limit of an inverse system of topological groups is just
the inverse limit of the groups together with the inverse limit topology on the
underlying topological space.
(1.1.3) Proposition. For a Hausdorff topological group G the following conditions are equivalent.
(i) G is the (topological) inverse limit of finite discrete groups.
(ii) G is compact and the unit element has a basis of neighbourhoods consisting of open and closed normal subgroups.
(iii) G is compact and totally disconnected.
Proof: (i) ⇒ (iii): The inverse limit of compact and totally disconnected
spaces is compact and totally disconnected.
(ii) ⇒ (i): Assume that U runs through a system of neighbourhoods of the unit
element e ∈ G, which consists of open normal subgroups. Then the canonical
homomorphism φ : G → lim
G/U is an isomorphism:
←−
U
To begin with, φ is injective, because G is Hausdorff. In order to show the
surjectivity, let x = {xU }U ∈ lim G/U . Denoting the canonical projection
←− U
by φU : G → G/U , we have the equality
φ−1 (x) =
\
φ−1
U (xU ).
U
The intersection on the right side is taken over nonempty compact spaces and
finite intersections of these are nonempty. Hence φ−1 (x) is nonempty, and
therefore φ is surjective. Furthermore, φ is open, hence a homeomorphism.
Finally, for every such U , the group G/U is discrete and compact, hence finite.
(iii) ⇒ (ii): By (1.1.1), the underlying topological space of G is profinite,
hence every point has a basis of neighbourhoods consisting of open and closed
subsets. Note that an open subgroup is automatically closed, because it is the
complement of the union of its (open) nontrivial cosets. Let U be an arbitrary
chosen, closed and open neighbourhood of the unit element e ∈ G. Set
V := {v
∈
U | Uv
⊆
U },
H := {h ∈ V | h−1
∈
V }.
We claim that H ⊆ U is an open (and closed) subgroup in G. We first show
that V is open. Fix a point v ∈ V . Then uv ∈ U for every u ∈ U and therefore
there exist neighbourhoods Uu of u and Vu of v, such that Uu Vu ⊆ U . The open
sets Uu cover the compact space U and therefore there exists a finite subcover,
Uu1 , . . . , Uun , say. Then
Vv := Vu1 ∩ · · · ∩ Vun
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Chapter I. Cohomology of Profinite Groups
is an open neighbourhood of v contained in V . Hence V is open and also
H := V ∩ V −1 , since the inversion map is a homeomorphism. It remains to
show that H is a subgroup. Trivially e ∈ H and H −1 = H by construction.
We now check that xy ∈ H if x, y ∈ H. First we have U xy ⊆ U y ⊆ U , and so
xy ∈ V . In the same way we obtain y −1 x−1 ∈ V , hence xy ∈ H. This proves
that H is an open subgroup of G contained in U . In particular, H has finite
index in G and there are only finitely many different conjugates of H. The
intersection of these finitely many conjugates is an open, closed and normal
subgroup of G contained in U .
2
(1.1.4) Definition. A Hausdorff topological group G satisfying the equivalent
conditions of (1.1.3) is called a profinite group.
Without further mention, homomorphisms between profinite groups are always assumed to be continuous and subgroups are assumed to be closed. Since
a subgroup is the complement of its nontrivial cosets and by the compactness
of G, we see that open subgroups are closed and a closed subgroup is open if
and only if it has finite index. If H is a (closed) subgroup of the profinite group
G, then the set G/H of coset classes with the quotient topology is a profinite
space. If H is normal, then the quotient G/H is a profinite group in a natural
way.
In principle, all objects and statements of the theory of finite groups have
their topological analogue in the theory of profinite groups. For example, the
profinite analogues of the Sylow theorems are true (see §6). We make the
following
(1.1.5) Definition. A supernatural number is a formal product
Y
pnp ,
p
where p runs through all prime numbers and, for each p, the exponent np is a
non-negative integer or the symbol ∞.
Using the unique decomposition into prime powers, we can view any natural
number as a supernatural number. We multiply supernatural numbers (even
infinitely many of them) by adding the exponents. By convention, the sum of
the exponents is ∞ if infinitely many summands are non-zero or if one of the
summands is ∞. We also have the notions of l.c.m. and g.c.d. of an arbitrary
family of supernatural numbers. In particular, any family of natural numbers
has an l.c.m., which is, in general, a supernatural number.
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§1. Profinite Spaces and Profinite Groups
(1.1.6) Definition. Let G be a profinite group and let A be an abelian torsion
group.
(i) The index of a closed subgroup H in G is the supernatural number
(G : H) = l.c.m.(G/U : H/H ∩ U ),
where U ranges over all open normal subgroups of G.
(ii) The order of G is defined by
#G = (G : 1) = l.c.m. #(G/U ).
U
(iii) The order of A is defined by
#A = l.c.m. #B ,
where B ranges over all finite subgroups of A.
Given closed subgroups N ⊆ H ⊆ G, we have
(G : N ) = (G : H)(H : N ).
Furthermore, the order #A of an abelian torsion group A is just the order of the
profinite group Hom(A, Q/ZZ).
(1.1.7) Definition. Let G be a profinite group. An abstract G-module M is
an abelian group M together with an action
G × M → M, (g, m) 7→ g(m)
such that 1(m) = m, (gh)(m) = g(h(m)) and g(m + n) = g(m) + g(n) for all
g, h ∈ G, m, n ∈ M .
A topological G-module M is an abelian Hausdorff topological group M
which is endowed with the structure of an abstract G-module such that the
action G × M → M is continuous.
For a closed subgroup H ⊆ G we denote the subgroup of H-invariant
elements in M by M H , i.e.
M H = {m ∈ M | h(m) = m for all h ∈ H}.
(1.1.8) Proposition. Let G be a profinite group and let M be an abstract
G-module. Then the following conditions are equivalent:
(i) M is a discrete G-module, i.e. the action G × M → M is continuous for
the discrete topology on M .
(ii) For every m ∈ M the subgroup Gm := {g
(iii) M =
S
∈
G | g(m) = m} is open.
M U , where U runs through the open subgroups of G.
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Chapter I. Cohomology of Profinite Groups
Proof: If we restrict the map G × M → M to G × {m}, then m ∈ M has
pre-image Gm × {m}. This shows (i)⇒(ii). The assertion (ii)⇒(iii) is trivial
because m ∈ M Gm . Finally, assume that (iii) holds. Let (g, m) ∈ G×M . There
exists an open subgroup U such that m ∈ M U . Therefore gU × {m} is an open
neighbourhood of (g, m) ∈ G × M mapping to g(m). This shows (i).
2
In this book we are mainly concerned with discrete modules and so the term
G-module, without the word “topological” or “abstract”, will always mean a
discrete module.
L
If (Ai )i∈I is a family of discrete G-modules, then their direct sum i∈I Ai ,
endowed with the componentwise G-action g((ai )i∈I ) = (g(ai ))i∈I , is again a
discrete G-module, but this is not necessarily true for the product. The tensor
product
A ⊗ B = A ⊗ ZZ B
of two discrete modules endowed with the diagonal action g(a⊗b) = g(a)⊗g(b)
is a discrete module. The set Hom(A, B) = Hom ZZ (A, B) becomes an abstract
G-module by setting g(φ)(a) = g(φ(g −1 (a))). Its subgroup of invariants
HomG (A, B) = Hom(A, B)G
is the set of G-homomorphisms from A to B. If A = AU for some open
subgroup U ⊆ G, then Hom(A, B) is a discrete G-module. This is the case,
for example, if G is finite or if A is finitely generated as a ZZ-module.
The groups ZZ, Q, ZZ/nZZ, IFq are always viewed as trivial discrete G-modules,
i.e. G-modules with trivial action of G.
So far we have considered totally disconnected compact groups. If A is any
topological group, then the connected component A0 (of the identity) of A is
a closed subgroup. We have the following general facts for which we refer to
[170] sec. 22, and [15] chap. III, §4.6.
(1.1.9) Proposition. Let A be a locally compact group. Then
(i) A0 is the intersection of all open normal subgroups of A, and A/A0 is the
largest totally disconnected quotient.
(ii) A0 is generated by every open neighbourhood of 1 in A0 .
(iii) If A → B is a continuous surjective homomorphism onto the locally
compact group B, then the closure of the image of A0 is the connected
component of 1 in B.
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§1. Profinite Spaces and Profinite Groups
9
An essential tool for working with locally compact abelian groups is a duality
theorem due to L. S. PONTRYAGIN. We consider the group IR/ZZ as a topological
group with the quotient topology inherited from IR.
(1.1.10) Definition. Let A be a Hausdorff, abelian and locally compact topological group. We call the group
A∨ := Homcts (A, IR/ZZ)
the Pontryagin dual of A.
Given locally compact topological spaces X, Y , the set of continuous maps
Mapcts (X, Y ) carries a natural topology, the compact-open topology. A subbasis of this topology is given by the sets
UK,U = {f
∈
Mapcts (X, Y ) | f (K) ⊆ U },
where K runs through the compact subsets of X and U runs through the open
subsets of Y . For the proof of the following theorem we refer to [170], th. 5.3
or [146], th. 23, [186], th. 1.7.2.
(1.1.11) Theorem (Pontryagin Duality). If A is a Hausdorff abelian locally
compact topological group, then the same is true for A∨ endowed with the
compact-open topology. The canonical homomorphism
A −→ (A∨ )∨ ,
given by a 7−→ τa : A∨ → IR/ZZ, φ 7→ φ(a), is an isomorphism of topological
groups. Thus ∨ defines an involutory contravariant autofunctor on the category of Hausdorff abelian locally compact topological groups which moreover
commutes with limits. Furthermore, ∨ induces equivalences of categories
∨
(abelian compact groups) ⇐⇒
(discrete abelian groups)
∨
(abelian profinite groups) ⇐⇒ (discrete abelian torsion groups).
For an (abstract) abelian group A we use the notation
A∗ = Hom(A, Q/ZZ).
Clearly, if A is a discrete torsion group, then A∨ ∼
= A∗ and we will frequently
∗
∨
also write A instead of A , at least if we are not interested in the topology
of the dual. If A is abelian and profinite, then it is an easy exercise to see
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Chapter I. Cohomology of Profinite Groups
that every continuous homomorphism φ : A → IR/ZZ has finite image. If,
moreover, A is topologically finitely generated, then every subgroup of finite
index is open in A, and hence also in this case A∨ ∼
= A∗ .
Now assume that we are given a family (Xi )i∈I of Hausdorff, abelian topological groups and let an open subgroup Yi ⊆ Xi be given for almost all i ∈ I
(i.e. for all but finitely many indices). For consistency of notation, we put
Yi = Xi for the remaining indices.
(1.1.12) Definition. The restricted product
Y
(Xi , Yi )
i∈I
Y
is the subgroup of Xi consisting of all (xi )i∈I such that xi ∈ Yi for almost all i.
The restricted product is a topological group, and a basis of neighbourhoods
of the identity is given by the products
Y
Uj ×
j ∈J
Y
Yi ,
i∈I\J
where J runs over the finite subsets of I and Uj runs over a basis of neighbourhoods of the identity of Xj .
Basic examples of restricted products are the product of groups (Yi = Xi for
all i) and the direct sum of discrete groups (Yi = 0 for all i). We will write
Y
Xi
i∈ I
for short if it is clear from the context what the Yi are. The restricted product
is again a Hausdorff, abelian topological group.
(1.1.13) Proposition. If all Xi are locally compact and almost all Yi are
compact, then the restricted product is again an abelian locally compact group.
For the Pontryagin dual of the restricted product, there is a canonical isomorphism
(
(Xi , Yi ) )∨ ∼
=
(Xi∨ , (Xi /Yi )∨ ).
Y
Y
i∈I
i∈ I
Proof: The product of compact topological spaces is compact, therefore the
restricted product is locally compact under the given conditions. Furthermore,
sinceY
Yi is compact and open in Xi , the same is true for (Xi /Yi )∨ in Xi∨ . Hence
also i∈I (Xi∨ , (Xi /Yi )∨ ) exists and is locally compact.
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§1. Profinite Spaces and Profinite Groups
11
A sufficiently small open neighbourhood of 0 ∈ IR/ZZ contains
no nontrivial
Y
(Xi , Yi ) → IR/ZZ
subgroups. Therefore a continuous homomorphism φ :
annihilates Yi for almost all i. In other words, the restriction of φ to Xi lies in
∨ ⊆
∨
the
Ysubgroup (Xi /Yi ) Y Xi for almost all i. This yields a bijection between
∨
( i∈I (Xi , Yi ) ) and i∈I (Xi∨ , (Xi /Yi )∨ ), which easily can be seen to be a
homeomorphism.
2
Exercise 1. Show that an injective (resp. surjective) continuous map between profinite spaces
may be represented as an inverse limit over a system of injective (resp. surjective) maps
between finite discrete spaces.
Exercise 2. Let X be a profinite space and let X0 ⊆ X be a closed subspace. Show that
every continuous map f : X0 → Y from X0 to a finite discrete space Y has a continuous
extension F : X → Y (i.e. F |X0 = f ) and that any two such extensions coincide on an open
neighbourhood of X0 in X.
Exercise 3. Let G, H be profinite groups. Show that
Hom(G, H) = lim lim Hom(G/U, H/V ),
←− −→
V ⊆H U ⊆G
where the limits are taken over all open normal subgroups V of H and U of G.
Exercise 4. If K ⊆ H are closed subgroups of the profinite group G, then the projection
π : G/K → G/H has a continuous section s : G/H → G/K.
Hint: Let X be the set of pairs (S, s), where S is a closed subgroup such that K ⊆ S ⊆ H and
s is a continuous section s : G/H → G/S. Write (S, s) ≤ (S 0 , s0 ) if S 0 ⊆ S and if s is the
composite of s0 and the projection G/S 0 → G/S. Then X is inductively ordered. By Zorn’s
lemma, there exists a maximal element (S, s) of X. Show that S = K.
Exercise 5. A morphism φ : X → Y in a category C is called a monomorphism if for every
object Z of C and for every pair of morphisms f, g : Z → X the implication "φ ◦ f = φ ◦ g
⇒ f = g" is true. The morphism φ is called an epimorphism if it is a monomorphism in the
opposite category C op (the category obtained from C by reversing all arrows).
(i) Show that the monomorphisms in the category of profinite groups are the injective homomorphisms.
(ii) Show that the epimorphisms in the category of profinite groups are the surjective homomorphisms.
Hint for (ii): First reduce the problem to the case of finite groups. Assume that there is an
epimorphism φ : G → H of finite groups which is not surjective. Assume that (H : φ(G)) ≥ 3
(otherwise φ(G) is normal in H) and choose two elements a, b ∈ H having different nontrivial
residue classes modulo φ(G). Let S be the (finite) group of set theoretic automorphisms of H.
Let s ∈ S be the map H → H which interchanges the cosets aφ(G) and bφ(G) and which is
the identity on the other left cosets modulo φ(G). Then consider the maps f and g defined by
−1
f (h1 )(h2 ) = h2 h−1
1 and by g(h) = s f (h)s.
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Chapter I. Cohomology of Profinite Groups
§2. Definition of the Cohomology Groups
The cohomology of a profinite group G arises from the diagram
−→ G × G −→
−→
· · · −→
−→
−→ G,
−→
−→ G × G × G −→
the arrows being the projections
di : Gn+1 −→ Gn ,
given by
i = 0, 1, . . . , n,
di (σ0 , . . . , σn ) = (σ0 , . . . , σ̂i , . . . , σn ),
where by σ̂i we indicate that we have omitted σi from the (n + 1)-tuple
(σ0 , . . . , σn ). G acts on Gn by left multiplication.
From now on, we assume all G-modules to be discrete. For every G-module
A we form the abelian group
X n = X n (G, A) = Map (Gn+1 , A)
of all continuous maps x : Gn+1 −→ A, i.e. of all continuous functions
x(σ0 , . . . , σn ) with values in A. X n is in a natural way a G-module by
(σx)(σ0 , . . . , σn ) = σx(σ −1 σ0 , . . . , σ −1 σn ).
The maps di : Gn+1 −→ Gn induce G-homomorphisms d∗i : X n−1 −→ X n and
we form the alternating sum
∂n =
n
X
(−1)i d∗i : X n−1 −→ X n .
i=0
We usually write ∂ in place of ∂ n . Thus for x ∈ X n−1 , ∂x is the function
(∂x)(σ0 , . . . , σn ) =
(∗)
n
X
(−1)i x(σ0 , . . . , σ̂i , . . . , σn ).
i=0
Moreover, we have the G-homomorphism ∂ 0 : A → X 0 , which associates to
a ∈ A the constant function x(σ0 ) = a.
(1.2.1) Proposition. The sequence
∂0
∂1
∂2
0 −→ A −→ X 0 −→ X 1 −→ X 2 −→ . . .
is exact.
Proof: We first show that it is a complex, i.e. ∂∂ = 0. ∂ 1 ◦ ∂ 0 = 0 is
clear. Let x ∈ X n−1 . Applying ∂ to (∗), we obtain summands of the form
x(σ0 , . . . , σ̂i , . . . , σ̂j , . . . , σn ) with certain signs. Each of these summands
arises twice, once where first σj and then σi is omitted, and again where first
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§2. Definition of the Cohomology Groups
σi and then σj is omitted. The first time the sign is (−1)i (−1)j and the second
time (−1)i (−1)j−1 . Hence the summands cancel to give zero.
For the exactness, we consider the map D−1 : X 0 → A, D−1 x = x(1), and
for n ≥ 0 the maps
Dn : X n+1 −→ X n ,
(Dn x)(σ0 , . . . , σn ) = x(1, σ0 , . . . , σn ).
These are homomorphisms of ZZ-modules, not of G-modules. An easy calculation shows that for n ≥ 0
(∗)
Dn ◦ ∂ n+1 + ∂ n ◦ Dn−1 = id.
If x ∈ ker(∂ n+1 ) then x = ∂ n (Dn−1 x), i.e. ker(∂ n+1 )
ker(∂ n+1 ) = im(∂ n ) because ∂ n+1 ◦ ∂ n = 0.
⊆
im(∂ n ) and thus
2
An exact sequence of G-modules 0 → A → X 0 → X 1 → X 2 → . . . is
called a resolution of A and a family (Dn )n≥−1 as in the proof with the property
(∗) is called a contracting homotopy of it. The above resolution is called the
standard resolution.
We now apply the functor “fixed module”. We set for n ≥ 0
C n (G, A) = X n (G, A)G .
C n (G, A) consists of the continuous functions x : Gn+1 → A such that
x(σσ0 , . . . , σσn ) = σx(σ0 , . . . , σn )
for all σ ∈ G. These functions are called the (homogeneous) n-cochains of
G with coefficients in A. From the standard resolution (1.2.1) we obtain a
sequence
∂1
∂2
C 0 (G, A) −→ C 1 (G, A) −→ C 2 (G, A) −→ . . . ,
which in general is no longer exact. But it is still a complex, i.e. ∂∂ = 0, and
is called the homogeneous cochain complex of G with coefficients in A.
We now set
∂ n+1
Z n (G, A) = ker (C n (G, A) −→ C n+1 (G, A)),
∂n
B n (G, A) = im (C n−1 (G, A) −→ C n (G, A))
and B 0 (G, A) = 0. The elements of Z n (G, A) and B n (G, A) are called the
(homogeneous) n-cocycles and n-coboundaries respectively. As ∂∂ = 0, we
have B n (G, A) ⊆ Z n (G, A).
(1.2.2) Definition. For n ≥ 0 the factor group
H n (G, A) = Z n (G, A)/B n (G, A)
is called the n-dimensional cohomology group of G with coefficients in A.
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Chapter I. Cohomology of Profinite Groups
For computational purposes, and for many applications, it is convenient to
pass to a modified definition of the cohomology groups, which reduces the
number of variables in the homogeneous cochains x(σ0 , . . . , σn ) by one. Let
C 0 (G, A) = A and C n (G, A), n ≥ 1, be the abelian group of all continuous
functions y : Gn −→ A. We then have the isomorphism
C 0 (G, A) −→ C 0 (G, A),
x(σ) 7−→ x(1),
and for n ≥ 1 the isomorphism
C n (G, A) −→ C n (G, A),
x(σ0 , . . . , σn ) 7→ y(σ1 , . . . , σn ) = x(1, σ1 , σ1 σ2 , . . . , σ1 · · · σn ),
whose inverse is given by
−1
y(σ1 , . . . , σn ) 7→ x(σ0 , . . . , σn ) = σ0 y(σ0−1 σ1 , σ1−1 σ2 , . . . , σn−1
σn ).
With these isomorphisms the coboundary operators ∂ n+1 : C n (G, A) −→
C n+1 (G, A) are transformed into the homomorphisms
∂ n+1 : C n (G, A) −→ C n+1 (G, A)
given by
for a ∈ A = C 0 (G, A),
(∂ 1 a)(σ) = σa − a
(∂ 2 y)(σ, τ ) = σy(τ ) − y(στ ) + y(σ)
for y
∈
C 1 (G, A),
(∂ n+1 y)(σ1 , . . . , σn+1 ) = σ1 y(σ2 , . . . , σn+1 )
n
X
+
(−1)i y(σ1 , . . . , σi−1 , σi σi+1 , σi+2 , . . . , σn+1 )
i=1
+(−1)n+1 y(σ1 , . . . , σn )
for y
∈
C n (G, A).
Setting
∂ n+1
Z n (G, A) = ker (C n (G, A) −→ C n+1 (G, A))
∂n
B n (G, A) = im (C n−1 (G, A) −→ C n (G, A)) ,
∼ C n (G, A) induce isomorphisms
the isomorphisms C n (G, A) −→
H n (G, A) ∼
= Z n (G, A)/B n (G, A).
The functions in C n (G, A), Z n (G, A), B n (G, A) are called the inhomogeneous n-cochains, n-cocycles and n-coboundaries. The inhomogeneous
coboundary operators ∂ n+1 are more complicated than the homogeneous ones,
but they have the advantage of dealing with only n variables instead of n + 1.
For n = 0, 1, 2 the groups H n (G, A) admit the following interpretations.
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§2. Definition of the Cohomology Groups
The group H 0 (G, A): We have a natural isomorphism C 0 (G, A) → A,
x 7→ x(1), by which we identify C 0 (G, A) with A. Then, for a ∈ A,
(∂ 1 a)(σ0 , σ1 ) = σ1 a − σ0 a, or (∂ 1 a)(σ) = σa − a in the inhomogeneous setting,
so that
H 0 (G, A) = AG .
The group H 1 (G, A): The inhomogeneous 1-cocycles are the continuous
functions x : G −→ A such that
x(στ ) = x(σ) + σx(τ )
for all σ, τ
They are also called crossed homomorphisms.
1-coboundaries are the functions
∈
G.
The inhomogeneous
x(σ) = σa − a
with a fixed a ∈ A. If G acts trivially on A, then
H 1 (G, A) = Homcts (G, A). ∗)
The group H 1 (G, A) occurs in a natural way if we pass from an exact sequence
i
j
0 −→ A −→ B −→ C −→ 0
of G-modules to the sequence of fixed modules. Then we lose the exactness
and are left only with the exactness of the sequence
0 −→ AG −→ B G −→ C G .
The group H 1 (G, A) now gives information about the deviation from exactness.
In fact we have a canonical homomorphism
δ : C G −→ H 1 (G, A)
extending the above exact sequence to a longer one. Namely, for c ∈ C G we
may choose an element b ∈ B such that jb = c. For each σ ∈ G there is an
aσ ∈ A such that iaσ = σb − b. The function σ 7→ aσ is a 1-cocycle and
we define δc to be the cohomology class of this 1-cocycle in H 1 (G, A). The
definition is easily seen to be independent of the choice of the element b. If
δc = 0, then aσ = i−1 (σb − b) = σa − a, a ∈ A, so that b0 = b − ia is an element
of B G with jb0 = c. This shows the exactness of the sequence
δ
0 −→ AG −→ B G −→ C G −→ H 1 (G, A).
We shall meet this again in a larger frame in §3.
The group H 1 (G, A) admits a concrete interpretation using the concept of
torsors. Since this concept may be more fully exploited in the framework of
non-abelian groups A, we generalize H 1 (G, A) as follows.
∗) Since G is automatically understood as a topological group, we usually write Hom(G, A)
instead of Homcts (G, A).
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Chapter I. Cohomology of Profinite Groups
A G-group A is a not necessarily abelian group with the discrete topology
on which G acts continuously. We denote the action of σ ∈ G on a ∈ A by
σ
a, so that σ(ab) = σa σb. A cocycle of G with coefficients in A is a continuous
function σ 7→ aσ on G with values in the group A such that
aστ = aσ σaτ .
The set of cocycles is denoted by Z 1 (G, A). Two cocycles a, a0 are said to
be cohomologous if there exists a b ∈ A such that a0σ = b−1 aσ σb. This is an
equivalence relation in Z 1 (G, A) and the quotient set is denoted by H 1 (G, A).
It has a distinguished element given by the cocycle aσ = 1.
A G-set is a discrete topological space X with a continuous action of G. Let
A be a G-group. An A-torsor is a G-set X with a simply transitive right action
X × A −→ X, (x, a) 7→ xa, of A which is compatible with the G-action on
X. This means that for every pair x, y ∈ X there is a unique a ∈ A such that
y = xa, and σ(xa) = σxσa. For example, if
j
1 −→ A −→ B −→ C −→ 1
is an exact sequence of G-groups, then the cosets j −1 (c) for c ∈ C G are typical
A-torsors. It is clear what we mean by an isomorphism of A-torsors. Let
now TORS (A) denote the set of isomorphism classes of A-torsors. It has a
distinguished element given by the A-torsor A, and is thus a pointed set.
(1.2.3) Proposition. We have a canonical bijection of pointed sets
H 1 (G, A) ∼
= TORS (A).
Proof: We define a map
λ : TORS (A) −→ H 1 (G, A)
as follows. Let X be an A-torsor and let x ∈ X. For every σ ∈ G there is a
unique aσ ∈ A such that σx = xaσ . One verifies at once that aσ is a cocycle.
Changing x to xb changes this cocycle to b−1 aσ σb, which is cohomologous.
We define λ(X) to be the class of aσ .
We define an inverse µ : H 1 (G, A) −→ TORS (A) as follows. Let the set X
be the group A. We let G act on X in the twisted form
σ0
x = aσ · σx.
The action of A on X is given by right multiplication. In this way, X becomes
an A-torsor and this defines the map µ. Replacing aσ by b−1 aσ σb, we have
an isomorphism x 7→ b−1 x of A-torsors. One now checks that λ ◦ µ = 1 and
µ ◦ λ = 1.
2
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§2. Definition of the Cohomology Groups
j
i
Remark: If 0 −→ A −→ B −→ C −→ 0 is an exact sequence of
G-modules and if we identify in the exact sequence
δ
0 −→ AG −→ B G −→ C G −→ H 1 (G, A)
H 1 (G, A) with TORS (A), then the map δ is given by δc = j −1 (c).
The group H 2 (G, A): We return to the case that A is abelian. The inhomogeneous 2-cocycles are the continuous functions x : G × G −→ A such that
∂x = 0, i.e.
x(στ, ρ) + x(σ, τ ) = x(σ, τ ρ) + σx(τ, ρ).
Among these we find the inhomogeneous 2-coboundaries as the functions
x(σ, τ ) = y(σ) − y(στ ) + σy(τ )
with an arbitrary 1-cochain y : G −→ A.
The 2-cocycles had been known before the development of group cohomology as factor systems and occurred in connection with group extensions. To
explain this, we assume that either A or G is finite, in order to avoid topological
problems (but see (2.7.7)).
The question is: how many groups Ĝ are there, which have the G-module A
as a normal subgroup and G as the factor group (we write A multiplicatively).
To be more precise, we consider all exact sequences
1 −→ A −→ Ĝ −→ G −→ 1
of topological groups (i.e. of profinite groups if A is finite, and of discrete
groups if G is finite), such that the action of G on A is given by
a = σ̂aσ̂ −1 ,
σ
where σ̂
∈
Ĝ is a pre-image of σ
1
A
∈
G. If
Ĝ0
G
1
G
1
f
1
A
Ĝ
is a commutative diagram of such sequences with a topological isomorphism f ,
then we call these sequences equivalent, and we denote the set of equivalence
classes [Ĝ] by EXT(G, A). This set has a distinguished element given by the
semi-direct product Ĝ = A o G (see ex.1 below).
(1.2.4) Theorem (SCHREIER). We have a canonical bijection of pointed sets
H 2 (G, A) ∼
= EXT(G, A).
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Chapter I. Cohomology of Profinite Groups
Proof: We define a map
λ : EXT(G, A) −→ H 2 (G, A)
as follows. Let the class [Ĝ] ∈ EXT(G, A) be represented by the exact sequence
1 −→ A −→ Ĝ −→ G −→ 1.
We choose a continuous section s : G −→ Ĝ of Ĝ −→ G, and we set σ̂ = s(σ).
Such a section exists (see §1, ex.4). Regarding A as a subgroup of Ĝ, every
γ̂ ∈ Ĝ has a unique representation
γ̂ = aσ̂,
a ∈ A, σ
∈
G,
and we have
σ̂a = σ̂aσ̂ −1 σ̂ = σaσ̂.
c are both mapped onto στ , i.e.
The elements σ̂ τ̂ and στ
c,
σ̂ τ̂ = x(σ, τ )στ
with an element x(σ, τ ) ∈ A such that x(σ, 1) = x(1, σ) = 1. Since σ̂ is a
continuous function of σ and A is closed in Ĝ, x(σ, τ ) is a continuous map
x : G × G −→ A. The associativity (σ̂ τ̂ )ρ̂ = σ̂(τ̂ ρ̂) yields that x(σ, τ ) is a
2-cocycle:
c ρ̂ = x(σ, τ )x(στ, ρ)(στ ρ)ˆ,
(σ̂ τ̂ )ρ̂ = x(σ, τ )στ
σ̂(τ̂ ρ̂) = σ̂x(τ, ρ)τcρ = σx(τ, ρ)σ̂ τcρ = σx(τ, ρ) x(σ, τ ρ)(στ ρ)ˆ,
i.e.
x(σ, τ )x(στ, ρ) = σx(τ, ρ) x(σ, τ ρ).
We thus get a cohomology class c = [x(σ, τ )] ∈ H 2 (G, A). This class does not
depend on the choice of the continuous section s : G −→ Ĝ. If s0 : G −→ Ĝ
is another one, and if we set σ̃ = s0 (σ), then σ̃ = y(σ)σ̂, y(σ) ∈ A, and
f . For the 2-cocycle x̃(σ, τ ) we obtain
σ̃ τ̃ = x̃(σ, τ )στ
c = x̃(σ, τ )y(στ )x(σ, τ )−1 σ̂ τ̂
σ̃ τ̃ = x̃(σ, τ )y(στ )στ
= x̃(σ, τ )x(σ, τ )−1 y(στ )y(σ)−1 σ̃y(τ )−1 τ̃
= x̃(σ, τ )x(σ, τ )−1 y(στ )y(σ)−1 σy(τ )−1 σ̃ τ̃ ,
i.e. x̃(σ, τ ) = x(σ, τ )y(σ, τ ) with the 2-coboundary
y(σ, τ ) = y(σ)y(στ )−1 σy(τ ).
The cohomology class c = [x(σ, τ )] also does not depend on the choice of the
representative 1 −→ A −→ Ĝ −→ G −→ 1 in the class [Ĝ]. Namely, if
1
A
Ĝ
G
1
G
1
f
1
A
Ĝ0
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§2. Definition of the Cohomology Groups
is a commutative diagram and σ̂ 0 = f (σ̂), then
c ) = x(σ, τ )(στ
c )0 ,
σ̂ 0 τ̂ 0 = f (σ̂)f (τ̂ ) = f (σ̂ τ̂ ) = f (x(σ, τ )στ
i.e. the group extensions Ĝ0 and Ĝ yield the same 2-cocycle x(σ, τ ). We thus
get a well-defined map
λ : EXT(G, A) −→ H 2 (G, A).
In order to prove the bijectivity, we construct an inverse µ : H 2 (G, A) −→
EXT(G, A). Every cohomology class c ∈ H 2 (G, A) contains a normalized
2-cocycle x(σ, τ ), i.e. a cocycle such that
x(σ, 1) = x(1, σ) = 1.
Namely, if x(σ, τ ) is any 2-cocycle in c, then we obtain from the equality
x(στ, ρ)x(σ, τ ) = x(σ, τ ρ) σx(τ, ρ) that
x(σ, 1) = σx(1, 1),
Setting y(σ) = x(1, 1) for all σ
∈
x(1, ρ) = x(1, 1).
G, we obtain a 2-coboundary
y(σ, τ ) = y(σ)y(στ )−1 σy(τ ) ,
and the 2-cocycle x0 (σ, τ ) = x(σ, τ )y(σ, τ )−1 has the property that
x0 (σ, 1) = x(σ, 1)(σx(1, 1))−1 = 1, x0 (1, τ ) = x(1, τ )x(1, 1)−1 = 1.
Let now x(σ, τ ) be a normalized 2-cocycle in c. On the set Ĝ = A × G with
the product topology we define the continuous multiplication
(a, σ)(b, τ ) = (x(σ, τ )a σb, στ ).
This product is associative because of the cocycle property:
((a, σ)(b, τ ))(c, ρ) = (x(σ, τ )a σb, στ )(c, ρ)
= (x(στ, ρ)x(σ, τ )a σb στc, στ ρ) = (x(σ, τ ρ) σx(τ, ρ)a σb στc, στ ρ)
= (a, σ)(x(τ, ρ)b τc, τ ρ) = (a, σ)((b, τ ), (c, ρ)).
(1,1) is an identity element:
(a, σ)(1, 1) = (x(σ, 1)a, σ) = (a, σ) = (x(1, σ)a, σ) = (1, 1)(a, σ)
σ −1
and ([
−1
x(σ, σ −1 ) σ a]−1 , σ −1 ) is an inverse of (a, σ) since
−1
−1
−1
(a, σ)([σ x(σ, σ −1 ) σ a]−1 , σ −1 ) = (a σ(σ a)−1 , σσ −1 ) = (1, 1).
In this way Ĝ = A × G becomes a group with the product topology, and the
maps a 7→ (a, 1) and (a, σ) 7→ σ yield an exact sequence
1 −→ A −→ Ĝ −→ G −→ 1.
−1
Setting σ̂ = (1, σ), we have σ̂ −1 = (σ x(σ, σ −1 )−1 , σ −1 ) and
−1
σ̂(a, 1)σ̂ −1 = (x(σ, 1) σa, σ)(σ x(σ, σ −1 )−1 , σ −1 ) = (σa, 1).
We thus obtain an element [Ĝ] in EXT(G, A). This element does not depend
on the choice of the normalized 2-cocycle x(σ, τ ) in c. For, if x0 (σ, τ ) =
x(σ, τ )y(σ, τ )−1 is another one, y(σ, τ ) = y(σ)y(στ )−1 σy(τ ) is a 2-coboundary,
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Chapter I. Cohomology of Profinite Groups
and if Ĝ0 is the group given by the multiplication on A×G via x0 (σ, τ ), then the
map f : (a, σ) 7→ (y(σ)a, σ) is an isomorphism from Ĝ to Ĝ0 and the diagram
1
A
Ĝ
G
1
f
1
A
Ĝ0
G
1
is commutative, noting that y(1) = 1 because 1 = x0 (1, σ) = x(1, σ)y(1)−1
= y(1)−1 . Therefore [Ĝ] = [Ĝ0 ], and we get a well-defined map
µ : H 2 (G, A) −→ EXT(G, A).
This map is inverse to the map λ constructed before. For, if x(σ, τ ) is the
2-cocycle produced by a section G −→ Ĝ, σ 7→ σ̂, of a group extension
1 −→ A −→ Ĝ −→ G −→ 1,
then the map f : (a, σ) 7→ aσ̂ is an isomorphism of the group A × G, endowed
with the multiplication given by x(σ, τ ), onto Ĝ. This proves the theorem.
2
It is a significant feature of cohomology theory that we don’t have concrete
interpretations of the groups H n (G, A) for dimensions n ≥ 3 in general. This
does, however, not at all mean that they are uninteresting. Besides their natural
appearance, the importance of the higher dimensional cohomology groups is
seen in the fact that the theory endows them with an abundance of homomorphic
connections, with which one obtains important isomorphism theorems. These
theorems give concrete results for the interesting lower dimensional groups,
whose proofs, however, have to take the cohomology groups of all dimensions
into account.
Next we show that the cohomology groups H n (G, A) of a profinite group G
with coefficients in a G-module A are built up in a simple way from those of
the finite factor groups of G. Let U, V run through the open normal subgroups
of G. If V ⊆ U , then the projections
Gn+1 −→ (G/V )n+1 −→ (G/U )n+1
induce homomorphisms
C n (G/U, AU ) −→ C n (G/V, AV ) −→ C n (G, A),
which commute with the operators ∂ n+1 . We therefore obtain homomorphisms
H n (G/U, AU ) −→ H n (G/V, AV ) −→ H n (G, A).
The groups H n (G/U, AU ) thus form a direct system and we have a canonical
homomorphism lim H n (G/U, AU ) −→ H n (G, A).
−→ U
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§2. Definition of the Cohomology Groups
(1.2.5) Proposition. The above homomorphism is an isomorphism:
∼ H n (G, A) .
lim H n (G/U, AU ) −→
−→
U
Proof: Already the homomorphism
.
.
lim C (G/U, AU ) −→ C (G, A)
−→
U
is an isomorphism of complexes. The injectivity is clear, since the maps
.
.
C (G/U, AU ) → C (G, A)
are injective.
Let conversely x : Gn+1 −→ A be an n-cochain of G. Since A is discrete, x
is locally constant. We conclude that there exists an open normal subgroup U0
of G such that x is constant on the cosets of U0n+1 in Gn+1 . It takes values in
AU0 , since for all σ ∈ U0 we have
x(σ0 , . . . , σn ) = x(σσ0 , . . . , σσn ) = σx(σ0 , . . . , σn ).
Hence x is the composite of
xU
0
Gn+1 −→ (G/U0 )n+1 −→
A U0
with an n-cochain xU0 of G/U0 , and is therefore the image of the element in
lim
C n (G/U, AU ) defined by xU0 . This shows the surjectivity. Since the
−→ U
functor lim is exact, we obtain the isomorphisms
−→
lim H n (G/U, AU ) ∼
= H n (lim C (G/U, AU ))
.
−→
U
−→
U
.
∼
= H n (C (G, A))
2
= H n (G, A).
Finally we introduce Tate cohomology. We do this for finite groups here,
and will extend the theory to profinite groups in §8. Let G be a finite group for
the remainder of this section.
We consider the norm residue group
Ĥ 0 (G, A) = AG /NG A,
where NG A is the image of the norm map∗)
NG : A −→ A,
NG a =
X
σa.
σ ∈G
∗) The name “norm” is chosen instead of “trace”, because in Galois cohomology this map
Q
will often be written multiplicatively, i.e. NG a =
σa.
σ ∈G
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Chapter I. Cohomology of Profinite Groups
We call the groups
(
n
Ĥ (G, A) =
AG /NG A
for n = 0,
H n (G, A)
for n ≥ 1
the modified cohomology groups. We also obtain these groups from a complex. Namely, we extend the standard complex (C n (G, A))n≥0 to
.
∂0
∂1
∂2
Ĉ (G, A) : C −1 (G, A) −→ C 0 (G, A) −→ C 1 (G, A) −→ . . . ,
where C −1 (G, A) = C 0 (G, A) and ∂ 0 x is the constant function with value
P
≥ 0 as
σ ∈G x(σ). We then obtain the modified cohomology groups for all n
the cohomology groups of this complex,
.
Ĥ n (G, A) = H n (Ĉ (G, A)).
Besides the fixed module AG , we have also a “cofixed module” AG = A/IG A,
where IG A is the subgroup of A generated by all elements of the form
σa − a, a ∈ A, σ ∈ G. AG is the largest quotient of A on which G acts
trivially. We set
H0 (G, A) = AG .
If G is a finite group, then IG A is contained in the group
NG A
= {a ∈ A | NG a = 0},
and we set
Ĥ0 (G, A) = NG A/IG A.
The norm NG : A −→ A induces a map NG : H0 (G, A) −→ H 0 (G, A), and
the proof of the following proposition is obvious.
(1.2.6) Proposition. We have an exact sequence
N
G
H 0 (G, A) −→ Ĥ 0 (G, A) −→ 0.
0 −→ Ĥ0 (G, A) −→ H0 (G, A) −→
The group Ĥ0 (G, A) is very often denoted by Ĥ −1 (G, A) for the following
reason. For a finite group G one can define cohomology groups Ĥ n (G, A) for
arbitrary integral dimensions n ∈ ZZ as follows:
For n ≥ 0, let ZZ[Gn+1 ] be the abelian group of all formal ZZ-linear combiP
nations a(σ0 ,...,σn ) (σ0 , . . . , σn ), σ0 , . . . , σn ∈ G, with its obvious G-module
structure. We consider the (homological) complete standard resolution of
ZZ, i.e. the sequence of G-modules X = X (G, ZZ)
.
∂
∂
.
∂
∂−1
0
2
1
. . . −→ X2 −→
X1 −→
X0 −→
X−1 −→ X−2 −→ . . .
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23
§2. Definition of the Cohomology Groups
where Xn = X−1−n = ZZ[Gn+1 ] for n
n > 0 by
∂n (σ0 , . . . , σn )
=
n
X
≥
0, and the differentials are defined for
(−1)i (σ0 , . . . , σi−1 , σi+1 , . . . , σn )
i=0
∂−n (σ0 , . . . , σn−1 ) =
n
XX
(−1)i (σ0 , . . . , σi−1 , τ, σi , . . . , σn−1 ) ,
τ ∈G i=0
while ∂0 : X0 → X−1 is given by
∂0 (σ0 ) =
X
τ.
τ ∈G
The (cohomological) complete standard resolution of A is defined as the
sequence of G-modules X = X (G, A) = Hom(X , A)
.
.
∂ −1
.
∂0
∂1
∂2
. . . −→ X −2 −→ X −1 −→ X 0 −→ X 1 −→ X 2 −→ . . .
where X −1−n = X n = Hom(Xn , A) = Map(Gn+1 , A) for n
Hom(∂n , A) for n ∈ ZZ. X is a complex. Using the maps
.
≥
0 and ∂ n =
D−n : X −n+1 −→ X −n
given by
(Dn x)(σ0 , . . . , σn ) = x(1, σ0 , . . . , σn )
∗)
(D−1 x)(σ0 ) = δσ0 ,1 x(1)
(D−n x)(σ0 , . . . , σn−1 ) = δσ0 ,1 x(σ1 , . . . , σn−1 )
for
n ≥ 0,
for
n = 1,
for
n ≥ 2,
we get
Dn ◦ ∂ n+1 + ∂ n ◦ Dn−1 = id
.
for all n ∈ ZZ. From this we conclude that the above complex X is exact.
For every n ∈ ZZ, we now define the n-th Tate cohomology group Ĥ n (G, A)
as the cohomology group of the complex
.
Ĉ (G, A) = ((X n )G )n∈ ZZ
at the place n:
.
Ĥ n (G, A) = H n (Ĉ (G, A)).
Clearly, for n ≥ 0 we get the previous (modified) cohomology groups, and it
is immediate to see that Ĥ −1 (G, A) is our group Ĥ0 (G, A) = NG A/IG A. More
generally, the Tate cohomology in negative dimensions can be identified with
homology (see §8).
∗) i.e. δ
σ,τ
= 0 if σ =/ τ and δσ,σ = 1.
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24
Chapter I. Cohomology of Profinite Groups
Exercise 1. Let G be a profinite group and A a G-group. Assume that either G or A is finite.
The semi-direct product is a group
Ĝ = A o G
containing A and G such that every element of Ĝ has a unique presentation aσ, a ∈ A, σ
and (aσ)(a0 σ 0 ) = a σa0 σσ 0 . We then have a group extension
∈
G,
π
1 → A → Ĝ → G → 1
and the inclusion G ,→ Ĝ is a homomorphic section of π. Two homomorphic sections
s, s0 : G → Ĝ of π are conjugate if there is an a ∈ A such that s0 (σ) = as(σ)a−1 for all σ ∈ G.
π
Let SEC (Ĝ → G) be the set of conjugacy classes of homomorphic sections of Ĝ → G. Then
there is a canonical bijection of pointed sets
H 1 (G, A) ∼
= SEC (Ĝ → G).
Exercise 2. There is the following interpretation of H 3 (G, A). Consider all possible exact
sequences
i
α
π
1 −→ A −→ N −→ Ĝ −→ G −→ 1,
where N is a group with an action σ̂ : ν 7→ σ̂ ν of Ĝ satisfying α(ν) ν 0 = νν 0 ν −1 , ν, ν 0 ∈ N , and
α(σ̂ ν) = σ̂α(ν)σ̂ −1 , ν ∈ N, σ̂ ∈ Ĝ. Impose on the set of all such exact sequences the smallest
equivalence relation such that
1 −→ A −→ N −→ Ĝ −→ G −→ 1
is equivalent to
1 −→ A −→ N 0 −→ Ĝ0 −→ G −→ 1,
whenever there is a commutative diagram
!"#$%&'()*
N
1
Ĝ
A
G
1
N0
Ĝ0
in which the vertical arrows are compatible with the actions of Ĝ and Ĝ0 on N and N 0 (but
need not be bijective). If EXT 2 (G, A) denotes the set of equivalence classes, then we have a
canonical bijection
EXT 2 (G, A) ∼
= H 3 (G, A)
(see [18], chap. IV, th. 5.4).
Exercise 3. Let G be finite and let (Ai )i∈I be a family of G-modules. Show that
Y
Y
H r (G,
Ai ) =
H r (G, Ai )
for all r
≥
0.
i∈I
i∈I
Exercise 4. An inhomogeneous cochain x ∈ C n (G, A), n ≥ 1, is called normalized if
x(σ1 , . . . , σn ) = 0 whenever one of the σi is equal to 1. Show that every class in H n (G, A) is
represented by a normalized cocycle.
Hint: Construct inductively cochains x0 , x1 , . . . , xn ∈ C n (G, A) and y1 , . . . , yn
C n−1 (G, A) such that
x0 = x, xi = xi−1 − ∂yi , i = 1, . . . , n,
yi (σ1 , . . . , σn−1 ) = (−1)i−1 xi−1 (σ1 , . . . , σi−1 , 1, σi , . . . , σn−1 ).
Then xn is normalized and x − xn is a coboundary.
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25
§3. The Exact Cohomology Sequence
§3. The Exact Cohomology Sequence
Having introduced the cohomology groups H n (G, A), we now turn to the
question of how they behave if we change the G-module A. If
f : A −→ B
is a homomorphism of G-modules, i.e. a homomorphism such that f (σa)
= σf (a) for a ∈ A, σ ∈ G, then we have the induced homomorphism
f : C n (G, A) → C n (G, B),
x(σ0 , . . . , σn ) 7→ f x(σ0 , . . . , σn ),
and the commutative diagram
· ·2+,10-./ ·
∂ n+1
C n (G, A)
C n+1 (G, A)
f
f
∂ n+1
C n (G, B)
···
···
C n+1 (G, B)
···
.
In other words, f : A −→ B induces a homomorphism
.
.
f : C (G, A) −→ C (G, B)
of complexes. Taking homology groups of these complexes, we obtain homomorphisms
f : H n (G, A) −→ H n (G, B).
Besides these homomorphisms there is another homomorphism, the “connecting homomorphism”, which is less obvious, but is of central importance in
cohomology theory. For its definition we make use of the following general
lemma, which should be seen as the crucial point of homological algebra.
(1.3.1) Snake lemma. Let
385674:9;
i
A
B
α
j
C
0
γ
β
j0
0
i
0
A0
B0
C0
be a commutative diagram of abelian groups with exact rows. We then have a
canonical exact sequence
ker(i)
ker(α)
i
ker(β)
j
ker(γ)
δ
FEBCD?@A><=
coker(α)
i0
coker(β)
j0
coker(γ)
coker(j 0 ).
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26
Chapter I. Cohomology of Profinite Groups
Proof: The existence and exactness of the upper and the lower sequence is
evident. The crucial cohomological phenomenon is the natural, but slightly
hidden, appearance of the homomorphism
δ : ker(γ) −→ coker(α).
It is obtained as follows. Let c ∈ ker(γ). Let b ∈ B and a0
such that
jb = c and i0 a0 = βb.
∈
A0 be elements
The element b exists since j is surjective and a0 exists (and is uniquely determined by b) since j 0 βb = γjb = γc = 0. We define
δc := a0 mod α(A).
This definition does not depend on the choice of b, since if b̃ ∈ B is another
element such that j b̃ = c and i0 ã0 = β b̃, ã0 ∈ A0 , then j(b̃ − b) = 0, i.e.
b̃ − b = ia, a ∈ A, so that i0 (ã0 − a0 ) = β(b̃ − b) = βia = i0 αa, and thus
ã0 − a0 = αa, i.e. ã0 ≡ a0 mod α(A).
Exactness at ker(γ): δc = 0 means a0 = αa, a ∈ A, which implies β(b − ia) =
i0 a0 − i0 αa = 0, i.e. b − ia ∈ ker(β) and j(b − ia) = c.
Exactness at coker(α): Let a0 ∈ A0 such that i0 a0 ≡ 0 mod β(B), i.e. i0 a0 =
βb, b ∈ B. Setting c = jb, we have by definition δc = a0 mod α(A).
2
We now show that every exact sequence of G-modules
j
i
0 −→ A −→ B −→ C −→ 0
gives rise to a canonical homomorphism
δ : H n (G, C) −→ H n+1 (G, A)
for every n ≥ 0. We consider the commutative diagram
0QGHIJKLMNOP
C n (G, A)
∂A
0
C n (G, B)
C n (G, C)
∂B
C n+1 (G, A)
C n+1 (G, B)
0
∂C
C n+1 (G, C)
0.
It is exact, which is seen by passing to the inhomogeneous cochains (see also
ex.1). By the snake lemma, we obtain a homomorphism
δ : ker(∂C ) −→ coker(∂A ).
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27
§3. The Exact Cohomology Sequence
(1.3.2) Theorem. For every exact sequence 0 −→ A → B → C → 0 of
G-modules, the above homomorphism δ induces a homomorphism
δ : H n (G, C) −→ H n+1 (G, A)
and we obtain an exact sequence
δ
0 −→ AG −→ B G −→ C G −→ H 1 (G, A) −→ · · ·
δ
· · · −→ H n (G, A) −→ H n (G, B) −→ H n (G, C) −→ H n+1 (G, A) −→ · · · .
Proof: Setting C n (G, A) = C n (G, A)/B n (G, A) and similarly for B and C
in place of A, we obtain from the above diagram the commutative diagram
VWXYZRSTU
C n (G, A)
C n (G, B)
∂A
Z n+1 (G, A)
0
C n (G, C)
∂B
Z n+1 (G, B)
0
∂C
Z n+1 (G, C),
which is obviously exact. Noting that
ker(∂A ) = H n (G, A) and coker(∂A ) = H n+1 (G, A),
the snake lemma yields an exact sequence
H n (G, A)
H n (G, B)
H n (G, C)
δ
`abc[\]^_
H n+1 (G, A)
H n+1 (G, B)
H n+1 (G, C).
2
This proves the theorem.
The homomorphism δ : H n (G, C) −→ H n+1 (G, A) is called the connecting
homomorphism, or simply the δ-homomorphism, and the exact sequence in
the theorem is called the long exact cohomology sequence.
Remark: If the group G is finite, then, using the unrestricted complex
. . . −→ X n−1 (G, A) −→ X n (G, A) −→ X n+1 (G, A) −→ . . .
(n ∈ ZZ)
mentioned in §2, we get by the same argument an unrestricted long exact
cohomology sequence
δ
δ
· · · −→ Ĥ n (G, A) −→ Ĥ n (G, B) −→ Ĥ n (G, C) −→ Ĥ n+1 (G, A) −→ · · · .
The connecting homomorphism δ has the following compatibility properties.
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28
Chapter I. Cohomology of Profinite Groups
(1.3.3) Proposition. If
0knijghdelmf
A
i
f
B
j
C
0
g
h
j0
0
i
0
A0
B0
C0
0
is an exact commutative diagram of G-modules, then the diagrams
orqp C)
H n (G,
δ
g
H n (G, C 0 )
H n+1 (G, A)
f
δ
H n+1 (G, A0 )
are commutative.
Proof: This follows immediately from the definition of δ. If cn ∈ H n (G, C)
and if bn ∈ C n (G, B) and an+1 ∈ C n+1 (G, A) are such that jbn = cn and ian+1 =
∂ n+1 bn , then δcn = an+1 , and f δcn = f an+1 = f an+1 . On the other hand, setting
c0n = gcn , b0n = hbn , a0n+1 = f an+1 , we have j 0 b0n = c0n , i0 a0n+1 = ∂ n+1 b0n , so
that
2
δgcn = δc0n = a0n+1 = f an+1 = f δcn .
In order to avoid repeated explanations it is convenient to introduce the
notion of δ-functor. Let A and B be abelian categories. An exact δ-functor
from A to B is a family H = {H n }n∈ ZZ of functors H n : A −→ B together
with homomorphisms
δ : H n (C) −→ H n+1 (A)
defined for each short exact sequence 0 −→ A −→ B −→ C −→ 0 in A with
the following properties:
(i) δ is functorial, i.e. if
0stuvwxyz{|}
A
B
C
0
0
A0
B0
C0
0
is a commutative diagram of short exact sequences in A, then
~€
H n (C)
δ
H n+1 (A)
H n (C 0 )
is a commutative diagram in B.
δ
H n+1 (A0 )
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29
§3. The Exact Cohomology Sequence
(ii) The sequence
δ
· · · −→ H n (A) −→ H n (B) −→ H n (C) −→ H n+1 (A) −→ · · ·
is exact for every exact sequence 0 → A → B → C → 0 in A.
If a family of functors H n is given only for an interval −∞ ≤ r ≤ n ≤ s ≤ ∞,
then one completes it tacitly by setting H n = 0 for n ∈/ [r, s].
In this sense the family of functors H n (G, −) (completed by H n (G, −) = 0
for n < 0) is a δ-functor from the category of G-modules into the category of
abelian groups. A curious property of δ-functors is their “anticommutativity”.
(1.3.4) Proposition. Let {H n } be an exact δ-functor from A to B. If
‚ƒ„…†‡ˆ‰Š‹ŒŽ‘’“”•–—˜™
0
0
0
0
A0
A
A00
0
0
B0
B
B 00
0
0
C0
C
C 00
0
0
0
0
is a commutative diagram in A with exact rows and columns, then
H n−1šœ› (C 00 )
δ
−δ
δ
H n (A00 )
H n (C 0 )
δ
H n+1 (A0 )
is a commutative diagram in B.
Proof: It simplifies the proof if we assume that A is a category whose objects
are abelian groups (together with some extra structure), as then we may prove
statements by “diagram chases” with elements. We may do this, since it can be
shown that every small abelian category may be fully embedded into a category
of modules over an appropriate ring in such a way that exactness relations are
preserved, and in any case we shall apply the proposition only to the category
of G-modules.
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30
Chapter I. Cohomology of Profinite Groups
Let D be the kernel of the composite map B −→ C 00 , so that the sequence
0 −→ D −→ B −→ C 00 −→ 0
is exact. Let
i : A0 −→ A ⊕ B 0
be the direct sum of the maps A0 −→ A and A0 −→ B 0 and let
j : A ⊕ B 0 −→ B
be the difference d1 − d2 of the maps d1 : A −→ B and d2 : B 0 −→ B. One
checks at once that we get an exact sequence
j
i
0 −→ A0 −→ A ⊕ B 0 −→ D −→ 0
and that the diagram
A°±²³­®¯¬«¤¥¦§¨©ª£žŸ ¡¢ 0
A00
A
B 00
C 00
pr1
id
A0
id
A ⊕ B0
D
C 00
B
−pr2
−id
A0
B0
id
C0
C 00
C
of solid arrows is commutative. This diagram can be commutatively completed by homomorphisms D → A00 and D → C 0 , since im(D → B 00 ) ⊆
im(A00 → B 00 ) and A00 → B 00 is injective, and since im(D → C) ⊆ im(C 0 → C)
and C 0 → C is injective. From this we obtain the commutative diagram
H n−1´º¹·¸¶µ¿¾¼½» (C 00 )
δ
H n (A00 )
δ
H n+1 (A0 )
id
id
H n−1 (C 00 )
δ
H n (D)
δ
H n+1 (A0 )
−id
id
H n−1 (C 00 )
δ
H n (C 0 )
δ
H n+1 (A0 )
and the proposition follows.
2
From the exact cohomology sequence, we often get important isomorphism
theorems. For example, if
0 −→ A −→ B −→ C −→ 0
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31
§3. The Exact Cohomology Sequence
is an exact sequence of G-modules and if H n (G, B) = H n+1 (G, B) = 0, then
δ : H n (G, C) −→ H n+1 (G, A)
is an isomorphism. For this reason it is very important to know which Gmodules are cohomologically trivial in the following sense.
(1.3.5) Definition. A G-module A is called acyclic if H n (G, A) = 0 for all
n > 0. A is called cohomologically trivial (welk in German, flasque in
French) if
H n (H, A) = 0
for all closed subgroups H of G and all n > 0.
Important examples of cohomologically trivial G-modules are the induced
G-modules given by
IndG (A) = Map (G, A),
where A is any G-module. The elements of IndG (A) are the continuous
functions x : G −→ A (with the discrete topology on A) and the action of
σ ∈ G on x is given by (σx)(τ ) = σx(σ −1 τ ).
If G is a finite group, then we have an isomorphism
IndG (A) ∼
= A ⊗ ZZ[G]
given by x 7→
X
x(σ) ⊗ σ, where ZZ[G] = {
X
nσ σ | nσ
∈
ZZ} is the group
σ ∈G
σ ∈G
ring of G.
(1.3.6) Proposition. (i) The functor A 7→ IndG (A) is exact.
(ii) An induced G-module A is also an induced H-module for every closed
subgroup H of G, and if H is normal, then AH is an induced G/H-module.
(iii) If one of the G-modules A and B is induced, then so is A ⊗ B. If G is
finite, the same holds for Hom(A, B).
(iv) If U runs through the open normal subgroups of G, then
IndG (A) = lim
IndG/U (AU ).
−→
U
We leave the simple proof to the reader (for (ii) use ex.4 of §1 to find a
homeomorphism G ∼
= H × G/H). As mentioned above, the very importance
of the induced G-modules lies in the following fact.
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32
Chapter I. Cohomology of Profinite Groups
(1.3.7) Proposition. The induced G-modules M = IndG (A) are cohomologically trivial. If G is finite, we have moreover Ĥ n (G, M ) = 0 for all n ∈ ZZ. ∗)
Proof: We consider the standard resolutions
.
.
X (G, A) and X (G, IndG (A))
of A and IndG (A). The map
X n (G, IndG (A))G −→ X n (G, A),
given by x(σ0 , . . . , σn ) 7→ y(σ0 , . . . , σn ) = x(σ0 , . . . , σn )(1) obviously commutes with ∂. Furthermore, it is an isomorphism, since it has the map
y(σ0 , . . . , σn ) 7→ x(σ0 , . . . , σn )(σ) = σy(σ −1 σ0 , . . . , σ −1 σn ) as inverse. We
thus have an isomorphism
.
.
.
C (G, IndG (A)) ∼
= X (G, A)
of complexes. But X (G, A) is exact by (1.2.1), so that
.
.
H n (G, IndG (A)) = H n (C (G, IndG (A))) = H n (X (G, A)) = 0
for n ≥ 1. If H is a closed subgroup of G, then by (1.3.6) we may write
IndG (A) = IndH (B) and get H n (H, IndG (A)) = 0. If G is finite, then the same
argument holds for the extended complex (X n )n∈ ZZ , hence Ĥ n (G, IndG (A)) = 0
for all n ∈ ZZ.
2
The above proposition allows us to adopt a technique, called dimension
shifting, by which definitions and proofs concerning the cohomology groups
for all G-modules A and all n, may be reduced to a single dimension n, e.g.
n = 0. Given A, define the G-module A1 by the exact sequence
i
0 −→ A −→ IndG (A) −→ A1 −→ 0,
where ia is the constant function (ia)(σ) = a. This is a sequence of G-modules.
If H is a closed subgroup of G, then H n (H, IndG (A)) = 0 for all n ≥ 1 by
(1.3.7), and the exact cohomology sequence shows that the homomorphism
δ : H n (H, A1 ) −→ H n+1 (H, A)
is surjective for n = 0 and bijective for n > 0. If we define A0 = A and
inductively
Ap = (Ap−1 )1 for p > 0,
then (1.3.7) yields inductively the
∗) We shall see in §7 that Ĥ n (G, A) = 0, n ∈ ZZ, for any cohomologically trivial G-module A.
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33
§3. The Exact Cohomology Sequence
(1.3.8) Proposition. For all n, p
canonical homomorphism
≥
0 and all subgroups H
⊆
G, we have a
δ p : H n (H, Ap ) −→ H n+p (H, A),
which is a surjection for n = 0 and an isomorphism for n > 0.
If G is a finite group, then we may also consider the exact sequence
ν
0 −→ A−1 −→ IndG (A) −→ A −→ 0,
P
where ν associates the element σ∈G x(σ) to an element x
define
Ap = (Ap+1 )−1 for p < 0 ,
∈
IndG (A). We
It is easy to see that
⊗p
Ap ∼
= A ⊗ JG
and
⊗p
A−p ∼
= A ⊗ IG
for p ≥ 0, where the G-modules IG and JG are given by the exact sequences
0ÆÇÀÁÂÃÄÅ
IG
0
ZZ
ZZ[G]
NG
ε
ZZ[G]
ZZ
0,
JG
0.
Here ε is the augmentation map
ε:
X
σ ∈G
aσ σ 7→
X
aσ ,
σ ∈G
P
and NG (1) = σ∈G σ. The G-module IG is called the augmentation ideal
of ZZ[G].
Noting that Ĥ n (H, IndG (A)) = 0, we obtain canonical isomorphisms
Ĥ n (H, A) ∼
= Ĥ n−p (H, Ap )
for all n, p ∈ ZZ. Furthermore, we observe the following rule for the G-module
Hom(A, Q/ZZ): if p ∈ ZZ, then
Hom(A, Q/ZZ)p ∼
= Hom(A−p , Q/ZZ) .
Let G again be a profinite group. The G-modules X n = X n (G, A) in the
standard resolution
0 −→ A −→ X 0 −→ X 1 −→ X 2 −→ . . .
are all induced G-modules, since X 0 = IndG (A) and X n = IndG (X n−1 ). They
are thus cohomologically trivial and, in particular, acyclic. We call a resolution
0 −→ A −→ Y 0 −→ Y 1 −→ Y 2 −→ . . .
of A acyclic (resp. resolution by cohomologically trivial G-modules) if the
Y n are acyclic (resp. cohomologically trivial). It is a remarkable fact that the
cohomology groups H n (G, A) can be obtained from any acyclic resolution.
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34
Chapter I. Cohomology of Profinite Groups
(1.3.9) Proposition. If
∂
∂
∂
0 −→ A −→ Y 0 −→ Y 1 −→ Y 2 −→ . . .
is an acyclic resolution of A, then canonically
H n (G, A) ∼
= H n (H 0 (G, Y )).
.
∂
Proof: Setting K p = ker(Y p −→ Y p+1 ), we obtain the short exact sequences
0 −→ A
0 −→ K
1
−→ Y 0
−→ K 1
−→ 0,
1
2
−→ 0,
−→ Y
−→ K
···
0 −→ K n−2 −→ Y n−2 −→ K n−1 −→ 0,
0 −→ K n−1 −→ Y n−1 −→ K n
−→ 0.
Since the Y n are acyclic, the exact cohomology sequence yields for n
isomorphisms
δ
δ
≥
1
δ
2
n−2
n
H 1 (G, K n−1 ) −→
) −→
∼ H (G, K
∼ · · · −→
∼ H (G, A).
On the other hand we have the exact sequence
H 0 (G, Y n−1 ) −→ H 0 (G, K n ) −→ H 1 (G, K n−1 ) −→ H 1 (G, Y n−1 ) = 0
and
H 0 (G, K n ) = ker(H 0 (G, Y n ) −→ H 0 (G, Y n+1 )),
im (H 0 (G, Y n−1 ) −→ H 0 (G, K n )) = im (H 0 (G, Y n−1 ) −→ H 0 (G, Y n )),
which proves that canonically
H n (G, A) ∼
= H 1 (G, K n−1 ) = H n (H 0 (G, Y )).
.
2
For example, if H is a closed subgroup of G, then the standard resolution
0 → A → X of a G-module A is also an acyclic resolution of the Hmodule A, hence H n (H, A) ∼
= H n (H 0 (H, X )). This isomorphism is also
obtained from the restriction map X (G, A)H → X (H, A)H , as one may see
by dimension shifting.
.
.
.
.
Remark: There exists a variant of (1.3.9) for the modified cohomology. If G
is a finite group, then, for all q ∈ ZZ and every G-module A, there are canonical
isomorphisms
Ĥ q (G, A) ∼
= H q (H 0 (G, Y )),
.
where Y
. is a complete acyclic resolution of A, i.e. a complex
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35
§3. The Exact Cohomology Sequence
· ·ÉÊÈËÐÑÏÎÍÌ ·
∂ −2
Y −2
∂ −1
Y −1
∂0
∂1
Y0
∂2
Y1
Y2
∂3
···
µ
ε
A
0
0
consisting of cohomologically trivial G-modules Y n , n
everywhere and ∂ 0 = µ ◦ ε.
∈
ZZ, which is exact
Exercise 1. The functor A 7→ C n (G, A) is exact.
Hint: C n (G, A) = X n (G, A)G , and X n (G, A) is induced.
f
g
Exercise 2. For any pair of maps A −→ B −→ C of abelian groups, there is an exact sequence
0
ker(f )
ker(g ◦ f )
ker(g)
ÒÓÔÕÖ×ØÙÚÛÜ
coker(f )
coker(g ◦ f )
coker(g)
0.
Exercise 3. Let G be a finite group and let
α
β
γ
0 −→ A −→ B −→ C −→ D −→ 0
be an exact sequence of G-modules. Define a homomorphism
δ 2 : Ĥ n−1 (G, D) −→ Ĥ n+1 (G, A).
Show that the following conditions are equivalent:
(i) δ 2 is an isomorphism for all n ∈ ZZ,
(ii) Ĥ n (G, B) → Ĥ n (G, C) is an isomorphism for all n ∈ ZZ.
Hint: Show this first under the assumption Ĥ n (G, B) = 0 for all n
to the exact commutative diagram
òóôîïðñíâãäåæçèéêëìàáßÝÞ
0
0
0
0
A
α
B
IndG B
ZZ. Then apply (1.3.4)
βB
0
C
0
(i,β)
i◦α
0
β
∈
(id,0)
IndG B ⊕ C
0+id
γ
Y
D
0
0
where X = coker (i ◦ α) and Y = coker (i, β).
0
0
X
0
Exercise 4. Let A be a G-module and let A0 be the trivial G-module with underlying abelian
group A. Then we have an isomorphism IndG (A) ∼
= IndG (A0 ) of G-modules.
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36
Chapter I. Cohomology of Profinite Groups
Exercise 5. Let G be a finite group and let 0 → A → I1 → I2 → · · · → Ip → B
→ 0 be an exact sequence of G-modules, where the I1 , . . . , Ip are acyclic G-modules. Then
Ĥ n (G, A) ∼
= Ĥ n−p (G, B) for n ≥ p.
.
.
Exercise 6. Let C = (C n , dn )n∈ ZZ and C 0 = (C 0n , d0n )n∈ ZZ be two complexes in an abelian
category and let f = (fn )n∈ ZZ and g = (gn )n∈ ZZ be two morphisms from C to C 0 . A
homotopy from f to g is a family h = (hn )n∈ ZZ of morphisms hn : C n+1 → C 0n such that
.
.
hn dn+1 + d0n hn−1 = fn − gn .
We say that f and g are homotopic and write f ' g if such a family exists.
Show that in this case f and g induce the same homomorphisms H n (C ) → H n (C 0 ) on
the homology.
.
.
Exercise 7. If G is finite and (Ai )i∈I is a projective system of finite G-modules, then
H n (G, lim Ai ) = lim H n (G, Ai ).
←− i
←− i
(Compare exercise 3 in §2.)
Exercise 8. If 1 → A → B → C → 1 is an exact sequence of G-groups, then
1 −→ AG −→ B G −→ C G −→ H 1 (G, A) −→ H 1 (G, B) −→ H 1 (G, C).
is an exact sequence of pointed sets, i.e. the image of a map is equal to the pre-image of the
distinguished element. If A is in the center of B, then the sequence extends exactly by an
δ
arrow → H 2 (G, A), given by cσ 7→ aσ,τ = cσ σcτ c−1
στ .
§4. The Cup-Product
If A and B are two G-modules, then A ⊗ ZZ B is also a G-module (by
σ(a ⊗ b) = σa ⊗ σb), and we obtain for every pair p, q ≥ 0 a bilinear map
(∗)
∪
C p (G, A) × C q (G, B) −→ C p+q (G, A ⊗ B)
by
(a ∪ b)(σ0 , . . . , σp+q ) = a(σ0 , . . . , σp ) ⊗ b(σp , . . . , σp+q ).
For this map, we have the following formula.
(1.4.1) Proposition. ∂(a ∪ b) = (∂a) ∪ b + (−1)p (a ∪ ∂b).
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37
§4. The Cup-Product
Proof: We have
p+q+1
X
(−1)i (a ∪ b)(σ0 , . . . , σ̂i , . . . , σp+q+1 )
∂(a ∪ b)(σ0 , . . . , σp+q+1 ) =
i=0
=
p
X
(−1)i a(σ0 , . . . , σ̂i , . . . , σp+1 ) ⊗ b(σp+1 , . . . , σp+q+1 )
i=0
p+q+1
X
(−1)i a(σ0 , . . . , σp ) ⊗ b(σp , . . . , σ̂i , . . . , σp+q+1 ).
+
i=p+1
On the other hand
(∂a ∪ b)(σ0 , . . . , σp+q+1 ) = (∂a)(σ0 , . . . , σp+1 ) ⊗ b(σp+1 , . . . , σp+q+1 )
=
p+1
X
(−1)i a(σ0 , . . . , σ̂i , . . . , σp+1 ) ⊗ b(σp+1 , . . . , σp+q+1 )
i=0
and
(a ∪ ∂b)(σ0 , . . . , σp+q+1 ) = a(σ0 , . . . , σp ) ⊗ (∂b)(σp , . . . , σp+q+1 )
= a(σ0 , . . . , σp ) ⊗
q+1
X
(−1)i b(σp , . . . , σ̂p+i , . . . , σp+q+1 ).
i=0
Now, in the second of these seven formula lines, let the index i run from 0 to
p + 1 and in the third from p to p + q + 1. The additional summands appearing
cancel to give the result claimed.
2
From this proposition, it follows that a ∪ b is a cocycle if a and b are cocycles,
and a coboundary if one of the cochains a and b is a coboundary and the other
a cocycle. Therefore the pairing (∗) induces a bilinear map
∪
H p (G, A) × H q (G, B) −→ H p+q (G, A ⊗ B),
(α, β) 7−→ α ∪ β.
This map is called the cup-product. For p = q = 0, we obtain the map
AG × B G −→(A ⊗ B)G ,
(a, b) 7−→ a ⊗ b.
We will see below that the cup-product induces a bilinear map
∪
Ĥ p (G, A) × Ĥ q (G, B) −→ Ĥ p+q (G, A ⊗ B)
on the modified cohomology of a finite group G for all p, q
∈
ZZ.
Whenever a new cohomological map is introduced, we must check its functoriality properties and also its compatibility with the cohomological maps
already defined. Directly from the definition follows the
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38
Chapter I. Cohomology of Profinite Groups
(1.4.2) Proposition. For two homomorphisms A → A0 , B → B 0 of Gmodules, we have the commutative diagram
∪
ö÷øùõ A) × H q (G, B)
H p (G,
H p+q (G, A ⊗ B)
H p (G, A0 ) × H q (G, B 0 )
∪
H p+q (G, A0 ⊗ B 0 ) .
The cup-product is very often used in a slightly more general form. Instead
of the bilinear map A × B → A ⊗ B, we may consider an arbitrary bilinear
pairing of G-modules A × B −→ C, (a, b) 7−→ ab. It factors through A ⊗ B,
and the composite
∪
H p (G, A) × H q (G, B) −→ H p+q (G, A ⊗ B) −→ H p+q (G, C)
is also called the cup-product. The compatibility with the δ-homomorphism is
given in the following proposition.
(1.4.3) Proposition. (i) Let
0 → A0 → A → A00 → 0 and 0 → C 0 → C → C 00 → 0
be exact sequences of G-modules. Let B be another G-module and suppose
we are given a pairing A × B → C which induces pairings A0 × B → C 0 and
A00 × B → C 00 . Then the diagram
∪
þýüûú A00 ) × H q (G, B)
H p (G,
H p+q (G, C 00 )
δ
δ
id
H p+1 (G, A0 ) × H q (G, B)
is commutative, i.e.
∪
H p+q+1 (G, C 0 )
δ(α00 ∪ β) = δα00 ∪ β.
(ii) Let
0 → B 0 → B → B 00 → 0 and 0 → C 0 → C → C 00 → 0
be exact sequences of G-modules and let A × B → C be a pairing which
induces pairings A × B 0 → C 0 and A × B 00 → C 00 . Then the diagram
∪
ÿ A) × H q (G, B 00 )
H p (G,
H p+q (G, C 00 )
id
δ
(−1)p δ
∪
H p (G, A) × H q+1 (G, B 0 )
H p+q+1 (G, C 0 )
is commutative, i.e.
(−1)p δ(α ∪ β 00 ) = α ∪ δβ 00 .
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39
§4. The Cup-Product
Proof: We show (ii). Let α = ā, β 00 = b̄00 , a ∈ Z p (G, A), b00 ∈ Z q (G, B 00 ).
Let b ∈ C q (G, B) be a pre-image of b00 (b exists by §3, ex.1). Identifying B 0 with
its image in B, δβ 00 is by definition represented by the cocycle ∂b ∈ Z q+1 (G, B 0 )
and δ(α ∪ β 00 ) by the cocycle ∂(a ∪ b) ∈ Z p+q+1 (G, C 0 ). Recalling that ∂a = 0,
we obtain from (1.4.1)
∂(a ∪ b) = (∂a) ∪ b + (−1)p (a ∪ ∂b) = (−1)p (a ∪ ∂b).
Passing to the cohomology classes, we get δ(α ∪ β 00 ) = (−1)p (α ∪ δβ 00 ).
(i) is proven in the same way.
2
As before, we make everywhere the identifications
(A ⊗ B) ⊗ C = A ⊗ (B ⊗ C)
and A ⊗ B = B ⊗ A.
(1.4.4) Proposition. The cup-product is associative and skew commutative,
i.e. for α ∈ H p (G, A), β ∈ H q (G, B), γ ∈ H r (G, C) we have
(α ∪ β) ∪ γ = α ∪ (β ∪ γ)
and
α ∪ β = (−1)pq (β ∪ α).
Proof: Let a, b, c be cocycles representing α, β, γ. Then
((a ∪ b) ∪ c)(σ0 , . . . , σp+q+r ) = (a ∪ b)(σ0 , . . . , σp+q ) ⊗ c(σp+q , . . . , σp+q+r )
= a(σ0 , . . . , σp ) ⊗ b(σp , . . . , σp+q ) ⊗ c(σp+q , . . . , σp+q+r )
= a(σ0 , . . . , σp ) ⊗ (b ∪ c)(σp , . . . , σp+q+r ) = (a ∪ (b ∪ c))(σ0 , . . . , σp+q+r ).
Passing to the cohomology classes gives (α ∪ β) ∪ γ = α ∪ (β ∪ γ).
The formula α ∪ β = (−1)pq (β ∪ α) is not so easy to prove at the level of
cocycles. We therefore use the method of dimension shifting introduced in §3.
By (1.3.8), we have the surjections δ n : H 0 (G, An ) −→ H n (G, A). Applying
(1.4.3) (i) p times (resp. (ii) q times), we obtain a commutative diagram
Ap ) × H 0 (G, Bq )
H 0 (G,
δp
∪
H p (G, (A ⊗ B)q ) = H p (G, A ⊗ Bq )
(−1)pq δ q
δq
H p (G, A) × H q (G, B)
H 0 (G, (A ⊗ Bq )p ) = H 0 (G, Ap ⊗ Bq )
δp
id
H p (G, A) × H 0 (G, Bq )
id
∪
∪
H p+q (G, A ⊗ B).
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40
Chapter I. Cohomology of Profinite Groups
For p = q = 0 the rule α ∪ β = β ∪ α is clear. Since the vertical arrows are
surjective, α ∪ β = (−1)pq (β ∪ α) follows for p, q ≥ 0.
2
(1.4.5) Proposition. Let
0 −→ A0 −→ A −→ A00 −→ 0
and 0 −→ B 0 −→ B −→ B 00 −→ 0
be exact sequences of G-modules. Suppose we are given a pairing
ϕ : A × B →C
into a G-module C such that ϕ(A0 × B 0 ) = 0, i.e. ϕ induces pairings ϕ0 and ϕ00
such that the diagram
ϕ0
A 0 × B 00
C
v
i
A × B
ϕ
u
j
A00 × B 0
commutes. Then the diagram
ϕ00
H p (G, A00 ) × H q (G, B 0 )
δ
C
C
∪
H p+q (G, C)
(−1)p+1
δ
H p+1 (G, A0 ) × H q−1 (G, B 00 )
∪
H p+q (G, C)
is commutative, i.e.
(δα) ∪ β + (−1)p (α ∪ δβ) = 0
for α ∈ H p (G, A00 ) and β
∈
H q−1 (G, B 00 ).
Proof: Let a00 ∈ Z p (G, A00 ) and b00 ∈ Z q−1 (G, B 00 ) be cocycles representing α
and β respectively, and let a ∈ C p (G, A) and b ∈ C q−1 (G, B) be pre-images.
Then there exist a0 ∈ C p+1 (G, A0 ) and b 0 ∈ C q (G, B 0 ) such that ia0 = ∂a and
ub 0 = ∂b. Now δα is represented by a0 and δβ by b 0 . It follows that
(δα) ∪ β + (−1)p (α ∪ δβ) = 0
since the left class is represented by
a0 ∪ b00 + (−1)p (a00 ∪ b 0 ) = a0 ∪ vb + (−1)p (ja ∪ b 0 )
= ia0 ∪ b + (−1)p (a ∪ ub 0 )
= ∂a ∪ b + (−1)p (a ∪ ∂b)
= ∂(a ∪ b),
which is a coboundary.
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41
§4. The Cup-Product
For discrete G-modules A and C, let G act on Hom(A, C) by
(gφ)(a) = gφ(g −1 a),
where φ ∈ Hom(A, C), g ∈ G and a ∈ A. If G is finite or if A is finitely
generated as a ZZ-module, then Hom(A, C) is a discrete G-module and the
canonical pairing
Hom(A, C) × A −→ C
induces the cup-product
∪
H p (G, Hom(A, C)) × H q (G, A) −→ H p+q (G, C).
We have the following
(1.4.6) Corollary. Let
0 −→ A0 −→ A −→ A00 −→ 0
be an exact sequence of G-modules and suppose that C is another G-module
such that the sequence
0 −→ Hom(A00 , C) −→ Hom(A, C) −→ Hom(A0 , C) −→ 0
is also exact. Assume further, that G is finite or that A, A0 and A00 are finitely
generated as ZZ-modules. Then the diagram
H p (G, Hom(A0 , C))
× H q (G, A0 )
δ
∪
H p+q (G, C)
(−1)p+1
δ
H p+1 (G, Hom(A00 , C)) × H q−1 (G, A00 )
∪
H p+q (G, C)
is commutative, i.e.
(δ α̂) ∪ α + (−1)p (α̂ ∪ δα) = 0
for α̂ ∈ H p (G, Hom(A0 , C)) and α ∈ H q−1 (G, A00 ).
Proof: This follows from (1.4.5) applied to the natural pairing
Hom(A, C) × A −→ C.
2
In the next proposition, whose proof is taken from [7], §7, we define the
cup-product in arbitrary integral dimensions if G is a finite group.
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42
Chapter I. Cohomology of Profinite Groups
(1.4.7) Proposition. Let G be a finite group. Then there exists a unique family
of bilinear maps
∪
Ĥ p (G, A) × Ĥ q (G, B) −→ Ĥ p+q (G, C),
defined for all integers p, q
such that:
∈
ZZ and all pairings A × B → C of G-modules,
(i) These bilinear maps are functorial with respect to the modules.
(ii) For p = q = 0 they are induced by the natural map
AG × B G −→ C G .
(iii) Let
0 → A0 → A → A00 → 0
and
0 → C 0 → C → C 00 → 0
be exact sequences of G-modules. Let B be another G-module and suppose
we are given a pairing A × B → C which induces pairings A0 × B → C 0 and
A00 × B → C 00 . Then the diagram
$"!# A00 ) × Ĥ q (G, B)
Ĥ p (G,
δ
∪
Ĥ p+q (G, C 00 )
δ
id
Ĥ p+1 (G, A0 ) × Ĥ q (G, B)
∪
Ĥ p+q+1 (G, C 0 )
is commutative, i.e. δ(α00 ∪ β) = δα00 ∪ β.
(iv) Let
0 → B 0 → B → B 00 → 0
and
0 → C 0 → C → C 00 → 0
be exact sequences of G-modules and let A × B → C be a pairing which
induces pairings A × B 0 → C 0 and A × B 00 → C 00 . Then the diagram
%)('& A) × Ĥ q (G, B 00 )
Ĥ p (G,
id
∪
(−1)p δ
δ
Ĥ p (G, A) × Ĥ q+1 (G, B 0 )
Ĥ p+q (G, C 00 )
∪
Ĥ p+q+1 (G, C 0 )
is commutative, i.e. (−1)p δ(α ∪ β 00 ) = α ∪ δβ 00 .
Proof: Without loss of generality, we may assume that C = A ⊗ B. If
X = X (G, ZZ) denotes the complete standard resolution of ZZ, cf. §2, p.22,
then we obtain a homomorphism of complexes
.
.
.
.
.
.
HomG (X , A) ⊗ HomG (X , B) −→ HomG (X ⊗ X , A ⊗ B) .
The proof of existence of the cup-product depends on constructing G-module
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§4. The Cup-Product
homomorphisms
ϕp,q : Xp+q −→ Xp ⊗ Xq ,
for all integers p, q, satisfying the following two conditions
(1)
ϕp,q ∂ = (∂ ⊗ 1)ϕp+1,q + (−1)p (1 ⊗ ∂)ϕp,q+1 ,
(2)
(ε ⊗ ε)ϕ0,0 = ε ,
where ε : X0 → ZZ is defined by ε(σ) = 1 for all σ ∈ G. The induced map of
complexes ϕ : X −→ X ⊗ X then defines a homomorphism
.
.
.
.
.
.
HomG (X , A) ⊗ HomG (X , B) −→ HomG (X , A ⊗ B)
(f, g) 7−→ f · g = (f ⊗ g)ϕ .
It follows from (1) that
∂(f · g) = (∂f ) · g + (−1)p f · (∂g) .
Hence if f, g are cocycles, so is f · g, and the cohomology class of f · g depends
only on the classes of f and g. Thus we obtain homomorphisms
∪
Ĥ p (G, A) ⊗ Ĥ q (G, B) −→ Ĥ p+q (G, A ⊗ B) ,
which obviously satisfy (i), and (ii) is a consequence of (2). The properties (iii)
and (iv) are proved as in (1.4.3). This gives us the existence of the cup-product
and the uniqueness is proved by starting with (ii) and shifting dimensions by
(iii) and (iv), as in §3, p.32. Observe that the exact sequences
0 → A → IndG (A) → A1 → 0
and
0 → A−1 → IndG (A) → A → 0
split over ZZ. Thus the result of tensoring these by any G-module B is still
exact and IndG (A) ⊗ B = IndG (A ⊗ B).
It remains to define the maps ϕp,q , which we will do as follows:
If p ≥ 0 and q ≥ 0,
ϕp,q (σ0 , . . . , σp+q ) = (σ0 , . . . , σp ) ⊗ (σp , . . . , σp+q ) .
If p ≥ 1 and q
≥
1,
ϕ−p,−q (σ1 , . . . , σp+q ) = (σ1 , . . . , σp ) ⊗ (σp+1 , . . . , σp+q ) .
If p ≥ 0 and q
≥
1,
ϕp,−p−q (σ1 , . . . , σq ) =
X
ϕ−p−q,p (σ1 , . . . , σq ) =
X
ϕp+q,−q (σ0 , . . . , σp ) =
X
ϕ−q,p+q (σ0 , . . . , σp ) =
X
(σ1 , τ1 , . . . , τp ) ⊗ (τp , . . . , τ1 , σ1 , . . . , σq ) ,
(σ1 , . . . , σq , τ1 , . . . , τp ) ⊗ (τp , . . . , τ1 , σq ) ,
(σ0 , . . . , σp , τ1 , . . . , τq ) ⊗ (τq , . . . , τ1 ) ,
(τ1 , . . . , τq ) ⊗ (τq , . . . , τ1 , σ0 , . . . , σp ) ,
where the τi on the right-hand side run independently through G. The verification that the ϕp,q satisfy the formulae above is straightforward.
2
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44
Chapter I. Cohomology of Profinite Groups
Remark: Under the assumption that G is finite, the propositions (1.4.4) and
(1.4.5) hold for all integers p, q, r ∈ ZZ.
From the proof of the last proposition, we obtain the following explicit
formulae.
(1.4.8) Proposition. Let G be a finite group. Let x̄ ∈ Ĥ p (G, A), p ≥ −1
and ȳ ∈ H q (G, B), q ≥ 1. Let y ∈ C q (G, B) be an inhomogeneous cocycle
representing ȳ. For p ≥ 0, let x ∈ C p (G, A) be a representing cocycle of x̄.
For p = −1, let x ∈ NG A represent x̄. Then the inhomogeneous cochains
(x ∪ y)(σ1 , . . . , σq ) = x ⊗ y(σ1 , . . . , σq )
(x ∪ y)(σ1 , . . . , σq−1 ) =
X
σ ∈G
for p = 0,
σx ⊗ σy(σ −1 , σ1 , . . . , σq−1 )
for p = −1,
(x ∪ y)(σ1 , . . . , σp+q ) = x(σ1 , . . . , σp ) ⊗ σ1 · · · σp y(σp+1 , . . . , σp+q )
for p > 0,
represent the cup-product x̄ ∪ ȳ
∈
Ĥ
p+q
(G, A ⊗ B).
Exercise 1. Let R be a G-ring, i.e. a ring with an action of G such that σ(a + b) = σa + σb and
σ(ab) = σaσb. Show that
M
H(G, R) :=
H n (G, R)
n≥ 0
is a graded ring with respect to the cup-product which is induced by the multiplication
R × R → R.
Exercise 2. Let A be an R-module with a G-operation compatible with the R-module structure.
Show that
M
H(G, A) :=
H n (G, A)
n≥ 0
is in a natural way an H(G, R)-module.
Exercise 3. Prove the following generalization of (1.4.3): Let
0 → A0 → A → A00 → 0,
0 → B 0 → B → B 00 → 0,
0 → C 0 → C → C 00 → 0
be exact sequences of G-modules. Suppose we are given a pairing ϕ: A × B → C such that
ϕ(A0 × B 0 ) = 0 ,
ϕ(A × B 0 ) ⊆ C 0
ϕ(A0 × B) ⊆ C 0 .
and
Then we get an induced pairing A00 ×B 00 → C 00 , and for α00
we have
∈
H p (G, A00 ) and β 00
∈
H q (G, B 00 )
δ(α00 ∪ β 00 ) = (δα00 ) ∪ β 00 + (−1)p α00 ∪ (δβ 00 ).
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45
§5. Change of the Group G
§5. Change of the Group G
We now turn to the question of what happens to the cohomology groups
H (G, A) if we change the group G. We put ourselves in the most general
situation if we consider two profinite groups G and G0 , a G-module A, a
G0 -module A0 and two homomorphisms
ϕ : G0 −→ G, f : A −→ A0 ,
such that f (ϕ(σ 0 )a) = σ 0 f (a). From such a compatible pair of homomorphisms,
we obtain a homomorphism
C n (G, A) −→ C n (G0 , A0 ), a 7→ f ◦ a ◦ ϕ.
Trivially, this commutes with ∂ and therefore induces a homomorphism
H n (G, A) −→ H n (G0 , A0 ).
If we have two compatible pairs of homomorphisms G00 → G0 → G,
A → A0 → A00 , then the homomorphism
H n (G, A) −→ H n (G00 , A00 ),
induced by the composites G00 → G and A → A00 , is the composite of the
homomorphisms
H n (G, A) −→ H n (G0 , A0 ) and H n (G0 , A0 ) −→ H n (G00 , A00 )
given by G0 → G, A → A0 and G00 → G0 , A0 → A00 . Thus the cohomology
groups H n (G, A) are functorial in G and A simultaneously.
n
Let (Gi )i∈I be a projective system of profinite groups and let (Ai )i∈I be a
direct system, where each Ai is a Gi -module and the transition maps
Gj → Gi , Ai → Aj
form compatible pairs. Then, with the induced homomorphisms
H n (Gi , Ai ) −→ H n (Gj , Aj ) ,
the cohomology groups H n (Gi , Ai ) form a direct system of abelian groups.
As a generalization of (1.2.5), we have the
(1.5.1) Proposition. If G = lim
←− i∈I
Gi and A = lim
−→ i∈I
Ai , then
H n (G, A) ∼
= lim H n (Gi , Ai ).
−→
i∈I
Proof: For every i ∈ I, the pair G → Gi , Ai → A is compatible. We
therefore have a canonical homomorphism κi : C n (Gi , Ai ) → C n (G, A),
hence a homomorphism
κ : lim
C n (Gi , Ai ) −→ C n (G, A),
−→
i∈I
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46
Chapter I. Cohomology of Profinite Groups
which obviously commutes with the ∂-homomorphism. It therefore suffices to
show that κ is an isomorphism.
For the surjectivity, let y : Gn → A be the inhomogeneous cochain associated to x ∈ C n (G, A). Since Gn is compact, A discrete and y continuous,
y takes only finitely many values and factors through ȳ : (G/U )n → A for a
suitable open normal subgroup U . The finitely many values are represented by
elements of some Ai , i.e. ȳ is the composite of a function ȳi : (G/U )n → Ai
with Ai → A. On the other hand, there exists a j ≥ i such that the projection
G → G/U factors through the canonical map Gj → G/U , i.e. we obtain an
inhomogeneous cochain yj : Gnj → Aj as the composite
ȳ
i
Gnj −→ (G/U )n −→
Ai −→ Aj ,
yj
such that the composite Gn −→ Gnj −→ Aj −→ A is y. If xj ∈ C n (Gj , Aj ) is
the homogeneous cochain associated to yj , then its image in C n (G, A) is x.
This shows the surjectivity of κ.
For the injectivity, let xi ∈ C n (Gi , Ai ) be a cochain which becomes zero in
C n (G, A), i.e. the composite
x
i
Gn+1 −→ Gn+1
−→
Ai −→ A
i
is zero. Since xi has only finitely many values, there exists a j
the composite
xi
Gn+1
−→ Gn+1
−→
Ai −→ Aj
j
i
≥
i such that
is already zero, i.e. xi becomes zero in C n (Gj , Aj ) and hence represents the
zero class in lim C n (Gi , Ai ). This shows the injectivity of κ.
2
−→
i
We shall have to deal mainly with three special cases of homomorphisms
H (G, A) → H n (G0 , A0 ) coming from compatible pairs G0 → G, A → A0 ,
and an additional case, arising in a different way.
n
1. Conjugation. Let H be a closed subgroup of G, A a G-module and B an
H-submodule of A. For σ, τ ∈ G we write τ σ = σ −1 τ σ and σH = σHσ −1 . The
two compatible homomorphisms
σ
H −→ H, τ 7−→ τ σ ,
B −→ σB, b 7−→ σb
induce isomorphisms
σ∗ : H n (H, B) −→ H n (σH, σB),
which are called conjugation. We have
1∗ = id
and
(στ )∗ = σ∗ τ∗ ,
from what we have said above about composition.
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47
§5. Change of the Group G
2. Inflation. Let H be a normal closed subgroup of G and A a G-module.
Then AH is a G/H-module. The projection and injection
G*+
AH
G/H,
A
form a compatible pair of homomorphisms, which induces a homomorphism
inf
G/H
G
: H n (G/H, AH ) −→ H n (G, A),
called inflation. The inflation is transitive, i.e. for two normal closed subgroups
H ⊆ F of G, we have
inf
G/H
G
◦ inf
G/F
G/H
G/F
G .
= inf
3. Restriction. For an arbitrary closed subgroup H of G and a G-module A,
we consider the two homomorphisms
H-, incl G,
A id A.
On the cochains they induce the restriction maps and we obtain homomorphisms on the cohomology
n
n
res G
H : H (G, A) −→ H (H, A),
called restriction. Clearly the restriction is transitive, i.e. for two closed
subgroups F ⊆ H, we have
G
G
res H
F ◦ res H = res F .
4. Corestriction. If H is an open subgroup of G, then besides the restriction,
we have another map in the opposite direction, which is a kind of norm
map and is called the corestriction: it arises from the standard resolution
A → X = X (G, A) of the G-module A, which is also an acyclic resolution
of A as an H-module, i.e.
.
.
.
H n (H, A) = H n ((X )H )
(see §3 p.34). For n ≥ 0, we have for the G-module X n the norm map
NG/H : (X n )H →(X n )G . It obviously commutes with ∂, hence we have a
morphism of complexes
.
.
NG/H : (X )H −→(X )G .
Taking homology groups of these complexes, we obtain canonical homomorphisms
n
n
cor H
G : H (H, A) −→ H (G, A).
For n = 0, this is the usual norm map
NG/H : AH −→ AG .
For two open subgroups F
implies the transitivity
⊆
H of G, the equation NG/H ◦ NH/F = NG/F
F
F
cor H
G ◦ cor H = cor G .
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48
Chapter I. Cohomology of Profinite Groups
On the level of cochains the corestriction is explicitly given as follows. From
every right coset c = Hσ ∈ H\G, we choose a fixed representative c̄ ∈ c and
define the homomorphism
cor : C n (H, A) −→ C n (G, A),
by
X
(cor x)(σ0 , . . . , σn ) =
c̄ −1 x(c̄σ0 cσ 0−1 , . . . , c̄σn cσ n−1 ).
c∈H\G
cor x is again G-linear. In fact, if σ
∈
G, then c̄σ =: τσ cσ for some τσ
∈
H and
−1
σ (cor x)(σσ0 , . . . , σσn )
=
X
σ −1 c̄ −1 x(c̄σσ0 cσσ 0−1 , . . .)
c∈H\G
=
X
−1
cσ −1 τσ−1 x(τσ cσσ0 (cσ)σ 0 , . . .)
c∈H\G
=
X
c̄ −1 x(c̄σ0 cσ 0−1 , . . .).
c∈H\G
Obviously cor ◦ ∂ = ∂ ◦ cor , so we get a homomorphism
cor : H n (H, A) −→ H n (G, A).
It is functorial in A and commutes with the δ-homomorphism, which we will
see in a moment. By dimension shifting we see that it coincides with the core≥ 0, we have the commutative
striction cor H
G constructed before: for each n
diagrams
n
H 0 (H,120/.3 An ) δ
H n (H, A)
cor H
G
cor H
G
cor
δn
H 0 (G, An )
cor
H n (G, A) ,
where the horizontal maps are surjective. The vertical arrows cor H
G and cor
≥
are the norm NG/H for n = 0, hence coincide for all n 0.
(1.5.2) Proposition. The maps σ∗ , inf, res, cor are functorial in the G-module
considered, and they commute with the δ-homomorphism.
Proof: The functoriality is seen already on the level of cochains. We show
the commutativity of the corestriction with δ, leaving the other cases to the
reader. Let
07456
A00
i
A
j
A0
0
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49
§5. Change of the Group G
be an exact sequence of G-modules. We then obtain a commutative exact
diagram
C n (G, A00 )
0
cor
∂
∂
∂
C n+1 (G, A0 )
C n+1 (G, A)
cor
C n+1 (H, A00 )
0
∂
C n+1 (G, A00 )
cor
0
∂
C n (H, A0 )
C n (H, A)
∂
0
0
cor
cor
C n (H, A00 )
0
C n (G, A0 )
C n (G, A)
0
cor
C n+1 (H, A0 )
C n+1 (H, A)
0
and from this, the commutativity of the diagram
=89:;<JBCDEFGHI>LMNOKA@?RSPQ
VUTW A0 )
H n (H,
δ
H n+1 (H, A00 )
cor
cor
δ
H n (G, A0 )
H n+1 (G, A00 ).
Namely, let a0 ∈ Z n (H, A0 ) be a cocycle in the class α0 ∈ H n (H, A0 ). If
a ∈ C n (H, A) is a pre-image of a0 , then ∂a is a cocycle in the class δα0 ∈
H n+1 (H, A00 ) and cor ∂a is a cocycle in the class cor δα0 . On the other hand,
cor ∂a = ∂cor a, and cor a is a pre-image of the cocycle cor a0 , which represents cor α0 ∈ H n (G, A0 ), so that ∂ cor a ∈ Z n+1 (G, A00 ) represents the class
δ cor α0 ∈ H n+1 (G, A00 ). Therefore we have δ ◦ cor = cor ◦ δ.
In the same way one proves δ ◦σ∗ = σ∗ ◦δ, δ ◦res = res ◦δ, δ ◦inf = inf ◦δ,
the latter if the sequence 0 → A00H → AH → A0H → 0 is also exact.
2
(1.5.3) Proposition. The maps σ∗ , inf, res, cor are compatible with the cupproduct as follows.
(i) σ∗ (α ∪ β) = σ∗ α ∪ σ∗ β
for α ∈ H p (H, A), β
∈
H q (H, B) and σ
∈
G.
(ii) inf (α ∪ β) = (inf α) ∪ (inf β)
for α ∈ H p (G/H, AH ), β
subgroup of G.
∈
H q (G/H, B H ), if H is a normal closed
(iii) res (α ∪ β) = (res α) ∪ (res β)
for α ∈ H p (G, A), β
∈
H q (G, B), if H is a closed subgroup of G.
(iv) cor (α ∪ res β) = (cor α) ∪ β
for α ∈ H p (H, A), β
∈
H q (G, B), if H is an open subgroup of G.
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50
Chapter I. Cohomology of Profinite Groups
Proof. (i), (ii) and (iii) are seen at once on the level of cochains. (iv) is
equivalent to the commutativity of the diagram
\[ZYX A) × H q (H, B)
H p (H,
∪
H p+q (H, A ⊗ B)
cor
res
cor
H p (G, A) × H q (G, B)
∪
H p+q (G, A ⊗ B).
We may by (1.2.5) assume that G is finite: apply lim to the diagram with G, H
−→ U
replaced by G/U, H/U , where U runs through the open normal subgroups U
contained in H. By dimension shifting, we may transform the above diagram
into the diagram
a`_^] Ap ) × Ĥ 0 (H, Bq )
Ĥ 0 (H,
∪
Ĥ 0 (H, Ap ⊗ Bq )
res
cor
cor
Ĥ 0 (G, Ap ) × Ĥ 0 (G, Bq )
∪
Ĥ 0 (G, Ap ⊗ Bq ),
which comes from the diagram
AbfdecH
× BqH
p
⊗
(Ap ⊗ Bq )H
NG/H
NG/H
G
AG
p × Bq
⊗
(Ap ⊗ Bq )G .
But this diagram is commutative:
NG/H (a ⊗ b) =
X
σa ⊗ σb =
X
σa ⊗ b = NG/H (a) ⊗ b.
2
σ ∈G/H
σ ∈G/H
The compatibilities of the maps σ∗ , inf, res, cor with each other are described
by the following propositions.
(1.5.4) Proposition. σ∗ commutes with inf, res, cor.
(1.5.5) Proposition. For two closed subgroups V
(i) inf
U/V
U
◦
G/V
res U/V
= res G
U ◦ inf
⊆
U
⊆
G, we have
G/V
G ,
if V is normal in G,
(ii) inf
G/V
G
U/V
◦ cor G/V = cor UG ◦ inf
U/V
U ,
if V is normal and U is open in G.
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§5. Change of the Group G
All this can be seen directly on the level of cochains. Another useful formula
is stated in the following
(1.5.6) Proposition. If U, V are closed subgroups of G, with V open, and if
σ runs through a system of representatives of the (finite) double coset decomposition
[
G = . U σV,
σ
then we have the double coset formula
V
res G
U ◦ cor G =
X
cor UU ∩σV σ
−1
◦ σ∗ ◦ res VV ∩σ−1 U σ .
σ
Proof: Because of (1.2.5), (1.5.4) and (1.5.5), we may assume that G is a
finite group. By dimension shifting we are then reduced to show the formula
only for n = 0, i.e. on H 0 (V, A) = AV . In this case res becomes the inclusion,
cor the norm and σ∗ the map a 7→ σa. Therefore we have to prove the formula
NG/V a =
X
NU/U ∩σV σ−1 (σa).
σ
For every σ, we choose a system τσ of left representatives of U/U ∩ σV σ −1 ,
i.e.
[
U = . τσ (U ∩ σV σ −1 ).
τσ
Then
[
G = . τσ σV,
σ,τσ
i.e. τσ σ runs through a system of representatives of G/V , and so
NG/V a =
XX
σ
τσ
τσ σa =
X
NU/U ∩σV σ−1 (σa).
2
σ
(1.5.7) Corollary. If U is an open subgroup of G, then
cor UG ◦ res G
U = (G : U ).
If U is normal, then
U
res G
U ◦ cor G = NG/U .
The first formula is trivial in dimension zero and follows for arbitrary dimension by dimension shifting. The second formula is the double coset formula
for the case V = U .
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52
Chapter I. Cohomology of Profinite Groups
(1.5.8) Corollary. If G = U V , then the diagram
jihg A)
H n (V,
res
H n (U ∩ V, A)
cor
cor
res
H n (G, A)
H n (U, A)
is commutative.
For a locally compact abelian group A, let A∨ denote the Pontryagin dual,
see §1, p.9. We have a canonical isomorphism
H 2 (G, ZZ)∨ ∼
= Gab
onto the abelianized group Gab = G/G0 , the quotient of G by the closure G0
of the commutator subgroup. For this, note that H n (G, Q) = 0 for n ≥ 1 (see
(1.6.2)(c)). Therefore the exact sequence 0 → ZZ → Q → Q/ZZ → 0 yields an
isomorphism
H 2 (G, ZZ) ∼
= H 1 (G, Q/ZZ) = Homcts (G, Q/ZZ) = (Gab )∨ ,
and, using Pontryagin duality,
H 2 (G, ZZ)∨ ∼
= (Gab )∨∨ = Gab .
For an open subgroup H of G, we may ask what homomorphism
Gab −→ H ab
is induced by the dual
cor ∨ : H 2 (G, ZZ)∨ −→ H 2 (H, ZZ)∨
of the corestriction. The answer is given by the transfer map (Verlagerung)
of group theory, which is defined as follows.
For each right coset c ∈ H\G, we choose a fixed representative c̄, 1̄ = 1, so
that c = H c̄. Then the transfer is the continuous homomorphism
Ver : Gab −→ H ab ,
σG0 7−→
Y
c̄σcσ −1 H 0 .
c∈H\G
(1.5.9) Proposition. The map cor ∨ : H 2 (G, ZZ)∨ → H 2 (H, ZZ)∨ induces the
transfer map
Ver : Gab −→ H ab .
If H is normal in G, the composite H ab → Gab → H ab is the norm NG/H .
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§5. Change of the Group G
Proof: On the homogeneous 1-cochains, the corestriction
cor : C 1 (H, Q/ZZ) → C 1 (G, Q/ZZ)
is given by
(cor x)(σ0 , σ1 ) =
X
x(c̄σ0 cσ 0−1 , c̄σ1 cσ 1−1 ).
c∈H\G
On the associated inhomogeneous cochains y(σ) = x(1, σ), it is thus given by
(cor y)(σ) = (cor x)(1, σ) =
X
y(c̄σcσ −1 ).
c∈H\G
Hence on the dual Gab of H 1 (G, Q/ZZ) = Z 1 (G, Q/ZZ) = Hom(Gab , Q/ZZ), it is
given by the transfer map.
Ver
Let H be normal in G. The composite H ab −→ Gab −→ H ab is dual to the
composite of
cor
res
H 2 (H, ZZ) −→ H 2 (G, ZZ) −→ H 2 (H, ZZ),
which by (1.5.7) is res ◦ cor = NG/H .
2
If A is a G-module, then for every pair V ⊆ U of open subgroups, we have
two maps
incl
Alk U
AV ,
NU/V
the inclusion and the norm. For the cohomology groups, we have similarly
two maps
nm A) res H n (V, A),
H n (U,
cor
and these satisfy the double coset formula (1.5.6). From this observation we
are led to a generalization of G-modules, which gives a conceptual explanation
of the double coset formula. We enlarge the totality of open subgroups U of
G as follows.
We consider the category B(G) of finite G-sets, i.e. of finite sets X with
a continuous action of G. For every open subgroup U of G, the quotient
G/U is a finite G-set by left multiplication. It is a connected, i.e. transitive,
G-set, and every connected finite G-set X is of this form. For, if x ∈ X and
Gx = {σ ∈ G | σx = x}, then G/Gx → X, σGx 7→ σx, is an isomorphism
of G-sets. The category B(G) has the advantage of containing, for every
two finite G-sets X, Y , the disjoint union X q Y (as categorical sum), and
for every pair of morphisms f : X → S, g : Y → S, the fibre product
X ×S Y = {(x, y) ∈ X × Y | f (x) = g(y)}.
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Chapter I. Cohomology of Profinite Groups
(1.5.10) Definition. A G-modulation is a pair of functors
A = (A∗ , A∗ ) : B(G) → Ab
into the category Ab of abelian groups, A∗ covariant, A∗ contravariant, such
that
A∗ (X) = A∗ (X) =: A(X)
for all X
∈
B(G), and that the following two conditions are satisfied:
(i) A(X q Y ) = A(X)⊕A(Y ).
(ii) If the left one of the diagrams
utsv
Xrqpo
g0
f0
f
Y
g
A(X)
X0
and
Y0
g 0∗
A(X 0 )
f∗0
f∗
A(Y )
g∗
A(Y 0 )
is cartesian (i.e. X ∼
= X 0 ×Y 0 Y ), then the right one commutes.
ϕ∗ = A∗ (ϕ), ϕ∗ = A∗ (ϕ) for a morphism ϕ in B(G). ∗)
Here
In this form the G-modulations were introduced by A. DRESS [43] under the
name “Mackey functors” (they were defined earlier by A. GREEN under still
another name, see [59]).
By condition (i), a G-modulation A is completely determined by its restriction to the full subcategory B0 (G) of the G-sets G/U , where U runs through
the open subgroups of G. One then writes A(U ) in place of A(G/U ). Every
morphism in B0 (G) is in a unique way the composite of a projection
πUV : G/V −→ G/U
(V
⊆
U)
and a “conjugation”
c(σ) : G/U −→ G/σU σ −1 ,
τ U 7−→ τ σ −1 (σU σ −1 ).
We set
res UV = A∗ (πUV ) : A(U ) −→ A(V ),
ind VU = A∗ (πUV ) : A(V ) −→ A(U ),
σ ∗ = A∗ (c(σ)) : A(σU σ −1 ) −→ A(U ),
σ∗ = A∗ (c(σ)) : A(U ) −→ A(σU σ −1 ).
∗) Clearly, this notion of G-modulations extends to G-modulations A : B(G) → A with
values in an arbitrary abelian category A.
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§5. Change of the Group G
(1.5.11) Proposition (Double Coset Formula). Let U, V ⊆ W be open subgroups of G and let R be a system of representatives of U \W/V . Then, for
every G-modulation A, we have the formula
V
res W
U ◦ ind W =
X
ind UU ∩σV σ
−1
◦ σ∗ ◦ res VV ∩σ−1 U σ .
σ ∈R
Proof: Let X = G/U, Y = G/V, S = G/W . For the fibre product X ×S Y
we have the orbit decomposition
[
X× Y = . C ,
S
σ
σ ∈R
U
where Cσ is the G-orbit of the element (U, σV ) ∈ X ×S Y . Let f = πW
,
V
g = πW and let p : X ×S Y → X, q : X ×S Y → Y be the projections and
pσ = p |Cσ , qσ = q |Cσ . Then by the properties (i) and (ii) in (1.5.10)
f ∗ ◦ g∗ =
(∗)
X
pσ∗ ◦ qσ∗ .
σ ∈R
The isotropy groups of the elements (U, σV ) and (σ −1 U, V ) of Cσ are U ∩
σV σ −1 and σ −1 U σ ∩ V , and we have the commutative diagram
G/U ∩ yzxw{}| σV σ −1
c(σ)
G/σ −1 U σ ∩ V
χ
Cσ
πU
pσ
G/U
πV
qσ
G/V.
V
−1
In the formula (∗) we have f ∗ = res W
U , g∗ = ind W , pσ∗ = πU ∗ ◦ c(σ)∗ ◦ χ∗ =
−1
ind UU ∩σV σ ◦ σ∗ ◦ χ∗ and qσ∗ = (πV ◦ χ−1 )∗ = (χ−1 )∗ ◦ πV∗ = χ∗−1 ◦ res VV ∩σ−1 U σ .
This gives the desired result.
2
Recall that a G-modulation A is completely determined by its restriction to
B0 (G), since every finite G-set is the disjoint union of connected G-sets, and
every connected G-set is isomorphic to a set G/U . Conversely, we have the
(1.5.12) Proposition. Let A = (A∗ , A∗ ) : B0 (G) → Ab be a pair of functors,
A∗ covariant, A∗ contravariant, such that A∗ (U ) = A∗ (U ) := A(U ). Assume
that the double coset formula (1.5.11) holds and that moreover σ ∗ ◦ σ∗ = id for
every σ ∈ G. Then A extends uniquely to a G-modulation on B(G).
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Chapter I. Cohomology of Profinite Groups
One obtains the extension of A to B(G) as follows. LetYX ∈ B(G) be
an arbitrary finite G-set. We let G act on the group AX =
A(Gx ) by the
x∈X
conjugation σx : A(Gx ) → A(Gσx ), and we set
A(X) = HomG (X, AX ),
i.e. A(X) is the group of all G-equivariant maps X → AX . The proof that this
becomes a G-modulation is left to the reader.
In order to give a G-modulation A, it thus suffices to define an abelian group
A(U ) for every open subgroup U of G and to establish, for every σ ∈ G and
every pair V ⊆ U of open subgroups, the maps
€~ )
A(U
σ∗
σ∗
A(σU σ −1 ),
A(V )
ind
res
A(U ),
which yield the functors A∗ and A∗ , and then to verify the double coset formula.
It is clear what is meant by a morphism A → B of G-modulations. So the
G-modulations form a category, which we denote by Mod(G). We mention
the following examples of G-modulations.
Example 1: G-modules. Let M be a G-module. For every open subgroup U
we set M (U ) = M (G/U ) = M U . For σ ∈ G we define the maps σ ∗ and σ∗ by
a 7→ σ −1 a and a 7→ σa, and for every pair V ⊆ U we define res UV and ind VU
as the inclusion M U ,→ M V and the norm M V → M U , a 7→ NU/V (a) =
Q
σ ∈U/V σa (M is written multiplicatively). In this way, every G-module M
becomes a G-modulation, denoted again by M , and we obtain an embedding
i : Mod(G) −→ Mod(G)
of the category Mod(G) of G-modules. With this embedding, Mod(G) becomes
the full subcategory of Mod(G) of the G-modulations A with Galois descent,
meaning that for every pair V E U of open subgroups the restriction
A(U ) −→ A(V )U/V
is an isomorphism.
We note that, in particular, every abelian group A gives rise to a constant
G-modulation U 7→ A(U ) = A with the maps σ ∗ = σ∗ = id, res UV = id and
ind VU = (U : V ).
The embedding i : Mod(G) → Mod(G) has as left adjoint the functor
Mod(G) → Mod(G), A 7→ Ā = lim A(U ), i.e. for every G-module M we
−→ U
have
HomMod(G) (A, iM ) = HomM od(G) (Ā, M ).
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57
§5. Change of the Group G
Example 2: Cohomology. For every G-module M and every n
cohomology H n yields a G-modulation
A : B0 (G) −→ Ab,
≥
0, the
G/U 7→ A(U ) = H n (U, M ),
if we choose for res UV and ind VU the cohomological restriction res UV and corestriction cor VU , and for σ ∗ , σ∗ the cohomological conjugations (σ −1 )∗ , σ∗ .
Example 3: The fundamental G-modulation π ab . We consider the map
B0 (G) −→ Ab,
G/U 7−→ π ab (U ) := U ab ,
which associates to every open subgroup U of G the abelianized group U ab =
U/U 0 , where U 0 is the closure of the commutator subgroup of U . For σ ∈ G the
maps σ∗ : U ab → (σU σ −1 )ab and σ ∗ : U ab → (σ −1 U σ)ab are the conjugations
x 7→ σxσ −1 and x 7→ σ −1 xσ, and for a pair V ⊆ U the map ind VU is induced by
the inclusion V ,→ U , whereas res UV is given by the transfer Ver : U ab → V ab .
Example 4: The representation ring. We consider all finite dimensional
complex representations V of the profinite group G, i.e. all continuous homomorphisms G → GL(V), where V is any finite dimensional C-vector space.
Such representations have finite images. Let R(G) be the set of isomorphism
classes {V} of such representations V. We define an addition in R(G) by
{V} + {V 0 } := {V ⊕ V 0 }.
Then R(G) becomes a commutative monoid. From R(G) we obtain an additive
group Rep(G) by setting
Rep(G) = (R(G) × R(G))/ ∼,
where the equivalence relation ∼ is defined by
({V}, {V 0 }) ∼ ({W}, {W 0 }) ⇐⇒ {V} + {W 0 } = {W} + {V 0 }.
R(G) becomes a submonoid of Rep(G) by identifying {V} with ({V}, O).
We may even turn Rep(G) into a ring if we define the multiplication by
{V}{W} = {V ⊗ W}. This ring is called the representation ring of G.
Forgetting the ring structure, we obtain a G-modulation
Rep : B0 (G) −→ Ab,
G/U 7−→ Rep(U ),
as follows. Let σ ∈ G and let U be an open subgroup of G. If U → GL(V) is
a representation of U , then the composite with the conjugation σU σ −1 → U ,
σuσ −1 7→ u, gives a representation σU σ −1 → GL(V). This assignment
extends to an isomorphism σ∗ : Rep(U ) → Rep(σU σ −1 ), and we set σ ∗ = σ∗−1 .
For a pair V ⊆ U of open subgroups, the map
res UV : Rep(U ) → Rep(V )
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Chapter I. Cohomology of Profinite Groups
is obtained by restricting a representation U → GL(V) to V − this is a
homomorphism (even a ring homomorphism). The map
ind VU : Rep(V ) −→ Rep(U )
is obtained by associating to a representation V of V the induced representation
ind VU (V) of U . The underlying vector space consists of all continuous maps
f : U → V such that f (τ σ) = τ f (σ) for τ ∈ V, σ ∈ U . The action of σ ∈ U on
ind VU (V) is given by f 7→ σf, (σf )(x) = f (xσ). In this way we obtain in fact
a G-modulation, because the double coset formula is in this case a theorem of
I. MACKEY. This brought about the name Mackey functor (see [43]).
For every G-modulation A, the groups A(U ) are in a canonical way topological groups: a basis of neighbourhoods of 0 is given by the groups ind VU A(V )
for V ⊆ U . It is easy to see that the maps σ ∗ , σ∗ , res , ind are continuous. A
is called quasi-compact, Hausdorff, compact etc. if all the groups A(U ) have
T
the corresponding property. A is Hausdorff if V ⊆U ind VU A(V ) = 0 for all U .
A is compact if it is quasi-compact and Hausdorff, and this is equivalent to
being profinite, i.e. all A(U ) are profinite groups.
To every G-modulation A there is associated a submodulation NA, called
the modulation of “universal norms”, which is given by
\
ind VU A(V ).
NA(U ) =
V ⊆U
ƒ‚ )
U , the homomorphisms A(V
A(U ) induce
res
ind
„… )
homomorphisms NA(V
NA(U ) is trivial for ind and follows for res from
res
the double coset formula. The quotient A/NA is a Hausdorff G-modulation
(see exercise 6).
For further results on G-modulations we refer to [147], [161], [235].
The fact that, for V
Exercise 1. For n
≥
ind
⊆
1 we have lim
−→ U
H n (U, A) = 0, where U runs over the open subgroups
of G and the limit is taken over the restriction maps res : H n (U, A) → H n (V, A), V
Exercise 2. If H is a normal subgroup of G and A a G-module, then res G
H
n
n
G/H
res G
:
H
(G,
A)
−→
H
(H,
A)
,
H
and if moreover H is open, then cor H
G yields a homomorphism
n
cor H
:
H
(H, A)G/H −→ H n (G, A).
G
⊆
U.
is a homomorphism
.
.
Exercise 3. Let H be an open subgroup of G, A a G-module and A → X = X (G, A) the
res
standard resolution of A. Then the restriction H n (G, A) −→ H n (H, A) is obtained by taking
the homology of the restriction map
.
res
.
(∗)
(X )G −→(X )H .
Hint: By (1.3.9), we obtain isomorphisms of δ-functors
H n (G, A) = H n (H 0 (G, X )), H n (H, A) ∼
= H n (H 0 (H, X )).
ι
(∗) induces a functorial map H n (G, A) → H n (H, A), which commutes with the δ-homomorphism. It coincides with res for n = 0 and for n > 0 by dimension shifting.
.
.
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59
§5. Change of the Group G
Exercise 4. Consider a diagram
1ŽŒ‹†‡ˆ‰Š
A
1
Ĝ
f
ϕ̂
A0
Ĝ0
G
1
ϕ
G0
1
0
with exact rows, where G, G are profinite groups and A (resp. A0 ) is a G-module (resp. a
G0 -module). Consider moreover the maps
ϕ∗
f∗
H 2 (G0 , A0 ) −→ H 2 (G, A0 ) ←− H 2 (G, A)
ϕ
id
f
id
given by the pairs (G → G0 , A0 → A0 ) and (G → G, A → A0 ). Let u0 ∈ H 2 (G0 , A0 ), u
H 2 (G, A) be the classes belonging to the upper and lower group extension respectively.
∈
(i) Show that the diagram of solid arrows can be commutatively completed by an arrow
ϕ̂ : Ĝ → Ĝ0 if and only if f is a G-homomorphism (G acting on A0 via ϕ) and
ϕ∗ (u0 ) = f∗ (u).
(ii) Two such arrows ϕ̂1 , ϕ̂2 are called equivalent if there exists an a0
ϕ̂2 (σ̂) = a0 ϕ̂1 (σ̂)a0−1
1
for all σ̂
∈
∈
A0 , such that
Ĝ.
0
Show that the set of equivalence classes is an H (G, A ) -torsor.
Exercise 5. Let U, V be open subgroups of G, and let A be a U -module. If V ⊆ U , then
A is also a V -module, which we denote by ResU
V A. For σ ∈ G we denote by σA the
σU σ −1 -module, whose underlying abelian group is A and the action of τ ∈ σU σ −1 is given
by a 7→ σ −1 τ σa.
Show that for any two open subgroups U, V and any U -module A, we have an isomorphism
of V -modules
M
−1
IndU ∩σV σ σResV −1 A,
ResG IndV A ∼
=
U
V ∩σ
U
G
Uσ
σ ∈R
where R is a set of representatives of U \G/V and where the modules Ind A are defined below
on p.61. In particular, if U is a normal open subgroup of G, then
M
σA.
ResG IndU A ∼
=
U
G
σ ∈G/U
Exercise 6. Let A be a G-modulation and let
\
NA(U ) =
ind VU A(V ).
V ⊆U
Show that U 7→ NA(U ) is a submodulation of A, and that the modulation A/NA is Hausdorff.
Show that we get also a G-modulation Â, the “completion of A”, by setting
Â(U ) = lim A(U )/NU/V AU .
←−
V ⊆U
Exercise 7. For any two G-modulations A, B we have G-modulations A⊗B and Hom(A, B).
In particular, we have the notion of a “dual” A∗ = Hom(A, Q/ZZ) of a G-modulation A.
Exercise 8. Let X be a finite G-set. A complex vector bundle on X is a continuous
representation G → GL(V) on a finite dimensional C-vector space V such that the projection
X × V → X is a morphism of G-sets. The vector bundle is called a line bundle if dim V = 1.
Define an abelian group Pic(X) of isomorphism classes of line bundles on X. Show that
X 7→ Pic(X) is actually a G-modulation Pic, and show that Pic = Hom(π ab , Q/ZZ).
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Chapter I. Cohomology of Profinite Groups
§6. Basic Properties
We collect in this section some basic properties of cohomology groups,
which will be used repeatedly. If we write Ĥ n (G, A) in the following for
profinite groups G, then for n ≥ 1 this means, as for finite groups, Ĥ n (G, A) =
H n (G, A). From the formula (1.5.7), cor UG ◦ res G
U = (G : U ), follows the
(1.6.1) Proposition. Let G be a profinite group and let U be an open subgroup.
Assume that G is finite or that n ≥ 1. Then for every discrete G-module A
such that Ĥ n (U, A) = 0, we have
(G : U )Ĥ n (G, A) = 0.
In particular, if G is finite, then Ĥ n (G, A) is annihilated by the order of G. If,
moreover, A is finitely generated as a ZZ-module, then Ĥ n (G, A) is finite.
We conclude that for arbitrary profinite groups G the cohomology groups
H n (G, A), n ≥ 1, are torsion groups, since by (1.2.5)
H n (G, A) = lim
H n (G/U, AU ),
−→
U
where U runs through the open normal subgroups of G.
(1.6.2) Proposition. Let G be a finite group and let A be a G-module. Assume
that multiplication by p is an automorphism of A for all prime numbers p | #G.
Then
Ĥ i (G, A) = 0 for all i ∈ ZZ.
If G is profinite, this remains true for i ≥ 1. In particular, A is a cohomologically trivial G-module in the following cases:
a) A is a torsion group whose supernatural order is prime to #G,
b) A is an abelian profinite group whose supernatural order is prime to #G,
c) A is uniquely divisible.
m
Proof: Let us first assume that G is finite. Putting m = #G, the map A → A
is an automorphism of A by assumption. It therefore induces an isomorphism
∼ Ĥ i (G, A)
m : Ĥ i (G, A) −→
which, by (1.6.1), implies that Ĥ i (G, A) = 0.
Now let G be profinite. We have H i (G, A) = lim H i (G/U, AU ) for i ≥ 1.
−→ U
Let U ⊆ G be a normal open subgroup and m = #(G/U ). By assumption, the
m-multiplication map m : A → A is an isomorphism. Taking U -invariants, we
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61
§6. Basic Properties
see that the m-multiplication map m : AU → AU is also an isomorphism. By
the first part of the proof, we obtain H i (G/U, AU ) = 0 for all i ≥ 1, showing
that H i (G, A) = 0 for all i ≥ 1. Finally note that the assumption on the
p-multiplication map for all p | #G is satisfied in the cases a)–c), and that
everything remains true after replacing G by a closed subgroup. This finishes
the proof.
2
Now let H be a closed normal subgroup of G. If A is a discrete G-module,
then the cohomology group H n (H, A) is a discrete G-module, too: each σ ∈ G
acts on it by conjugation σ∗ .
(1.6.3) Proposition. The closed normal subgroup H acts trivially on the
cohomology group H n (H, A), i.e. H n (H, A) is a G/H-module. In particular,
the conjugation σ∗ : H n (G, A) → H n (G, A) is the identity for all σ ∈ G.
Proof: The assertion is trivial for H 0 (H, A) = AH . For n > 0 it follows by
dimension shifting from the commutative diagram (see (1.3.8))
Ĥ 0 (H,’‘”“ An )
δn
σ∗
Ĥ 0 (H, An )
H n (H, A)
σ∗
δn
H n (H, A).
2
Now let H be an arbitrary closed subgroup of G. For every H-module A,
we consider the G-module
M = IndH
G (A)
consisting of all continuous maps x : G → A such that x(τ σ) = τ x(σ) for all
τ ∈ H. The action of ρ ∈ G on M is given by x(σ) 7→ (ρx)(σ) = x(σρ). The
module M is said to be obtained by inducing A from H to G. ∗)
∗) The terminology induced module is commonly used, but strictly speaking it is slightly
inaccurate. From a categorical point of view the situation is as follows. Given a pair H ⊆ G
of abstract groups, the forgetful functor Res: G-Mod→H-Mod admits the left adjoint functor
Ind: H-Mod→G-Mod, A 7→ ZZ[G] ⊗ ZZ [H] A, and the right adjoint functor Coind: H-Mod→
G-Mod, A 7→MapH (G, A). If H has finite index in G, both functors are isomorphic. However,
with respect to the groups, the functor Ind is covariant while Coind is contravariant. This
phenomenon can be viewed as the reason for the existence of cor . In the case of profinite
groups, we have the functor Coind on discrete modules, but we write IndH
G A for Coind(A).
Furthermore, we have the functor Ind (“compact induction”) on compact modules .
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Chapter I. Cohomology of Profinite Groups
We have a canonical projection
π : IndH
G (A) −→ A,
x 7→ x(1).
This is a homomorphism of H-modules, which maps the H-submodule A0 =
{x : G → A | x(τ ) = 0 for all τ ∈/ H} isomorphically onto A. We identify
A0 with A. When H is of finite index in G and σ1 , . . . , σn is a system of
representatives of G/H, then
M=
n
M
σi A .
i=1
If A is a G-module, then IndH
G (A) is canonically isomorphic to the G-module
Map(G/H, A) of all continuous functions y : G/H → A, where the action of
ρ ∈ G on y is given by (ρy)(σH) = ρy(ρ−1 σH). The isomorphism
IndH (A) ∼
= Map(G/H, A)
G
is given by x(σ) 7→ y(σH) = σx(σ −1 ). In particular, when H = 1 we have a
canonical isomorphism
∼
IndH
G (A) = IndG (A)
with the G-module IndG (A) = Map(G, A) on which G acts by (ρx)(σ) =
ρx(ρ−1 σ) (see §3). We have seen in (1.3.7) that
H n (G, IndG (A)) = 0
for all n > 0.
We generalize this fact with the following proposition, commonly cited as
Shapiro’s lemma.
(1.6.4) Proposition. Let H be a closed subgroup of G and let A be an Hmodule. Then, for all n ≥ 0, we have a canonical isomorphism
n
∼
sh : H n (G, IndH
G (A)) −→ H (H, A).
Proof: The groups H n (G, IndH
G (A)) are the homology groups of the complex
G
X (G, IndH
(A))
.
The
canonical
homomorphism
G
.
π : IndH
G (A) −→ A
,
x 7→ x(1),
of H-modules induces an isomorphism
0
∼
H 0 (G, IndH
G (A)) −→ H (H, A) ,
and for each n ≥ 0, a homomorphism
G
n
H
X n (G, IndH
G (A)) −→ X (G, A) .
This is actually an isomorphism, since it has as inverse the map which associates to a function y(σ0 , . . . , σn ) in X n (G, A) the function x(σ0 , . . . , σn )(σ)
= y(σσ0 , . . . , σσn ). This is readily checked. We thus have
H n (G, IndH (A)) ∼
= H n (X (G, A)H ).
G
.
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§6. Basic Properties
The complex 0 → A → X 0 (G, A) → X 1 (G, A) → · · · is an acyclic resolution
of the H-module A, since, by the remark following (1.3.8), the X n (G, A) are
induced G-modules. Therefore, by (1.3.9), we obtain a canonical isomorphism
H n (X (G, A)H ) ∼
2
= H n (H, A).
.
Remark: If G is finite, the same argument yields isomorphisms
Ĥ n (G, IndH (A)) ∼
= Ĥ n (H, A)
G
for all n ∈ ZZ.
If A is a G-module, then we have an injective G-homomorphism
i : A −→ IndH
G (A),
(ia)(σ) = σa .
If, moreover, H is an open subgroup of G, then we have a G-homomorphism
ν : IndH
G (A) −→ A,
ν(x) =
X
σx(σ −1 ),
σ ∈G/H
where σ runs through a system of representatives of G/H. ∗) By this and by
the lemma of Shapiro, we get the following interpretation of the restriction and
the corestriction.
(1.6.5) Proposition. We have commutative diagrams
œ™—˜–•š› H (A))
H n (G, Ind
G
ν∗
H n (G, A)
sh
H n (H, A)
cor
H n (G, A),
H n (G, IndH
G (A))
i∗
H n (G, A)
sh
H n (H, A)
res
H n (G, A).
Proof: For the restriction this is obvious, so we prove it only for the corestriction. By dimension shifting, it suffices to consider the case n = 1. For each
class c ∈ H \ G, let c̄ ∈ c be a fixed representative. Let x ∈ Z 1 (G, IndH
G (A)).
∗) These homomorphisms are given by the Frobenius reciprocity. On the one hand we
have the isomorphism
G
∼
HomG (A, IndH
G (B)) = HomH (ResH (A), B) ,
where H is a closed subgroup of G, A is a G-module and B is an H-module. Let B = ResG
H (A);
G
then i : A → IndH
Res
(A)
is
the
unit
of
the
adjunction
Res
a
Ind.
If,
moreover,
H
is
an
open
G
H
subgroup of G, then we have the isomorphism
G
∼
HomG (IndH
G (B), A) = HomH (B, ResH (A))
L
(where IndH
G (B) is identified with
σ ∈G/H σB, see the footnote on p.61). Thus we obtain
H
G
ν : IndG ResH (A) → A as the counit of the adjunction Ind a Res.
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Chapter I. Cohomology of Profinite Groups
By definition, sh maps the class of x to the class of the cocycle y = sh(x)
Z 1 (H, A) given by
y(σ0 , σ1 ) = x(σ0 , σ1 )(1).
∈
Noting that the action of σ ∈ G on f ∈ IndH
G (A) is given by (σf )(τ ) = f (τ σ),
we obtain
X
cor sh(x)(σ0 , σ1 ) = c̄−1 x(c̄σ0 cσ 0−1 , c̄σ1 cσ 1−1 )(1)
c̄
=
X
c̄−1 x(σ0 cσ 0−1 , σ1 cσ 1−1 )(c̄).
c̄
On the other hand
ν∗ (x)(σ0 , σ1 ) =
X
c̄−1 x(σ0 , σ1 )(c̄).
c̄
Since x is a cocycle, we have
x(σ0 cσ 0−1 , σ1 cσ 1−1 ) = x(σ0 cσ 0−1 , σ0 ) + x(σ0 , σ1 cσ 1−1 )
= x(σ0 cσ 0−1 , σ0 ) + x(σ0 , σ1 ) + x(σ1 , σ1 cσ 1−1 )
= x(σ0 cσ 0−1 , σ0 ) − x(σ1 cσ 1−1 , σ1 ) + x(σ0 , σ1 ).
We therefore have to show that the function
f (σ0 , σ1 ) =
X
c̄−1 [x(cσ −1
0 , 1)(c̄σ0 )] −
X
c̄−1 [x(cσ −1
1 , 1)(c̄σ1 )]
c̄
c̄
is a coboundary. Noting that
X
x(cσ −1
i , 1) ∈
IndH
G (A)
and c̄σi cσ −1
i
∈
H, we have
c̄−1 [x(cσ −1
i , 1)(c̄σi )]
c̄
c̄−1 c̄σi cσ i−1 [x(cσ i−1 , 1)(cσ i σi−1 c̄−1 c̄σi )]
=
X
=
X
X
c̄
c̄
c̄
σi cσ i−1 [x(cσ i−1 , 1)(cσ i )] = σi
c̄−1 x(c̄−1 , 1)(c̄)
since cσi runs through H\G as c does. This shows that f (σ0 , σ1 ) = σ1 a − σ0 a
P
where a = − c̄−1 x(c̄−1 , 1)(c̄), so is a coboundary.
2
Proposition (1.6.5) (as well as (1.6.4)) will follow without computation from
a general uniqueness theorem for δ-functors, which we will prove in II §6 (see
(2.6.3)). Here, it says that ν∗ and cor ◦ sh are morphisms between the δn
functors H n (G, IndH
G (−)) and H (H, −) which coincide for n = 0 and hence
for all n. In the same way one can show the following fact:
If H is a normal closed subgroup of G and A is a G-module, then we have
−1
a further action of G on IndH
G (A) given by x 7→ σ∗ x, (σ∗ x)(ρ) = σx(σ ρ).
H
σ∗ is even an automorphism of the G-module IndG (A) (not only of the abelian
group), and is the identity for σ ∈ H. It therefore induces an automorphism on
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§6. Basic Properties
the cohomology group H n (G, IndH
G (A)), which in this way becomes a G/Hmodule. Considering the G/H-action on H n (H, A) given by conjugation (see
§5), the map
n
sh : H n (G, IndH
G (A)) −→ H (H, A)
becomes an isomorphism of G/H-modules.
Besides the maps inf, res, cor, we sometimes have to consider still another,
more subtle map, whose meaning is best understood in the framework of
spectral sequences (see II §1), but which can explicitly be defined as follows.
(1.6.6) Proposition. Let H be a normal subgroup of G and A a G-module.
Then there is a canonical homomorphism
tg : H 1 (H, A)G/H −→ H 2 (G/H, AH ),
called transgression, which is given as follows. If x : H → A is an inhomogeneous 1-cocycle in a class [x] ∈ H 1 (H, A)G/H , there exists a 1-cochain
y : G → A such that y|H = x and that (∂y)(σ1 , σ2 ) is contained in AH and
depends only on the cosets σ1 H, σ2 H, i.e. may be regarded as a cocycle of
G/H. For each such cochain y,
tg[x] = [∂y].
Proof: We construct a cochain y : G → A with the following properties
(i) y|H = x,
(ii) y(στ ) = y(σ) + σy(τ )
for σ
∈
G, τ
∈
H,
(iii) y(τ σ) = y(τ ) + τ y(σ)
for σ
∈
G, τ
∈
H.
Let s : G/H → G, γ 7→ sγ, be a continuous section of the projection G →
G/H such that s1 = 1. Such a section exists by ex.4 of §1. Since [x] is
invariant under every γ ∈ G/H, we have
(1)
sγx((sγ)−1 τ sγ) − x(τ ) = τ y(sγ) − y(sγ)
for an element y(sγ) ∈ A. We may assume that y(1) = 0 and that γ 7→ y(sγ) is
continuous. In fact, there exists an open normal subgroup U of G such that x(τ )
depends only on the cosets τ (H ∩ U ) and is contained in AU . Therefore the left
side takes the same value for all elements sγ in a coset mod U . So we may
choose for y(sγ) the same value within a coset of G mod U , and this means
that y(sγ) is continuous as a function of γ. For an arbitrary σ = sγτ ∈ G, we
now set
y(σ) = y(sγ) + sγx(τ ).
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Chapter I. Cohomology of Profinite Groups
Then y|H = x. Let τ, τ 0
∈
H, g = sγ and σ = gτ 0 . Then
y(στ ) = y(gτ 0 τ ) = y(g) + gx(τ 0 τ )
= y(g) + gx(τ 0 ) + gτ 0 x(τ ) = y(σ) + σx(τ ).
This proves (ii). Using (1), we obtain
y(τ g) = y(gτ g ) = y(g) + gx(τ g )
= y(g) + x(τ ) + τ y(g) − y(g) = x(τ ) + τ y(g),
and for arbitrary σ = gτ 0 ,
y(τ σ) = y(τ gτ 0 ) = y(τ g) + τ gx(τ 0 )
= x(τ ) + τ y(g) + τ gx(τ 0 )
= y(τ ) + τ y(σ).
This proves (iii).
The function
∂y(σ1 , σ2 ) = σ1 y(σ2 ) − y(σ1 σ2 ) + y(σ1 )
depends only on the cosets σ1 H, σ2 H, i.e. ∂y(σ1 , σ2 τ ) and ∂y(σ1 τ, σ2 ) are
independent of τ ∈ H. In fact,
∂y(σ1 , σ2 τ ) = σ1 y(σ2 τ ) − y(σ1 σ2 τ ) + y(σ1 )
= σ1 σ2 y(τ ) + σ1 y(σ2 ) − y(σ1 σ2 τ ) + y(σ1 )
= σ1 y(σ2 ) − y(σ1 σ2 ) + y(σ1 ) = ∂y(σ1 , σ2 ),
∂y(σ1 τ, σ2 ) = σ1 τ y(σ2 ) − y(σ1 τ σ2 ) + y(σ1 τ )
= σ1 y(τ σ2 ) − σ1 y(τ ) − y(σ1 τ σ2 ) + y(σ1 τ )
= σ1 y(τ σ2 ) − y(σ1 τ σ2 ) + y(σ1 )
= ∂y(σ1 , τ σ2 ) = ∂y(σ1 , σ2 τ σ2 ) = ∂y(σ1 , σ2 ).
From ∂∂y = 0 and ∂y(τ, σ) = ∂y(1, σ) = 0, we now obtain
τ ∂y(σ1 , σ2 ) = ∂y(τ σ1 , σ2 ) − ∂y(τ, σ1 σ2 ) + ∂y(τ, σ1 ) = ∂y(σ1 , σ2 ),
i.e. ∂y(σ1 , σ2 ) ∈ AH .
Finally, let y 0 : G → A be any cochain such that y 0 |H = x and that ∂y 0 (σ1 , σ2 )
takes values in AH and depends only on the cosets σ1 H, σ2 H. Then, for the
function z = y − y 0 , we have z(τ ) = 0 for τ ∈ H and ∂z(τ, σ) = ∂z(1, τ σ) = 0,
i.e. τ z(σ) − z(σ) = 0, so that z(σ) ∈ AH , and ∂z(1, τ σ) = ∂z(1, σ), so
that z(τ σ) = z(σ). Therefore ∂y and ∂y 0 may be viewed as cocycles in
Z 2 (G/H, AH ) and differ by the coboundary ∂z ∈ B 2 (G/H, AH ), i.e. define
the same cohomology class in H 2 (G/H, AH ). Thus defining tg[x] = [∂y]
gives a well-defined homomorphism.
2
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§6. Basic Properties
(1.6.7) Proposition (Five Term Exact Sequence). Let H be a closed normal
subgroup of G and let A be a G-module. We then have an exact sequence
inf
res
0 −→ H 1 (G/H, AH ) −→ H 1 (G, A) −→ H 1 (H, A)G/H
tg
inf
−→ H 2 (G/H, AH ) −→ H 2 (G, A).
Moreover, if H i (H, A) = 0 for i = 1, . . . , n − 1, we have an exact sequence
inf
res
0 −→ H n (G/H, AH ) −→ H n (G, A) −→ H n (H, A)G/H
tg
inf
−→ H n+1 (G/H, AH ) −→ H n+1 (G, A).
Proof: Consider the first sequence. The image of res is contained in
res
H 1 (H, A)G/H , since the map H 1 (G, A) −→ H 1 (H, A) is a homomorphism
of G-modules and H 1 (G, A)G = H 1 (G, A) by (1.6.3).
Exactness at H 1 (G/H, AH ). For the injectivity of the first map inf , let x :
G/H → AH be an inhomogeneous 1-cocycle such that the composite inf x :
G → G/H → A is a coboundary, (inf x)(σ) = σa − a. For all τ ∈ H, we
have σa − a = στ a − a, hence a ∈ AH . Therefore x(σH) = σHa − a is a
1-coboundary.
Exactness at H 1 (G, A). Let x : G/H → AH be an inhomogeneous 1-cocycle.
Then for τ ∈ H,
(res ◦ inf x)(τ ) = (inf x)(τ ) = x(τ H) = x(H) = x(1) = 0,
i.e. im(inf ) ⊆ ker(res ). Conversely, let x : G → A be an inhomogeneous
1-cocycle such that res x is a coboundary, i.e. x(τ ) = τ a − a for τ ∈ H. The
1-cocycle x0 (σ) = x(σ) − (σa − a) of G defines the same cohomology class as
x and satisfies x0 (τ ) = 0 for τ ∈ H. Hence
x0 (στ ) = x0 (σ) + σx0 (τ ) = x0 (σ),
and also
x0 (τ σ) = x0 (τ ) + τ x0 (σ) = τ x0 (σ).
We now define y : G/H → A by y(σH) = x0 (σ). Then y(σH) ∈ AH , because
y(σH) = y(τ σH) = τ y(σH) for all τ ∈ H, and we obtain a 1-cocycle with
inf y = x0 . This shows ker(res ) ⊆ im(inf ).
Exactness at H 1 (H, A)G/H . If y ∈ Z 1 (G, A) and x = res y represents a class
[x] in H 1 (H, A)G/H , then tg[x] = [∂y] = 0, i.e. im(res ) ⊆ ker(tg). Conversely,
let x ∈ Z 1 (H, A) represent a class [x] ∈ H 1 (H, A)G/H such that tg[x] = 0. Let
y ∈ C 1 (G, A) be a cochain as in (1.6.6). Viewing ∂y as a 2-cocycle of G/H,
then [∂y] = tg[x] = 0, hence ∂y = ∂z, where z ∈ C 1 (G/H, AH ). Viewing y
and z as functions on G, we have y − z ∈ Z 1 (G, A). Since res (y − z) and
res y = x are 1-cocycles of H, so is res z, and since z is constant on H, we
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Chapter I. Cohomology of Profinite Groups
have res z = 0. Thus res (y − z) = res y = x, i.e. [x] = res [y − z]. This proves
ker(tg) ⊆ im(res ).
Exactness at H 2 (G/H, AH ). Let x ∈ Z 1 (H, A) be an inhomogeneous cocycle,
representing a class [x] ∈ H 1 (H, A)G/H . Then, by (1.6.6), there is a cocycle z ∈
Z 2 (G/H, AH ) such that inf z = ∂y and tg[x] = [z]. Thus inf z ∈ B 2 (G, A),
hence inf tg[x] = [inf z] = 0, showing im(tg) ⊆ ker(inf ). Conversely, let
z ∈ Z 2 (G/H, AH ) be a normalized cocycle, i.e. z(1, σ) = z(σ, 1) = 0 (see
p.19), such that inf [z] = [inf z] = 0. Then inf z = ∂y with y ∈ C 1 (G, A).
Setting x = res y we have ∂x = res ∂y = res inf z = 0 and, regarding ∂y as the
2-cocycle z of G/H, tg[x] = [∂y] = [z]. This proves ker(inf ) ⊆ im(tg).
The exact sequence
0 −→ H n (G/H, AH ) −→ H n (G, A) −→ H n (H, A)G/H
tg
−→ H n+1 (G/H, AH ) −→ H n+1 (G, A)
for G-modules A such that H i (H, A) = 0 for i = 1, . . . , n − 1, is obtained by
induction. We have it for n = 1, and we assume it for n ≥ 1. Let A be a
G-module, such that H i (H, A) = 0 for i = 1, . . . , n. Consider the sequence
0 −→ AH −→ ĀH −→ AH
1 −→ 0,
where Ā = IndG (A), which is exact since H 1 (H, A) = 0. We therefore have a
commutative diagram
0¦¥¢¡ «ª©¨§£¤Ÿž
H n (G/H, AH
1 )
inf
tg
H n+1 (G/H, AH )
inf
H n+1 (G, A)
H n+1 (G/H, AH
1 )
inf
H n+1 (G, A1 )
H n+2 (G/H, AH )
H n (H, A1 )G/H
δ
res
H n+1 (H, A)G/H
δ
δ
tg
res
δ
δ
0
H n (G, A1 )
inf
H n+2 (G, A),
where the δ’s are isomorphisms, since Ā is G-induced and H-induced, and ĀH
is G/H-induced. The lower map tg is defined by the upper one. Furthermore,
H i (H, A1 ) ∼
= H i+1 (H, A) = 0 for i = 1, . . . , n − 1.
Therefore, by assumption, the upper sequence is exact, hence also the lower
one.
2
Let p be a prime number. A profinite group G is called a pro-p-group if its
supernatural order is a p-power, i.e. for every open normal subgroup U of G,
the finite group G/U is a p-group. Equivalently, a pro-p-group is the projective
limit of finite p-groups.
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§6. Basic Properties
(1.6.8) Definition. A p-Sylow subgroup of a profinite group G is a closed
subgroup Gp which is a pro-p-group such that the index (G : Gp ) is prime to p.
The Sylow theorems for finite groups hold as well for profinite groups.
(1.6.9) Theorem. Let G be a profinite group and p a prime number.
(i)
There exists a p-Sylow subgroup Gp .
(ii) Every pro-p-subgroup is contained in a p-Sylow subgroup.
(iii) The p-Sylow subgroups of G are conjugate.
Proof: Let U run through the open normal subgroups of G and let Σp (U )
denote the finite, nonempty set of all p-Sylow subgroups of G/U . If V ⊆ U are
two open normal subgroups, then the projection G/V → G/U maps p-Sylow
subgroups onto p-Sylow subgroups and induces a surjection Σp (V ) → Σp (U ),
so that the Σp (U ) form a projective system of nonempty finite sets. The
projective limit lim Σp (U ) is a compact, nonempty topological space (see
←−
[160], chap.IV).
(i) Now let (SU ) ∈ lim Σp (U ). For V ⊆ U we have the surjective projection
←−
SV → SU , i.e. the system (SU ) is a projective system of finite p-groups. The
projective limit Gp = lim SU is then a p-Sylow subgroup of G by definition.
←−
(ii) Let H be a pro-p-subgroup of G and let HU be its image under G → G/U .
Let Σ0p (U ) be the set of p-Sylow subgroups of G/U , containing HU . Again,
lim
Σ0p (U ) is nonempty. Let (SU ) ∈ lim
Σ0p (U ). Then the inclusions HU ,→
←−
←−
SU form a morphism of projective systems, and we obtain
H = lim HU
←−
⊆
lim SU = Gp .
←−
(iii) Let Gp and G0p be two p-Sylow subgroups of G and let SU and SU0 be
their images in G/U . Let C(U ) be the set of elements σU ∈ G/U such that
σU SU σU−1 = SU0 . The C(U ) form again a projective system of finite, nonempty
sets. The projective limit lim
C(U ) is nonempty. If σ = (σU ) ∈ lim
C(U ) ⊆ G,
←−
←−
then clearly σGp σ −1 = G0p .
2
The p-primary part A(p) of an abelian torsion group A is the subgroup
L
consisting of all elements of A of p-power order. We have A = p A(p). For
the p-primary part of the torsion group Ĥ n (G, A), we have the
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Chapter I. Cohomology of Profinite Groups
(1.6.10) Proposition. Let A be a discrete G-module and let Gp a p-Sylow
subgroup of G. Assume that G is finite or that n ≥ 1. Then the homomorphism
res : Ĥ n (G, A)(p) −→ Ĥ n (Gp , A)
is injective, and if Gp is open in G, the homomorphism
cor : Ĥ n (Gp , A) −→ Ĥ n (G, A)(p)
is surjective.
Proof: The proposition holds for any closed subgroup H of G of index
(G : H) prime to p. Indeed, if H is open, then by (1.5.7), cor ◦ res = (G : H).
Since (G : H) is prime to p, the map
Ĥ n (G,¬ A)(p) cor ◦ res Ĥ n (G, A)(p)
is an automorphism, so that res must be injective and cor surjective. If H is
not open, the injectivity of res follows from (1.5.1).
2
(1.6.11) Corollary. Assume that G is finite or that n
for all prime numbers p, then
Ĥ n (G, A) = 0.
≥
1. If Ĥ n (Gp , A) = 0
Because of the above proposition, we are often reduced to the cohomology
of pro-p-groups. Its most frequently used property is the assertion (1.6.13)
which we first prove for finite p-groups.
(1.6.12) Proposition. Let G be a finite p-group and let A be a p-primary
G-module. If H 0 (G, A) = 0 or H0 (G, A) = 0, then A = 0.
Proof: For the proof of H 0 (G, A) = 0 ⇒ A = 0, we may assume that A
is finite, since every element of the p-primary G-module A generates a finite
G-module. A r AG is a disjoint union of G-orbits Ga not consisting of only
one point, hence #Ga ≡ 0 mod p and #A ≡ #AG mod p. If AG = 0, then
#A ≡ 1 mod p, and hence A = 0. If H0 (G, A) = AG = 0, then, setting
A∗ = Hom(A, Qp /ZZp ), we get (A∗ )G ∼
= (AG )∗ = 0, whence A∗ = 0 and A = 0.
2
Recall that a G-module A =/ 0 is said to be simple if it does not contain any
submodule other than 0 and A itself. Since the set {ga | g ∈ G} ⊆ A is finite
for every a ∈ A, a simple module is finitely generated as an abelian group.
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§6. Basic Properties
p
Therefore there exists a prime number p such that the p-multiplication A −→ A
is not surjective, whence pA = 0. We conclude that a simple module is finite
and there exists a unique prime number p such that pA = 0.
(1.6.13) Corollary. Let G be a pro-p-group. Then every discrete simple
p-primary G-module A is isomorphic to ZZ/pZZ (with trivial G-action). In
particular, if A is a p-primary G-module, then
A = 0 if and only if AG = 0 .
Proof: Let A be a discrete simple p-primary G-module. Then pA = 0.
Further, AU =/ 0 for some open normal subgroup U ⊆ G, and by (1.6.12) we
obtain 0 =/ H 0 (G/U, AU ) = AG ⊆ A. Hence A = AG by simplicity. We
conclude that A is an IFp -vector space with trivial G-action and therefore of
dimension 1.
2
(1.6.14) Proposition.
(i) The maximal closed subgroups of a pro-p-group G are normal of index p.
(ii) A homomorphism G → G0 of pro-p-groups is surjective if and only if the
induced homomorphism H 1 (G0 , ZZ/pZZ) → H 1 (G, ZZ/pZZ) is injective. In
particular,
G = 1 ⇔ H 1 (G, ZZ/pZZ) = 0.
Proof: (i) Let H be a maximal closed subgroup of G. Then there exists
an open normal subgroup U of G such that H U/U =/ G/U , since otherwise
H = G. Clearly, H U/U is a maximal subgroup of the finite group G/U and
is therefore normal of index p. This is a well-known result in group theory
which follows from the first Sylow theorem for finite groups (see [73], chap.4,
(4.2.2)). Since H is maximal, it is the pre-image of H U/U under G → G/U
and is thus also normal of index p.
(ii) If G → G0 is surjective, then obviously H 1 (G0 , ZZ/pZZ) −→ H 1 (G, ZZ/pZZ)
is injective. Conversely, assume the latter. If G → G0 were not surjective, the
image would be contained in a maximal subgroup H of G0 which is normal of
index p. The composition G0 → G0 /H ∼
= ZZ/pZZ would be an element χ =/ 0 in
H 1 (G0 , ZZ/pZZ) which becomes zero in H 1 (G, ZZ/pZZ), a contradiction.
2
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Chapter I. Cohomology of Profinite Groups
(1.6.15) Proposition. Let
φ : G0 −→ G
be a homomorphism of pro-p-groups. Let A be ZZ/pr ZZ, r ∈ IN, or Qp /ZZp
with trivial action. Then φ is an isomorphism if and only if the induced
homomorphism
H i (φ, A) : H i (G, A) −→ H i (G0 , A)
is an isomorphism for i = 1 and injective for i = 2.
Proof: Let us show the nontrivial direction. ZZ/pZZ is a subgroup of A in a
canonical way, and we obtain, for any group H, an inclusion
H 1 (H, ZZ/pZZ) ⊂ H 1 (H, A).
In particular, H 1 (φ, ZZ/pZZ) is injective if H 1 (φ, A) is, and therefore (1.6.14)
implies that φ is surjective. Put K = ker(φ) and consider the 5-term exact
sequence
H 1 (φ)
H 2 (φ)
0 → H 1 (G, A) → H 1 (G0 , A) → H 1 (K, A)G → H 2 (G, A) → H 2 (G0 , A).
Our assumptions imply H 1 (K, A)G = 0, thus H 1 (K, A) = 0 by (1.6.13). We
conclude that H 1 (K, ZZ/pZZ) = 0, and K = 0 by (1.6.14)(ii).
2
(1.6.16) Proposition. Let G and G0 be pro-p-groups and assume that Gab
and (G0 )ab are torsion groups. Let pr be a common exponent of the factor
commutator groups and assume that H 2 (G, Qp /ZZp ) = 0 = H 2 (G0 , Qp /ZZp ).
Let φ : G0 → G be a homomorphism. Then the following assertions are
equivalent.
(i) φ is an isomorphism.
(ii) H 1 (φ) : H 1 (G, ZZ/pr ZZ) → H 1 (G0 , ZZ/pr ZZ) is an isomorphism.
(iii) H 2 (φ) : H 2 (G, ZZ/pr ZZ) → H 2 (G0 , ZZ/pr ZZ) is an isomorphism.
Proof: Clearly, (i) implies (ii) and (iii) and, by (1.6.15), (ii) and (iii) together imply (i). It remains to show that (ii) and (iii) are equivalent. Since
H 2 (G, Qp /ZZp ) = 0, the exact sequence
pr
0 −→ ZZ/pr ZZ −→ Qp /ZZp −→ Qp /ZZp → 0
induces the four term exact sequence
α
β
0 → H 1 (G, ZZ/pr ZZ) → H 1 (G, Qp /ZZp ) →
γ
H 1 (G, Qp /ZZp ) → H 2 (G, ZZ/pr ZZ) → 0.
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§6. Basic Properties
By the assumption on Gab , the map α is an isomorphism. Hence β is zero and
γ is an isomorphism. The same argument also applies to G0 and therefore (ii)
and (iii) are both equivalent to
(iv) H 1 (φ, Qp /ZZp ) : H 1 (G, Qp /ZZp ) → H 1 (G0 , Qp /ZZp ) is an isomorphism.
This concludes the proof.
2
Exercise 1. Let G be a profinite group, H a closed normal subgroup, A a discrete G-module,
and let s : G/H → G be a continuous section of the projection G → G/H. Let x : H → A
be a cocycle representing a class [x] ∈ H 1 (H, A)G/H , and set y(σ) = sγx(τ ) for σ = sγτ ,
γ ∈ G/H, τ ∈ H.
Show that if H acts trivially on A, then y : G → A is a cochain as in (1.6.6), i.e. y|H = x and
∂y(σ1 , σ2 ) depends only on the cosets σ1 H, σ2 H, so that tg[x] = [∂y].
Exercise 2. Let G be finite, H a subgroup, A a G-module, and let H 0 be the commutator
subgroup of H. The group extension
1 −→ H ab −→ G/H 0 −→ G/H −→ 1
defines a class u ∈ H 2 (G/H, H ab ).
Assume that H acts trivially on A. Then H 1 (H, A)G/H = H 0 (G/H, Hom (H ab , A)), and
the cup-product
∪
H 2 (G/H, H ab ) × H 0 (G/H, Hom (H ab , A)) −→ H 2 (G/H, A)
yields a homomorphism
u∪ : H 1 (H, A)G/H −→ H 2 (G/H, A).
Show that this homomorphism coincides with −tg. Generalize this result to profinite groups.
Hint: Let x be a cocycle representing a class [x] ∈ H 1 (H, A)G/H . Then xσ = x for all σ ∈ G.
Let y be the function y(σ) = sγx(τ ) (σ = sγτ ) as in ex.1. The class u ∈ H 2 (G/H, H ab )
is represented by the cocycle τ (γ1 , γ2 ) = s̄γ1 s̄γ2 s̄(γ1 γ2 )−1 , where s̄ is the composite of
s
G/H → G → G/H 0 . Show that
∂y(γ1 , γ2 ) = −x(τ (γ1 , γ2 )).
Exercise 3. Let G be a finite group, H a subgroup, and let A be a G-module. Consider the
exact sequence
0 −→ A −→ Ā −→ A1 −→ 0
with Ā = IndG (A), the induced G-module. The associated long exact cohomology sequence
yields the exact sequence
δ
1
0 −→ AH −→ ĀH −→ AH
1 −→ H (H, A) −→ 0.
We split this up into the two exact sequences
(1)
0 −→ AH −→ ĀH −→ B −→ 0,
(2)
1
0 −→ B −→ AH
1 −→ H (H, A) −→ 0,
H
where B denotes the image of ĀH in AH
is an induced G/H-module, we get
1 . Since Ā
from (1)
δ2
2
H
H 1 (G/H, B) −→
∼ H (G/H, A ).
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Chapter I. Cohomology of Profinite Groups
Show that the composite of
δ
δ
1
2
2
H
H 1 (G/H, B) −→
H 1 (H, A)G/H −→
∼ H (G/H, A ),
where δ1 is obtained from (2), is the transgression.
Exercise 4. Define the cohomology groups of a pair H
⊆
G and a G-module A by
H 0 (G, H, A) = 0 and H n (G, H, A) = H n−1 (G, Γ(A)) for n ≥ 1,
where Γ(A) is defined by the exact sequence
0 −→ A −→ IndH
G (A) −→ Γ(A) −→ 0.
We then have a “relative exact cohomology sequence” (see [178])
δ
. . . → H n (G, A) → H n (H, A) → H n+1 (G, H, A) → H n+1 (G, A) → H n+1 (H, A) → . . . .
Exercise 5. For two pairs H ⊆ G and L
ϕL ⊆ H, we have functorial maps
⊆
K and a homomorphism ϕ : K → G such that
ϕn : H n (G, H, A) −→ H n (K, L, A).
§7. Cohomology of Cyclic Groups
In this section we will compute the cohomology of cyclic groups and we
will introduce the concept of the Herbrand quotient. Let G be a finite cyclic
group. Recall that for every G-module A we have
Ĥ 0 (G, A) = AG /NG A
and
Ĥ −1 (G, A) = NG A/IG A.
If σ is a generator of the cyclic group G, then IG A = (σ − 1)A, since for all
i ≥ 1 we have the equality
σ i − 1 = (σ − 1)(σ i−1 + · · · + σ + 1).
(1.7.1) Proposition. Let G be a finite cyclic group. Then the group Ĥ 2 (G, ZZ)
is cyclic of the same order as G. Let χ ∈ Ĥ 2 (G, ZZ) be any generator. Then the
cup-product induces isomorphisms
∼ Ĥ n+2 (G, A)
χ∪ : Ĥ n (G, A) −→
for all n ∈ ZZ and every G-module A. In particular, we have isomorphisms
Ĥ 2n (G, A) ∼
= AG /NG A,
Ĥ 2n−1 (G, A) ∼
= NG A/IG A
for all G-modules A and all n ∈ ZZ.
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§7. Cohomology of Cyclic Groups
Proof: Let σ ∈ G be a generator and let N = #G. Consider the four-term
exact sequence
µ
±°¯­®
(∗)
0
ZZ
ZZ[G] σ−1 ZZ[G] ε ZZ
0
where ε is the augmentation map ai σ i 7→ ai and µ(a) = a(1 + σ + · · · +
σ N −1 ). It induces an isomorphism (see the remark following (1.3.8))
P
P
∼ Ĥ 2 (G, Z
δ 2 : Ĥ 0 (G, ZZ) −→
Z).
Since
Ĥ 0 (G, ZZ) = ZZ/(1 + σ + · · · + σ N −1 )ZZ = ZZ/N ZZ,
we see that Ĥ 2 (G, ZZ) is cyclic of order N and every generator is of the form
χ = δ 2 (m), m ∈ (ZZ/N ZZ)× .
Now let A be a G-module. Since all objects in (∗) are ZZ-free, it remains
exact when tensored by A. Hence for every n ∈ ZZ, we obtain an isomorphism
∼ Ĥ n+2 (G, A), which fits into the commutative diagram
δ 2 : Ĥ n (G, A) →
µ´²³ A)
Ĥ n (G,
Ĥ n (G, A)
χ∪
m
Ĥ n (G, A)
δ2
Ĥ n+2 (G, A).
It remains to show that multiplication by m induces an automorphism on
Ĥ n (G, A). But this is clear, because by (1.6.1), Ĥ n (G, A) is an abelian group
which is annihilated by N = #G, hence a ZZ/N ZZ-module.
2
(1.7.2) Proposition. Let G be a finite cyclic group. If 0 → A → B → C → 0
is an exact sequence of G-modules, then we have an exact hexagon
¶·¸¹º»
Ĥ 0 (G, A)
Ĥ 0 (G, B)
Ĥ −1 (G, C)
Ĥ 0 (G, C)
Ĥ −1 (G, B)
Ĥ −1 (G, A).
Proof: All maps in the hexagon above are the canonical ones, except the homomorphism Ĥ 0 (G, C) → Ĥ −1 (G, A) which is the composite of the connecting homomorphism Ĥ 0 (G, C) → Ĥ 1 (G, A) and the inverse of the isomorphism
∼ Ĥ 1 (G, A) obtained in (1.7.1). In order to prove exactness, we
Ĥ −1 (G, A) →
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Chapter I. Cohomology of Profinite Groups
only have to show that the diagram
¼½¾¿ A)
Ĥ −1 (G,
Ĥ −1 (G, B)
Ĥ 1 (G, A)
Ĥ 1 (G, B)
commutes, where the vertical maps are the isomorphisms obtained in (1.7.1).
But this is obvious.
2
A very useful concept for calculating indices and orders of abelian groups
is the Herbrand quotient, given by
(1.7.3) Definition. Let A be an abelian group and let f, g be endomorphisms
of A such that f g = gf = 0, i.e. im g ⊆ ker f and im f ⊆ ker g. Then the
Herbrand quotient with respect to f and g is defined to be
qf,g (A) =
(ker f : im g)
(ker g : im f )
provided that both indices are finite.
The following special case is of great importance for the cohomology of
cyclic groups.
(1.7.4) Definition. If G is a finite cyclic group and A a G-module, then the
Herbrand quotient of A is defined to be
h(G, A) =
#Ĥ 0 (G, A)
,
#Ĥ −1 (G, A)
provided that both orders are finite.
If σ is a generator of the cyclic group G of order n, then the endomorphisms
D =σ−1
and
NG = 1 + σ + . . . + σ n−1
have the property that D · NG = NG · D = 0. As Ĥ 0 (G, A) = AG /NG A and
Ĥ −1 (G, A) = NG A/IG A, we see that h(G, A) = qD,N (A).
The salient property of the Herbrand quotient is its multiplicativity.
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§7. Cohomology of Cyclic Groups
(1.7.5) Proposition. Let G be a finite cyclic group. If 0 → A → B → C → 0
is an exact sequence of G-modules, then
h(G, B) = h(G, A) · h(G, C)
in the sense that, whenever two of these quotients are defined, so is the third
and the identity holds.
Proof: Let f : Ĥ 0 (G, A) → Ĥ 0 (G, B). Using (1.7.2), we obtain an exact
sequence
0 → ker f → Ĥ 0 (G, A) → Ĥ 0 (G, B) → . . . → Ĥ −1 (G, C) → ker f → 0,
and therefore
#Ĥ 0 (G, A)·#Ĥ 0 (G, C)·#Ĥ −1 (G, B) = #Ĥ 0 (G, B)·#Ĥ −1 (G, A)·#Ĥ −1 (G, C).
At the same time, we see that if any two of the quotients are well-defined, then
so is the third. From the last equation we obtain the desired equality.
2
Another special case for the Herbrand quotient is the following: let f = 0
and let g be the multiplication by a natural number n. Then
(A : nA)
.
q0,n (A) =
# nA
Considering A as a trivial G-module, where G is a cyclic group of order n, we have q0,n (A) = h(G, A). In particular, for an exact sequence
0 → A → B → C → 0 of abelian groups, we obtain
q0,n (B) = q0,n (A) · q0,n (C),
again in the sense that whenever two of these quotients are defined, so is the
third and the identity holds.
(1.7.6) Proposition. Let A be a finite abelian group and let f, g be endomorphisms of A such that f g = gf = 0. Then
qf,g (A) = 1.
In particular, if G is a finite cyclic group and A is a finite G-module, then
h(G, A) = 1.
Proof: Since #A = # ker f · #im f = # ker g · #im g , the result follows.
2
From the last result and from (1.7.5) we obtain h(G, A) = h(G, B) for any
G-submodule B of A of finite index.
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Chapter I. Cohomology of Profinite Groups
A procyclic group is a profinite group G which is topologically generated
by a single element σ, i.e. G is the closure of the subgroup (σ) = {σ n | n ∈ ZZ}.
For example,
ZZp = lim ZZ/pn ZZ
ẐZ = lim ZZ/nZZ
and
←−
n∈IN
←−
n∈IN
are (additive) procyclic groups, and every procyclic group G is a quotient of
ẐZ (see [160], chap.IV, § 2, example 7). As
ẐZ =
Y
ZZp ,
p
where p runs through the prime numbers, every procyclic group G is of the
form
Y
Y
G∼
ZZ/pep ZZ ×
ZZp ,
=
p∈T
p∈S
where T and S are disjoint sets of prime numbers and ep , p ∈ T , are natural
numbers.
Q
We assume now that G is torsion-free, i.e. G = p∈S ZZp . Let IN(S) be the
set of natural numbers not divisible by prime numbers p ∈/ S. Then the group
Gn of n-th powers, n ∈ IN(S), are the open subgroups of G and thus
G = lim G/Gn ∼
= lim ZZ/nZZ.
←−
n∈IN(S)
←−
n∈IN(S)
We say that an abelian group
X is S-divisible if X = nX for all n ∈ IN(S),
[
and is S-torsion if X =
n X, where n X = {x ∈ X | nx = 0}.
n∈IN(S)
(1.7.7) Proposition. Let G =
let A be a discrete G-module.
Q
p∈S
ZZp be a torsion-free procyclic group and
(i) If A is S-torsion, then H 1 (G, A) ∼
= AG .
(ii) If A is torsion or S-divisible, then H n (G, A) = 0 for n ≥ 2.
σ−1
Proof: (i) Let σ be a topological generator of G. Then AG = ker(A −→ A)
σ−1
and AG = coker(A −→ A). Let Nn = 1 + σ + · · · + σ n−1 . Then
n
H 1 (G, A) = lim H 1 (G/Gn , AG )
−→
n∈IN(S)
∼
= lim
−→
NnA
Gn
n
/(σ − 1)AG = A0 /(σ − 1)A,
n∈IN(S)
where A0 = {a ∈ A | Nn a = 0 for some n ∈ IN(S)}. The isomorphism
H 1 (G, A) ∼
= A0 /(σ − 1)A
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§7. Cohomology of Cyclic Groups
is given by associating to a 1-cocycle x : G → A the value x(σ). If A is
S-torsion, then A0 = A. In fact, for a ∈ A there exist n, m ∈ IN(S), such that
na = 0 and σ m a = a. From this, it follows that
(1 + σ + · · · + σ mn−1 )a = n(1 + σ + · · · + σ m−1 )a = 0,
i.e. a ∈ A0 . This proves (i).
(ii) By (1.7.1), we have
H 2 (G/Gn , AG ) ∼
= AG /Nn AG .
n
n
A careful analysis of the definition of this isomorphism shows that the inn
nm
flation map H 2 (G/Gn , AG ) → H 2 (G/Gnm , AG ) corresponds to the homomorphism
n
nm
AG /Nn AG −→ AG /Nnm AG ,
given as multiplication by m. If A is finite and m is a multiple of the order of
A, then this homomorphism is zero, hence
n
H 2 (G, A) = lim H 2 (G/Gn , AG ) = 0.
−→
If A is torsion, then A = lim Aα , where Aα runs through the finite G-submo−→
dules of A, hence H 2 (G, A) = lim
H 2 (G, Aα ) = 0.
−→
Assume now inductively that H n (G, A) = 0, n ≥ 2, for all torsion modules
A. For any torsion module A, consider the exact sequence
0 −→ A −→ IndG (A) −→ A1 −→ 0,
where the left arrow associates to a ∈ A the constant function x(τ ) = a.
Clearly, as A is torsion, so are IndG (A) and A1 (note that every continuous map
x : G → A has finite image). By (1.3.7), IndG (A) is cohomologically trivial,
so that
H n+1 (G, A) ∼
= H n (G, A1 ) = 0.
Now let A be S-divisible and let m ∈ IN(S). From the exact sequence
m
0 −→ m A −→ A −→ A −→ 0 ,
we get the exact cohomology sequence
m
H n (G, m A) −→ H n (G, A) −→ H n (G, A).
Since m A is torsion, H n (G, m A) = 0 for n ≥ 2, i.e. multiplication by m is
injective on H n (G, A) for n ≥ 2. But these groups are S-torsion by (1.6.1),
which finishes the proof.
2
Exercise 1. Let f and g be two commuting endomorphisms of an abelian group A. Show that
q0,gf (A) = q0,g (A) · q0,f (A),
provided all quotients are defined.
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Chapter I. Cohomology of Profinite Groups
Exercise 2. Let G be a cyclic group of prime order p, and let A be a G-module such that
q0,p (A) is defined. Show that
h(G, A)p−1 = q0,p (AG )p /q0,p (A).
∼ ZZ[ζ], ζ a primitive
Hint: Let σ be a generator of G. Show that in the ring ZZ[G]/ZZNG =
p-th root of unity, one has p = (σ − 1)p−1 ε, where ε is a unit in ZZ[G]/ZZNG . Then use the
exact sequence 0 → AG → A → Aσ−1 → 0.
§8. Cohomological Triviality
Let G be a profinite group. For every prime number p, let Gp be a p-Sylow
subgroup of G. We have called a discrete G-module A cohomologically trivial
if H n (H, A) = 0 for all n > 0 and all closed subgroups H of G. We have seen
(cf. (1.3.7)) that induced G-modules are cohomologically trivial. We will give
now further criteria for cohomological triviality.
(1.8.1) Proposition. A discrete G-module A is cohomologically trivial if and
only if for every prime number p it is a cohomologically trivial Gp -module.
Proof: This follows easily from (1.6.11).
2
(1.8.2) Proposition. A discrete G-module A is cohomologically trivial if
and only if for every open normal subgroup U of G, the G/U -module AU is
cohomologically trivial.
Proof: If the AU are cohomologically trivial G/U -modules, then by (1.5.1),
H n (H, A) = lim H n (HU/U, AU ) = 0
−→
U
for n > 0 and every closed subgroup H, i.e. A is a cohomologically trivial
G-module. For a closed subgroup H/U of G/U , the sequence
0 −→ H n (H/U, AU ) −→ H n (H, A) −→ H n (U, A)
is exact if H i (U, A) = 0 for i = 1, . . . , n − 1 (see (1.6.7)). If A is cohomologically trivial, this is true for all n > 0 and we get H n (H/U, AU ) = 0 for n > 0,
showing that AU is a cohomologically trivial G/U -module.
2
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§8. Cohomological Triviality
(1.8.3) Proposition. Let G be a finite p-group and let A be a p-primary Gmodule. If pA = 0 and if Ĥ q (G, A) = 0 for one q, then A is an induced
G-module and, in particular, cohomologically trivial.
Proof: Let Λ = IFp [G], I a basis of the IFp -vector space AG and V =
In the exact sequence of G-modules
L
I
Λ.
0 −→ Hom (A/AG , V ) −→ Hom (A, V ) −→ Hom(AG , V ) −→ 0 ,
B = Hom(A/AG , V ) is an induced G-module by (1.3.6)(iii), so that we have
H 1 (G, B) = 0, and the homomorphism
HomG (A, V ) −→ HomG (AG , V ) = Hom(AG , V G )
is surjective. We have canonically ΛG ∼
= IFp , hence an isomorphism AG ∼
= V G,
which by the above argument extends to a G-homomorphism
j : A −→ V.
By (1.6.12), the equality ker(j|AG ) = ker(j)G = 0 implies that j is injective . If
C is the cokernel of j, then we have an exact sequence
∼
0 −→ AG −→
V G −→ C G −→ H 1 (G, A) −→ 0.
Hence if H 1 (G, A) = 0, then C G = 0 and thus C = 0, i.e. A ∼
= V is an induced
G-module. If Ĥ q (G, A) = 0 for some q, then H 1 (G, Aq−1 ) ∼
= Ĥ q (G, A) = 0,
i.e., by the argument as above, Aq−1 is an induced G-module. But this implies
H 1 (G, A) = Ĥ 2−q (G, Aq−1 ) = 0, and hence A is induced.
2
(1.8.4) Proposition. Let G be a finite group and let A be a G-module such
that for every prime number p there exists a dimension np ∈ ZZ with
Ĥ np (Gp , A) = Ĥ np +1 (Gp , A) = 0.
Then A is a cohomologically trivial G-module. If A is ZZ-free, it is a direct
summand of a free ZZ[G]-module. Conversely, for every cohomologically
trivial G-module A, we have
Ĥ n (H, A) = 0
for all n ∈ ZZ and all subgroups H of G.
Proof: Let
(1)
0 −→ R −→ F −→ A −→ 0
be an exact sequence with a free ZZ[G]-module F . We claim that, for every
prime number p, R/pR is an induced Gp -module. In fact, since F is an induced
Gp -module and Ĥ np (Gp , A) = Ĥ np +1 (Gp , A) = 0, we obtain
(2)
Ĥ np +1 (Gp , R) = Ĥ np +2 (Gp , R) = 0,
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Chapter I. Cohomology of Profinite Groups
and the exact sequence
p
0 −→ R −→ R −→ R/pR −→ 0
(3)
yields Ĥ np +1 (Gp , R/pR) = 0, giving the claim by (1.8.3).
Now we assume that A is ZZ-free. Then from (1) we obtain an exact sequence
0 −→ Hom(A, R) −→ Hom(A, F ) −→ Hom(A, A) −→ 0.
If we knew that H 1 (G, Hom(A, R)) = 0, then the homomorphism
HomG (A, F ) −→ HomG (A, A)
would be surjective, so that the identity map of A could be extended to a
G-embedding A ,→ F , making A a direct summand of F . We thus have to
show H 1 (G, M ) = 0 for M = Hom(A, R). Since A is ZZ-free, we get from (3)
an exact sequence
p
0 −→ M −→ M −→ Hom(A, R/pR) −→ 0,
i.e. M/pM ∼
= Hom(A, R/pR) is an induced Gp -module, since R/pR is an
induced Gp -module. From this it follows that
p
H 1 (Gp , M ) −→ H 1 (Gp , M )
is an isomorphism, whence H 1 (Gp , M ) = 0 for all p. By (1.6.11), we get
H 1 (G, M ) = 0, as required.
Now let A be an arbitrary G-module. Since R is ZZ-free, it follows from
(2) and from what we have just seen, that R is a direct summand of a free
ZZ[G]-module, i.e. of an induced G-module. Therefore R is cohomologically
trivial. As F is cohomologically trivial, so is A.
Conversely, we have seen that for every cohomologically trivial G-module
A and every surjection ε : F A with F free, the module R := ker(ε) is a
direct summand in a free module. Hence
Ĥ n (G, A) ∼
= Ĥ n+1 (G, R) = 0
for all n ∈ ZZ.
2
(1.8.5) Corollary. Let G be a finite group and let A, B be G-modules. If either
A is cohomologically trivial and B is divisible, or if B is cohomologically
trivial and A is ZZ-free, then Hom(A, B) is cohomologically trivial.
Proof: Let A be cohomologically trivial and B divisible. We consider an
exact sequence
0 −→ R −→ F −→ A −→ 0,
where F is a free ZZ[G]-module. Since F and A are cohomologically trivial,
so is R, and since R is ZZ-free, it is moreover a direct summand of a free
ZZ[G]-module, say F 0 . Therefore Hom(R, B) is a direct summand of the
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§9. Tate Cohomology of Profinite Groups
induced G-module Hom(F 0 , B) and is thus cohomologically trivial. Since B
is divisible, i.e. an injective ZZ-module, the sequence
0 −→ Hom(A, B) −→ Hom(F, B) −→ Hom(R, B) −→ 0
is exact. The last two modules are cohomologically trivial, hence also the
module Hom(A, B).
If B is cohomologically trivial and A is ZZ-free, the same argument applies
if we exchange the roles of A and B.
2
Exercise 1. Let G be a finite group and let A, B be G-modules. Show that A ⊗ B is
cohomologically trivial if A is cohomologically trivial and either A or B is without p-torsion
for all primes p | #G.
Exercise 2. Let f : A → B be a homomorphism of G-modules and let fn : H n (H, A)
→ H n (H, B) be the induced homomorphisms on the cohomology (H ⊆ G). Assume, for
some q ∈ ZZ, that fq−1 is injective, fq is bijective, fq+1 is surjective, for all subgroups H of G.
Then fn is an isomorphism for all n ∈ ZZ and all subgroups H of G (see [124]).
§9. Tate Cohomology of Profinite Groups
In this section we extend the definition of the modified cohomology groups,
defined in §2 for finite groups, to profinite groups. We start with the definition
of homology of finite groups.
Let G be a finite group and let A be a G-module. Recall from §2 the cofixed
module AG = A/IG A, which is the largest quotient of A on which G acts
trivially. We consider the homological complex P• given by
Pn = ZZ[Gn+1 ],
and the differential ∂n : Pn → Pn−1 given by
∂n (σ0 , . . . , σn ) =
n
X
(−1)i (σ0 , . . . , σi−1 , σi+1 , . . . , σn ).
i=0
Moreover, we have the homomorphism ∂0 : P0 → ZZ, which is defined by
P
P
aσ σ 7→ aσ . Using the contracting homotopy given by Dn (σ0 , . . . , σn ) =
(1, σ0 , . . . , σn ), one verifies in a similar way as in the proof of (1.2.1) that the
complex
∂
∂
∂
0
2
1
· · · −→ P2 −→
P1 −→
P0 −→
ZZ −→ 0
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Chapter I. Cohomology of Profinite Groups
is exact. Moreover, this complex consists of torsion-free abelian groups and
therefore remains exact when tensored by any G-module A. Now we apply
the functor “cofixed module”. For n ≥ 0, we set
Cn (G, A) = (Pn ⊗ A)G
and we call the group
Hn (G, A) = Hn (C• (G, A))
the n-dimensional homology group of G with coefficients in A. From the
exact sequence above it follows that
H0 (G, A) = AG
which is consistent with our definition in §2. In contrast to cohomology, which
is contravariant, homology is covariantly functorial in the group. As a functor
in the module, homology is dual to cohomology in the following sense.
(1.9.1) Proposition. Let G be a finite group and let A be a G-module. If
A∗ = Hom(A, Q/ZZ), then we have a natural isomorphism
H i (G, A∗ ) ∼
= Hi (G, A)∗ , i ≥ 0.
Proof: Applying the exact functor Hom(−, Q/ZZ) to the exact sequence
∂
∂
∂
0
2
1
A −→ 0,
· · · −→
P1 ⊗ A −→
P0 ⊗ A −→
we obtain the exact sequence
0 −→ A∗ −→(P0 ⊗ A)∗ −→(P1 ⊗ A)∗ −→ · · · ,
which, by (1.3.6)(iii), is an induced, hence acyclic, resolution of A∗ . By
(1.3.9), we obtain isomorphisms
H i (G, A∗ ) ∼
= H i (H 0 (G, (P• ⊗ A)∗ )) ∼
= Hi (H0 (G, P• ⊗ A))∗ ∼
= Hi (G, A)∗ .
2
All cohomological notions have their homological counterpart. For example,
for a normal subgroup U of G, we have the coinflation maps
coinf : Hn (G, A) −→ Hn (G/U, AU ),
which are induced by the projections ZZ[Gn+1 ] → ZZ[(G/U )n+1 ] and A → AU .
Recalling the modified homology group Ĥ0 (G, A) = NG A/IG A from §2, we
call the groups
(
for n = 0,
NG A/IG A
Ĥn (G, A) =
Hn (G, A) for n ≥ 1
the modified homology groups. In §2 we identified the group Ĥ0 (G, A) with
Ĥ −1 (G, A). The following proposition extends this to arbitrary dimension.
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§9. Tate Cohomology of Profinite Groups
(1.9.2) Proposition. Let G be a finite group and let A be a G-module. Then
we have natural isomorphisms
Ĥ (G, A) ∼
=
(
n
Ĥ n (G, A)
Ĥ−n−1 (G, A)
for n ≥ 0,
for n ≤ −1.
Proof: We have seen this for n ≥ −1 in §2. In order to deal with the remaining
dimensions, we consider the dual G-modules
(X−1−n )+ = Hom(X−1−n , ZZ)
for n ≥ 0. Let ((σ0 , . . . , σn )∗ )σ0 ,...,σn ∈G , be the dual basis of (X−1−n− )+ , i.e.
(σ0 , . . . , σn )∗ maps (σ0 , . . . , σn ) to 1 and all other basis elements of X−1−n =
ZZ[Gn+1 ] to zero. We consider the homomorphisms
ϕn : Pn −→(X−1−n )+
defined by sending (σ0 , . . . , σn ) to (σ0 , . . . , σn )∗ . A straightforward calculation
shows that the ϕn are G-homomorphisms and that the diagram
PÃÂÁÀ n
ϕn
(X−1−n )+
(∂−n )+
∂n
ϕ
n−1
Pn−1
(X−n )+
commutes for n ≥ 1. Therefore, for n ≥ 1, Hn (G, A) is the cohomology of the
complex
((X• )+ ⊗ A)G
in dimension (−1 − n). Since the abelian groups Xi are free of finite rank, the
duality maps Di : (Xi )+ ⊗ A −→ Hom(Xi , A), given by Di (f ⊗ a)(x) = f (x)a,
are isomorphisms. These isomorphisms are easily seen to be G-invariant, and
we obtain an isomorphism of complexes
∼
D : ((X• )+ ⊗ A)G −→ Hom(X• , A)G .
Finally, by (1.3.6)(iii), the modules Hom(Xi , A) are induced. By (1.2.6), the
norm induces an isomorphism of complexes
∼
NG : Hom(X• , A)G −→ HomG (X• , A) = Ĉ • (G, A).
2
This finishes the proof.
Now let G be a profinite group and let A be a discrete G-module. We want
to define Tate cohomology groups Ĥ i (G, A), i ∈ ZZ. For i ≤ 0 this requires the
deflation map
def : Ĥ i (G/V, AV ) −→ Ĥ i (G/U, AU ),
for open normal subgroups V
⊆
i ≤ 0,
U , which is defined as follows:
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Chapter I. Cohomology of Profinite Groups
Consider, for j
≥
1, the composition
coinf
N
∗
Hj (G/V, AV ) −→ Hj (G/U, (AV )U ) −→
Hj (G/U, AU ),
where N∗ is the map which is induced on homology by the norm
NU/V : (AV )U −→ AU .
Via the identification Ĥ i = H−i−1 , this defines the deflation in dimension
i ≤ −2. In dimensions i = 0, −1, the map def is induced via the identifications
Ĥ 0 (G/U, AU ) ∼
= AG /NG/U AU and Ĥ −1 (G/U, AU ) ∼
= N AU /IG/U AU
G/U
by the identity and the norm map, respectively.
(1.9.3) Definition (Tate cohomology). Let G be a profinite group and let A
be a discrete G-module. For i > 0, we set Ĥ i (G, A) = H i (G, A). For i ≤ 0,
we set
Ĥ i (G, A) = lim Ĥ i (G/U, AU ),
←−
U, def
where U runs through the open normal subgroups of G.
Our next goal is a cup-product pairing for Tate cohomology. We start by
specifying a map on the chain level on the negative part of the complete standard
resolution which defines the deflation map on cohomology in dimension ≤ −2.
For a finite group G, let X•(G) denote the (homological) complete standard
resolution for G.
(1.9.4) Lemma. Let G be a finite group and let U ⊆ G be a normal subgroup.
(G/U )
(G)
In negative dimension, let the map α−i : X−i
−→ X−i
be given by
(σ0 U, . . . , σi−1 U ) 7−→
X
(σ0 τ0 , . . . , σi−1 τi−1 ),
τ0 ,...,τi−1 ∈U
where the τ0 , . . . , τi−1 on the right-hand side run independently through U .
Then in dimension ≤ −2 the deflation map is induced by a cocycle map
def : HomG (X•(G) , A) −→ HomG/U (X•(G/U ) , AU ),
which is uniquely defined by the commutative diagram
ÇÅÄÆ (G) , A)G
Hom(X−i
NG
(α∗−i ,(NU )∗ )
(G/U )
Hom(X−i
, AU )G/U
(G)
HomG (X−i
, A)
def
NG/U
(G/U )
HomG/U (X−i
, AU ).
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§9. Tate Cohomology of Profinite Groups
Proof: First note that the map α−i commutes with the differential on the
negative part of the standard complex. Let ((σ0 , . . . , σi−1 )∗ )σ0 ,...,σi−1 ∈G denote
(G/U )
(G) +
(G)
the dual basis of (X−i
) = Hom(X−i
, ZZ). The dual basis of (X−i )+ is
denoted by ((σ0 U, . . . , σi−1 U )∗ )σ0 U,...,σi−1 U ∈G/U . Consider the diagram
(G) +ÓÒÈËÊÉÑÏÎÍÌÐ
((X−i
) ⊗ A)G
Di
(G)
Hom(X−i
, A)G
(G/U ) +
) ⊗ AU )G/U
Di
(G/U )
Hom(X−i
(G/U ) +
, AU )G/U
β
NG/U
(G/U )
HomG/U (X−i
(id∗ ,(NU )∗ )
id⊗NU
((X−i
(G)
HomG (X−i
, A)
(α∗−i ,pr∗ )
coinf
((X−i
NG
) ⊗ AU )G/U
Di
(G/U )
Hom(X−i
, AU )G/U
, AU )
(id∗ ,(NU )∗ )
NG/U
(G/U )
HomG/U (X−i
, AU ).
Here, pr : A → AU is the natural projection, coinf is the map which is induced
(G/U )
(G) +
by pr and by the map (X−i
) → (X−i )+ which sends (σ0 , . . . , σi−1 )∗ to
(σ0 U, . . . , σi−1 U )∗ . Via the identification
Hi−1 (G, A) = H −i ((X• )+ ⊗ A)G ,
i ≥ 2,
(see the proof of (1.9.2)) it induces coinflation on homology. The maps D−i are
induced by the canonical duality isomorphisms (X−i )+ ⊗ A ∼
= Hom(X−i , A).
The upper left square commutes and we define β in order to make the upper
right square commute. The lower squares obviously commute. By definition,
def is induced by the composition of the two vertical arrows on the left. This
completes the proof of the lemma.
2
(1.9.5) Proposition. Let i ≥ 2 and q ≥ 1. Let G be a finite group, U a normal
subgroup of G and let A and B be G-modules. Then
x̄ ∪ inf ȳ = inf (def x̄ ∪ ȳ) ∈ H q (G, A ⊗ B),
for x̄ ∈ Ĥ −i (G, A) and ȳ
∈
H q+i (G/U, B U ).
(G/U )
(G)
Proof: Let x ∈ HomG (X−i
, A) and y ∈ HomG/U (Xq+i , A) be cocycles
representing x̄ and ȳ, respectively. We calculate both sides on the level of
cochains. Let us start with the left-hand side. For (σ0 , . . . , σq ) ∈ Xq(G) we
obtain
(x ∪ inf y)(σ0 , . . . , σq ) =
X
x(τ1 , . . . , τi ) ⊗ y(τi U, . . . , τ1 U, σ0 U, . . . , σq U ),
τ1 ,...,τi ∈G
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Chapter I. Cohomology of Profinite Groups
see the definition of the maps ϕp,q in the proof of (1.4.7). In order to compute
(G)
the right-hand side, choose a z ∈ Hom(X−i
, A) with x = NG z. Then, using
again the maps ϕp,q of (1.4.7), by (1.9.4) we have
∗
inf (def x ∪ y)(σ0 , . . . , σq ) = (NG/U (α−i
, (NU )∗ )z) ∪ y (σ0 U, . . . , σq U ).
The right-hand side is equal to
X
∗
(NG/U (α−i
, (NU )∗ )z)(τ1 U, . . . , τi U )⊗y(τi U, . . . , τ1 U, σ0 U, . . . , σq U ).
τ1 U,...,τi U ∈G/U
Using the definition of α−i and of NU this transforms to
X
X
X
X
τ U ∈G/U u∈U τ1 U,...,τi U ∈G/U u1 ,...,ui ∈U
τ u z(τ −1 τ1 u1 , . . . , τ −1 τi ui ) ⊗ y(τi U, . . . , τ1 U, σ0 U, . . . , σq U ),
which coincides with
X
(NG z)(τ1 , . . . , τi ) ⊗ y(τi U, . . . , τ1 U, σ0 U, . . . , σq U ).
τ1 ,...,τi ∈G
2
This shows the proposition.
(1.9.6) Proposition. Let G be a finite group and let U be a normal subgroup
of G. Let A and B be G-modules. Then, for i ≤ 0 and q ≥ 1, the diagram
Ø×ÖÕÔ A)
Ĥ i (G,
(∗)
×
def
∪
H q−i (G, B)
H q (G, A ⊗ B)
inf
inf
Ĥ i (G/U, AU ) × H q−i (G/U, B U )
∪
H q (G/U, (A ⊗ B)U )
commutes.
Proof: For i ≤ −2 this follows from (1.9.5). Via the isomorphism
∼
−→
Ĥ −1 (G, A)
a 7−→ xa : ZZ[G] → A, σ 7→ σ(a),
NG A/IG A
the cup product is given on the chain level by
xa ∪ y(σ0 , . . . , σq ) =
X
σa ⊗ y(σ, σ0 , . . . , σq ).
σ ∈G
The commutativity of (∗) for i = −1 follows immediately, since def is induced
by the norm
NU :
NG A/IG A
−→
U
U
NG/U A /IG/U A .
The case i = 0 is obvious.
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§9. Tate Cohomology of Profinite Groups
Now let G be a profinite group and let A be a discrete G-module which
is finitely generated as a ZZ-module. The subgroup of elements of G that act
trivially on A is open, since for all a ∈ A the group Ga = {σ ∈ G | σa = a} is
open. Hence AU = A for U sufficiently small. In this case
Hom(A, B U ) = Hom(A, B)U
for any G-module B. By passing to the limit of the diagram (∗) of (1.9.6), we
obtain the
(1.9.7) Corollary. Let G be a profinite group and let A and B be discrete
G-modules. Then we have a cup-product pairing
Ĥ i (G, A) × H q−i (G, B) −→ H q (G, A ⊗ B)
for i ≤ 0 and q
≥
1.
For an abelian profinite group A = lim Ai , Ai finite, and a prime number p,
←−
the p-part A(p) of A is defined by
A(p) = lim Ai (p).
←−
i
Let G be a profinite group and let A be a discrete G-module which is finitely
generated as a ZZ-module. By (1.6.1), Ĥ i (G/U, AU ) is finite for all i and all
open normal subgroups U of G. Therefore the abelian groups
Ĥ i (G, A) = lim Ĥ i (G/U, AU ),
←−
U, def
i ≤ 0,
are naturally equipped with a profinite topology. For i ≥ 1, we equip the
abelian torsion groups Ĥ i (G, A) with the discrete topology.
(1.9.8) Lemma. Let G be a profinite group and let A be a discrete G-module
which is finitely generated as a ZZ-module. If p is a prime number with p∞ |#G,
then we have a canonical isomorphism
Ĥ 0 (G, A)(p) ∼
= AG ⊗ ZZp .
Proof: It suffices to show that for arbitrarily given n ∈ IN the subgroup
NG/U AU is contained in pn AG for sufficiently small U . Let U be an open
normal subgroup such that AU = A. Then for every normal open subgroup
V ⊆ U with pn |(U : V ) and for every a ∈ AV = AU = A, we have
NG/V a = NG/U NU/V a = (U : V )NG/U a ∈ pn AG .
This shows the lemma.
2
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Chapter I. Cohomology of Profinite Groups
(1.9.9) Proposition. Let
0 −→ A0 −→ A −→ A00 −→ 0
be a short exact sequence of discrete G-modules which are finitely generated
as ZZ-modules. Then there is an associated long exact cohomology sequence
· · · −→ Ĥ −n (G, A0 ) −→ Ĥ −n (G, A) −→ Ĥ −n (G, A00 ) −→ · · ·
ending with · · · → Ĥ 0 (G, A0 ) → Ĥ 0 (G, A) → Ĥ 0 (G, A00 ).
For every prime number p with p∞ |#G, we obtain a long exact sequence
· · · −→ Ĥ i (G, A0 )(p) −→ Ĥ i (G, A)(p) −→ Ĥ i (G, A00 )(p) −→ · · ·
which is unbounded in both directions. The groups are compact for i
discrete for i > 0 and all homomorphisms are continuous.
≤
0,
Proof: Since AU = A for small U , we obtain for small U a long exact
sequence
· · · −→ Ĥ 0 (G/U, A0U ) −→ Ĥ 0 (G/U, AU ) −→ Ĥ 0 (G/U, A00U )
of finite abelian groups. Passing to the projective limit, we obtain the negative
part of our long exact sequence. Now let p be a prime number with p∞ |#G and
consider for small U the long exact sequence
0 −→ A0G −→ AG −→ A00G −→ Ĥ 1 (G/U, A0U ) −→ · · · .
Tensoring by ZZp and passing to the direct limit over U , we obtain the right-hand
part of our long exact sequence. The left-hand part is obtained by taking the
p-part of the long exact sequence which we obtained above. Both fit together
by (1.9.8). The continuity of the maps is clear from their definitions.
2
Let G be a profinite group and let A be a discrete G-module. For each pair
of open normal subgroups V ⊆ U of G, the exact sequences
(∗)
0 −→ NG/U AU −→ AG −→ Ĥ 0 (G/U, AU ) −→ 0,
(∗∗)
0 −→ IG/U AU −→ NG/U AU −→ Ĥ −1 (G/U, AU ) −→ 0
induce exact sequences
0 −→ NG A −→ AG −→ Ĥ 0 (G, A),
0 −→ IG A −→ NG A −→ Ĥ −1 (G, A),
where we have set
\
NG A = lim NG/U AU = NG/U AU ,
←−
U
NG A
= lim
←−
U
U
U
NG/U A
, IG A = lim IG/U AU .
←−
U
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§9. Tate Cohomology of Profinite Groups
The group NG A is called the group of universal norms. The last exact
sequences are in general not exact on the right, unless we make assumptions
on the module. This motivates the following
(1.9.10) Definition. Let G be a profinite group. A level-compact G-module
is a discrete G-module A which is endowed with the following additional
topological structure: For each open subgroup U ⊆ G the group AU carries
a compact group topology, such that for all open subgroups U and V and all
σ ∈ G with V ⊆ σU σ −1 , the natural map
σ : AU −→ AV , a 7−→ σa,
is continuous.
In particular, for V ⊆ U , the group AU carries the subgroup topology of AV
with respect to the inclusion AU ⊆ AV , and the norm map NU/V : AV → AU
is continuous.
Assume now that the discrete G-module A is level-compact. In this case, the
additional topological structure on A induces a compact group topology on all
groups in the above sequences (∗) and (∗∗). Furthermore, all maps occurring
are continuous. Since the projective limit is exact on compact groups, we
obtain the
(1.9.11) Lemma. Let A be a level-compact G-module. Then
Ĥ 0 (G, A) = AG /NG A
and
Ĥ −1 (G, A) = NG A/IG A.
If A is level-compact, then the groups
Ĥ i (G, A) = lim
Ĥ i (G/U, AU ),
←−
i ≤ 0,
U, def
are abelian profinite groups in a natural way: For each open normal subgroup
U ⊆ G, the group Ĥ i (G/U, AU ) inherits a natural compact topology from AU
via the standard complex. Furthermore, this group is annihilated by (G : U ),
hence Ĥ i (G/U, AU ) is profinite. Since def is continuous, the group Ĥ i (G, A)
is profinite as an inverse limit of abelian profinite groups. For i > 0 we give
Ĥ i (G, A) = H i (G, A) the discrete topology.
We will frequently make use of the following fact: For a level-compact
G-module A, the natural map
\
lim AU −→ NG A =
NG/U AU
←−
U,Norm
U
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Chapter I. Cohomology of Profinite Groups
is surjective. This follows easily from the fact that a filtered inverse limit of
nonempty compact spaces is nonempty, cf. [160], chap.IV, (2.3). In particular,
the norm map NG/U : NU A → NG A is surjective for all open normal subgroups
U ⊆ G.
The next lemma shows that we have some flexibility when calculating the
Tate cohomology of level-compact modules.
(1.9.12) Lemma. Let A be a level-compact G-module. Suppose that for every
normal open subgroup U ⊆ G a closed (with respect to the additional topology)
G-submodule
A(U ) ⊆ AU
is given in such a way that the following conditions hold
(i) NU A ⊆ A(U ) for all U ,
(ii) for V
⊆
U , NU/V : AV → AU maps A(V ) to A(U ).
Then
Ĥ i (G, A) ∼
= lim Ĥ i (G/U, A(U ))
←−
U
for all i ≤ −1.
Proof: The transition maps in the above inverse system are defined by an
obvious modification of def (possible by condition (ii)). We have seen in (1.9.4)
that the deflation maps in negative dimensions are given by a map on the chain
level. As projective limits are exact on compact groups, we see that for i ≤ −2
the group Ĥ i (G, A) can also be calculated as the quotient of the inverse limit
of the cocycles modulo the inverse limit of the coboundaries. These, however,
take values in the groups of universal norms on the corresponding levels, i.e.,
by condition (i), we may take the limit over the groups A(U ) instead of AU as
well. This shows the lemma for i ≤ −2.
Now we show the assertion for i = −1. We have Ĥ −1 (G, A) = NG A/IG A
with NG A = lim NG/U AU and IG A = lim I G/U AU . Recalling that the
←− U
←− U
functor lim is exact on compact groups, we have to show
←−
NG A
Obviously, lim
←−
= lim
←−
U
NG/U A(U )
NG/U A(U ) ⊆ NG A.
V
the norm maps NU/V : A
and
IG A = lim IG/U A(U ).
Let (aU )
U
←−
U
∈ NG A.
−→ A , we have aU
∈
Since lim
is taken over
←−
\
V
A(U ). This proves NG A = lim
←−
NU/V AV = NU A
⊆U
NG/U A(U ).
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§9. Tate Cohomology of Profinite Groups
The inclusion A(U ) ⊆ AU yields the injection
IG/U AU = IG A.
IG/U A(U ) −→ lim
lim
←−
←−
U
U
For the surjectivity, let (aU ) ∈ lim
←−
U
IG/U AU . The projective limit is taken over
the maps NU/V : IG/V AV −→ IG/U AU given by (σ −1)a 7−→ (σ̄ −1)NU/V (a).
Therefore
\
(∗)
aU ∈
IG/U NU/V AV .
V ⊆U
We show aU ∈ IG/U NU A for any fixed U . Let V run through the open normal
Q
subgroups of G contained in U . Let ÃV = σ∈G/U AV and let
fV : ÃV −→ IG/U AU
be the composite of the maps
ÃV
Q
where ÑU/V = σ∈G/U NU/V
groups are compact and fV
is a nonempty, closed and
lim
fV−1 (aU ) =/ ∅.
←− V ⊆U
Let (ãV )
hence
∈
lim
←− V ⊆U
ÑU/V
f
U
−→ ÃU −→
IG/U AU ,
P
and fU : (aσ )σ∈G/U 7→ σ∈G/U (σ − 1)aσ . All
is continuous. Therefore, using (∗), fV−1 (aU )
thus compact subset of ÃV . It follows that
fV−1 (aU ). Then for ãU = (aσ )σ∈G/U we have aσ
aU = fU (ãU ) =
X
(σ − 1)aσ
∈
∈
NU A,
IG/U NU A.
σ ∈G/U
This proves that (aU )
lim
IG/U A(U ).
←−
∈
lim
←− U
IG/U NU A
⊆
lim
←− U
U
IG/U A(U ), whence IG A =
2
In everything that follows, we tacitly assume maps between level-compact
modules to be continuous with respect to the additional topology.
(1.9.13) Proposition. Let
i
j
0 −→ A0 −→ A −→ A00
be an exact sequence of level-compact G-modules, such that the induced map
NUÙ A U NU A00
is surjective for a cofinal system of open normal subgroups U of G (and
hence for all open normal subgroups). Then there is an associated long exact
cohomology sequence
N j
· · · −→ Ĥ −n (G, A0 ) −→ Ĥ −n (G, A) −→ Ĥ −n (G, A00 ) −→ · · ·
ending with · · · → Ĥ 0 (G, A0 ) → Ĥ 0 (G, A) → Ĥ 0 (G, A00 ). Moreover, if j is
surjective, we obtain the long exact cohomology sequence unbounded in both
directions, i.e. from −∞ to +∞.
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94
Chapter I. Cohomology of Profinite Groups
Proof: For every open normal subgroup U in G we consider the kernel
NU j
A0 (U ) := ker(NU A −→
NU A00 ). We have inclusions NU A0 ⊆ A0 (U ) ⊆ A0U ,
and obtain the exact and commutative diagram
0áâãàßÚÛÜÝÞ
A0 (U )
NU A
NG/U
NG/U
A0G
0
NU A00
AG
0
NG/U
A00G .
Consider the long exact cohomology sequence
· · · → Ĥ i (G/U, A0 (U )) → Ĥ i (G/U, NU A) → Ĥ i (G/U, NU A00 ) → · · ·
associated to the upper line. It consists of compact abelian groups, is clearly
exact and all homomorphisms including the boundary maps are continuous
(for example use the snake lemma in the abelian category of compact abelian
groups). Passing to the inverse limit over U , we obtain, using (1.9.12), the
asserted long exact sequence up to dimension −1.
By compactness, the image of NG/U : NU A −→ AG is NG A and hence, by
(1.9.11), the cokernel of this map is Ĥ 0 (G, A). We denote its kernel by
X(A, U ) =
NG/U NU A,
which contains Y (A, U ) := IG/U NU A, and the same holds for A00 . The snake
lemma implies an exact commutative diagram
Y (A,éêåæçèä U )
j
X(A, U )
Y (A00 , U )
X(A00 , U )
δ
A0G /NG/U A0 (U )
Ĥ 0 (G, A)
Ĥ 0 (G, A00 ).
Observe that
lim
(A0G /NG/U A0 (U )) = Ĥ 0 (G, A0 ).
←−
U
Furthermore, the upper map j : IG/U NU A → IG/U NU A00 is obviously surjective and NG/U : NU A → AG maps IG/U NU A to zero. Therefore δ :
X(A00 , U ) → A0G /NG/U A0 (U ) maps Y (A00 , U ) to zero by the definition of δ.
This means that we may replace in the last diagram the group X(A, U ) by
Ĥ −1 (G/U, NU A) = X(A, U )/Y (A, U ) and X(A00 , U ) by Ĥ −1 (G/U, NU A00 ) =
X(A00 , U )/Y (A00 , U ) and obtain an exact sequence of compact groups and continuous homomorphisms. Taking projective limits over U , we obtain the exact
sequence
δ
Ĥ −1 (G, A) → Ĥ −1 (G, A00 ) → Ĥ 0 (G, A0 ) → Ĥ 0 (G, A) → Ĥ 0 (G, A00 ).
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95
§9. Tate Cohomology of Profinite Groups
Now suppose that j is surjective. Consider the commutative and exact diagram
0ö÷ëìíîïðñòóôõøù
0
Ĥ 0 (G, A)
Ĥ 0 (G, A00 )
AG
A00G
H 1 (G, A0 )
H 1 (G, A)
0
NG A
NG j
00
NG A
0
0.
The vertical sequences are exact by (1.9.11). We deduce the existence of the
dotted arrow, which glues the already proven long exact sequence in negative
dimension with the long exact sequence
H 1 (G, A0 ) −→ H 1 (G, A) −→ H 1 (G, A00 ) −→ · · · .
2
The homomorphisms in the negative part of the long exact sequence are
easily seen to be continuous. The maps in dimensions at least 1 are continuous
because the groups are discrete. But the boundary map δ 0 : Ĥ 0 (G, A00 ) →
H 1 (G, A0 ) may be discontinuous. Therefore the next proposition is nontrivial.
(1.9.14) Proposition. If p is a prime number, then the long exact sequence of
lemma (1.9.13) also induces a long exact sequence of the p-parts.
Proof: All occurring cohomology groups are either abelian profinite groups
or abelian discrete torsion groups and therefore they naturally decompose into
the direct sum of their p-parts and their prime-to-p-parts. In order to prove the
corollary, it suffices to show that also the differentials decompose into a direct sum of homomorphisms. This is trivially true for continuous differentials,
hence it remains to consider the homomorphism δ 0 : Ĥ 0 (G, A00 ) → H 1 (G, A0 ).
Let x ∈ Ĥ 0 (G, A00 )(p). Choose a pre-image m00 ∈ A00G of x and let m ∈ A
be a pre-image of m00 . Furthermore, let U ⊆ G be an open normal subgroup in G such that m ∈ AU . The closed subgroup generated by m in
AU maps onto the closed subgroup generated by x which coincides with
ZZp· x ⊆ Ĥ 0 (G, A00 )(p). We conclude that δ 0 (λx) ∈ H 1 (G/U, A0U ) ⊆ H 1 (G, A0 )
for all λ ∈ ZZp . Writing #(G/U ) = N pk with (N, p) = 1, we obtain that
δ 0 (x) = N δ 0 (N −1 x) ∈ N ·H 1 (G/U, A0U ) = H 1 (G/U, A0U )(p) ⊆ H 1 (G, A0 )(p).
A similar argument shows that δ 0 sends the prime-to-p-part of Ĥ 0 (G, A00 ) to
the prime-to-p-part of H 1 (G, A0 ).
2
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96
Chapter I. Cohomology of Profinite Groups
Applying the above results to the case of a finite module, we obtain the
(1.9.15) Proposition. Let A be a finite G-module. Assume that `∞ | #G for
all prime numbers ` dividing the order of A. Then
Ĥ 0 (G, A) = AG
and
Ĥ i (G, A) = 0 for all i ≤ −1.
Proof: Let U ⊆ G be an open subgroup with A = AU and let V ⊆ U be an
open normal subgroup. Then the norm map NU/V from A = AV into A = AU
is just the multiplication by (U : V ). As `∞ | #G for all prime numbers `
dividing the order of A, we conclude that NU A = 0 for all open subgroups U
of G. Applying (1.9.13) to the exact sequence
0 −→ 0 −→ 0 −→ A,
i
we obtain Ĥ (G, A) = 0 for all i
likewise from (1.9.8) or (1.9.11).
≤
−1. The statement on Ĥ 0 (G, A) follows
2
Remark: Tate cohomology of profinite groups was first introduced by
G. POITOU in [168]. The presentation of this section essentially follows [196].
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Chapter II
Some Homological Algebra
§1. Spectral Sequences
If G is a profinite group and H a closed normal subgroup, then one may
ask whether the cohomology groups H n (G, A) can be computed from the
cohomology groups of the smaller groups H and G/H. We have already seen
a relation of this type, namely the isomorphism
H n (G/H, H 0 (H, A)) ∼
= H n (G, A)
if H n (H, A) = 0 for all n ≥ 1, which follows from (1.6.7). There is a quite
general relation, which is denoted by
H p (G/H, H q (H, A)) ⇒ H n (G, A)
and is called a “spectral sequence”. The situation is slightly involved. It
roughly says that there is a canonical decreasing filtration of H n = H n (G, A),
H n = F 0H n
⊇
F 1H n
⊇
· · · ⊇ F nH n
⊇
F n+1 H n = 0,
such that the quotient F p H n /F p+1 H n is isomorphic not directly to the group
H p (G/H, H n−p (H, A)), but to a certain subquotient of it. The notion of
spectral sequence is very general and of utmost importance in cohomology
theory. The general set-up, which can be generalized in several directions if
the underlying category has inductive limits, is the following.
Let A be an abelian category. A (decreasing) filtration of an object A is a
family (F p A)p∈ ZZ of subobjects F p A of A such that F p A ⊇ F p+1 A for all p.
Write
grp A = F p A/F p+1 A.
By convention, we put F ∞ A = 0 and F −∞ A = A. We say that the filtration
is finite if there exist n, m ∈ ZZ with F m A = 0 and F n A = A. Given filtered
objects A and B in A, a morphism f : A → B is said to be compatible with the
filtration if f (F p A) ⊆ F p B for all p ∈ ZZ.
Let m be a natural number. An Em -spectral sequence∗) in A is a system
E = (Erpq , E n ),
∗) In the applications usually only E - and E -spectral sequences occur.
1
2
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98
Chapter II. Some Homological Algebra
consisting of
a) objects Erpq
b)
∈
A for all (p, q) ∈ ZZ × ZZ and any integer r
dpq
r
morphisms d =
: Erpq −→ Erp+r,q−r+1 with d ◦ d = 0
each fixed pair (p, q) ∈ ZZ × ZZ the morphisms dpq
r and
≥
m,
and such that for
drp−r,q+r−1 vanish
for sufficiently large r,
p−r,q+r−1
∼ E pq ,
c) isomorphisms αrpq : ker (dpq
) −→
r+1
r )/im(dr
d) finitely filtered objects E n
∈
A for all n ∈ ZZ,
pq ∼
e) isomorphisms β pq : E∞
−→ grp E p+q .
By b) and c), the objects Erpq are independent of r for r sufficiently large and
pq
. These are the objects occurring in e).
are then denoted by E∞
In other words, for each r ≥ m, the system Erpq is a system of complexes
pq
whose cohomology groups are the objects Er+1
of the next system. A spectral
pq
pq
pq
sequence is like a book with (infinitely many) pages Em
, Em+1
, Em+2
, . . . and
n
a limit page
E
at
the
end.
úûüýþÿ
!"#
E1pq
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
$%&'()*+,-.
E3pq
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
E2pq
/01234
E4pq
For an Em -spectral sequence E = (Erpq , E n ), one usually writes
pq
Em
⇒ E p+q
pq
pq
or Em
⇒ E n . The Em
are called the initial terms, the E n the limit terms
0
and the dpq
r differentials. By forgetting the first m −m pages, an Em -spectral
sequence induces an Em0 -spectral sequence for all m0 ≥ m in a natural way.
A morphism of Em -spectral sequences
ϕ : E = (Erpq , E n ) −→ E 0 = (Er0pq , E 0n )
in A is a system of morphisms
0pq
pq
ϕpq
ϕn : E n −→ E 0n ,
r : Er −→ Er ,
n
where the ϕn are compatible with the filtrations of E n and E 0n and the ϕpq
r ,ϕ
pq
pq
commute with dpq
r , αr and β .
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99
§1. Spectral Sequences
If Erpq = 0 for p < 0 or q < 0, one speaks of a first quadrant spectral
sequence. In this case we have
pq
for r > max(p, q + 1), r ≥ m.
Erpq = E∞
Once a first quadrant spectral sequence is given, we obtain a realm of
homomorphic connections. We restrict to the most important case of an E2 spectral sequence. Of basic importance are the two homomorphisms
E2n,0 −→ E n −→ E20,n ,
the so-called edge morphisms . The first one is the composite of the morphisms
n,0
E2n,0 −→ E3n,0 −→ . . . −→ E∞
−→ E n ,
which are well-defined because F n+1 E n = 0 and Ern+r,−r+1 = 0 for r ≥ 2
n,0
is a quotient of Ern,0 ). The second one is the composite of the
(so that Er+1
morphisms
0,n
E n −→ E∞
−→ . . . −→ E30,n −→ E20,n ,
which are well-defined because F 0 E n = E n and Er−r,n+r−1 = 0 for r ≥ 2 (so
0,n
that Er+1
is embedded in Er0,n ). A direct consequence of the definition of the
edge morphisms is the following
(2.1.1) Proposition. For any first quadrant E2 -spectral sequence the sequence
d
0 −→ E21,0 −→ E 1 −→ E20,1 −→ E22,0 −→ E 2
is exact. It is called the associated five term exact sequence.
We get a generalization of this result under the assumption that E2pq = 0 for
0,n ∼
∼ E 0,n
0 < q < n. Namely, in this case we have isomorphisms En+1
→ ··· →
2
n+1,0
0,n
n+1,0
n+1,0 ∼
∼ E
and E2
→ ··· →
n+1 . Therefore the differential En+1 → En+1 induces
a homomorphism
d
E20,n −→ E2n+1,0 .
We obtain the following
(2.1.2) Lemma. Assume that, in a first quadrant E2 -spectral sequence, the
terms E2pq vanish for 0 < q < n and all p. Then
E2m,0 ∼
= Em
for m < n and the sequence
d
0 −→ E2n,0 −→ E n −→ E20,n −→ E2n+1,0 −→ E n+1
is exact.
The proof of this lemma (and also of the results below) is elementary and we
refer to [21], chap.XV, §5. The most frequent application of spectral sequences
are in the following special cases.
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100
Chapter II. Some Homological Algebra
(2.1.3) Lemma. Assume that a first quadrant E2 -spectral sequence is given.
(i) If E2pq = 0 for all q > 1 and all p, then we have a long exact sequence
0
BCDEF;<=>?@A56789:
E21,0
E1
E22,0
E2
E23,0
E3
E20,1
d
E21,1
d
E22,1
···
.
(ii) If E2pq = 0 for all p > 1 and all q, then the sequences
0 −→ E21,n−1 −→ E n −→ E20,n −→ 0
are exact for all n ≥ 1.
(2.1.4) Lemma. Assume that, in a first quadrant E2 -spectral sequence, the
term E2pq vanishes for all (p, q) with (p − m)·(q − n) < 0. Then
E mn ∼
= E m+n .
2
In particular, if
E2pq
= 0 for all q > 0, then
E2m,0 ∼
= Em
for all m.
Proof. If p > m and q < n or if p < m and q > n, then Erpq = 0 for
pq
all r ≥ 2, since it is a subquotient of E2pq , and hence E∞
= 0. Therefore on
mn
pq
the line p + q = m + n, all terms E∞ are zero up to E∞ and consequently
mn ∼
E∞
= E m+n . The maps
d
d
r
r
Erm−r,n+r−1 −→
Erm,n −→
Erm+r,n−r+1
are zero for all r
≥
mn
.
2, hence E2mn = E3mn = · · · = Ermn = E∞
2
In practice, the differentials of a spectral sequence are often difficult to
calculate. We are in a rather comfortable situation if they vanish from a certain
point on. In this case one says that the spectral sequence degenerates. The
precise definition is the following.
(2.1.5) Definition. An Em -spectral sequence degenerates at Em0 for some
m0 ≥ m if the differentials
pq
p+r,q−r+1
dpq
r : Er −→ Er
vanish for all r
≥
m0 and all (p, q) ∈ ZZ × ZZ. Hence, in this case
pq
pq
pq
pq
Em
0 = Em0 +1 = Em0 +2 = · · · = E∞
for all p, q
∈
ZZ × ZZ.
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101
§2. Filtered Cochain Complexes
§2. Filtered Cochain Complexes
Many spectral sequences arise from filtered cochain complexes. Let A be an
abelian category. By a (cochain) complex A• in A we understand a sequence
A• = (An , dn )n∈ ZZ of objects and homomorphisms
dn−1
dn
dn+1
· · · −→ An−1 −→ An −→ An+1 −→ An+2 −→ · · ·
with dn+1 ◦ dn = 0 for all n ∈ ZZ. The d’s are called differentials. We say that
A• is bounded below (resp. bounded above, resp. bounded) if An = 0 for
n 0 (resp. for n 0, resp. for n 0 and n 0). We set
dn
Z n (A• ) = ker (An −→ An+1 ),
dn−1
B n (A• ) = im (An−1 −→ An ).
The elements of Z n (A• ) and B n (A• ) are called the n-cocycles and n-coboundaries, respectively. As d ◦ d = 0, we have B n (A• ) ⊆ Z n (A• ). The factor
group
H n (A• ) = Z n (A• )/B n (A• )
is called the n-dimensional cohomology group of A• .
A homomorphism of complexes f : A• → B • is a sequence f = (f n )n∈ ZZ
of homomorphisms f n : An → B n with f n+1 ◦ dn = dn ◦ f n for all n ∈ ZZ. A
homomorphism f : A• → B • of complexes induces homomorphisms
H n (f ): H n (A• ) −→ H n (B • )
on the cohomology, and we call f a quasi-isomorphism if H n (f ) is an isomorphism for all n.
A filtration by subcomplexes of A• is a filtration F • An of An for all n ∈ ZZ
such that for each n, F n A• is a subcomplex of A• . We say that the filtration
F • A• is biregular, if, for each n ∈ ZZ, the filtration F • An is finite.
Examples: For any complex A• we have the following filtrations
1. The trivial filtration tr• A• defined by
(
A• , for p ≤ 0,
trp A• =
0, for p ≥ 1.
We have
(
H q (A• ), for n ≤ 0,
q
n •
H (tr A ) =
0,
for n ≥ 1.
2. Consider for p ∈ ZZ the subcomplex τ≤p (A• ) of A• given by
An ,
for n ≤ p−1,
d
τ≤p (A• )n = ker(An →
An+1 ), for n = p,


0,
for n ≥ p + 1.



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Chapter II. Some Homological Algebra
We have
(
q
•
H (τ≤p (A )) =
H q (A• ), for q
0,
for q
≤
≥
p,
p + 1.
The canonical filtration τ • A• on A• is the decreasing filtration defined by
τ p An = τ≤−p (A• )n .
A biregular filtration F • A• induces an E1 -spectral sequence in the following
way.
(2.2.1) Proposition. Let F • A• be a biregularly filtered cochain complex. For
(p, q) ∈ ZZ × ZZ and r ∈ ZZ ∪ {∞}, we put
Zrpq = ker F p Ap+q → Ap+q+1 /F p+r Ap+q+1 ,
Brpq = d(F p−r Ap+q−1 ) ∩ F p Ap+q ,
pq
p+1,q−1
+ Zr−1
),
Erpq = Zrpq /(Br−1
F p H p+q (A• ) = im (H p+q (F p A• ) → H p+q (A• )) .
Then the differential d of the complex A• induces homomorphisms
pq
p+r,q−r+1
d = dpq
r : Er −→ Er
for all r
∈
ZZ in a natural way. There are canonical isomorphisms
pq
p−r,q+r−1
∼
αrpq : ker (dpq
) −→
Er+1
r )/im(dr
p−r,q+r−1
for all r ∈ ZZ. For fixed (p, q) ∈ ZZ × ZZ, the morphisms dpq
r and dr
vanish for sufficiently large r and we have natural isomorphisms
∼ E pq ,
Erpq −→
∞
r0.
Finally, there exist natural isomorphisms
pq ∼
β pq : E∞
−→ grp H p+q (A• ).
In particular, these data define a spectral sequence
E1pq ⇒ H p+q (A• ).
Remark: We have Erpq = grp Ap+q for all r ≤ 0. For r ≤ −1 the differentials
pq
pq
≥ 0
dpq
r are zero and αr = id grp Ap+q . The sequence of isomorphisms αr for r
starts with
∼ E pq .
α0pq : H q (E0p• ) = H q (grp Ap+• ) −→
1
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§2. Filtered Cochain Complexes
Proof of (2.2.1): The fact that d induces homomorphisms dpq
r and the existence of natural isomorphism αrpq can be easily verified using the definition
of the objects occurring (cf. [21], I.3.1.5). As the filtration is biregular, for
p−r,q+r−1
each fixed pair (p, q) ∈ ZZ × ZZ the morphisms dpq
vanish for
r and dr
pq
pq
for r 0 and
sufficiently large r. The natural isomorphisms Er = E∞
pq ∼
→ grp H p+q (A• ) can be read off directly from the definition, using the
E∞
biregularity of the filtration.
2
(2.2.2) Definition. We call the spectral sequence of (2.2.1) the spectral sequence associated to the biregularly filtered complex F • A• .
Examples: 1. For the trivial filtration tr• A• , the associated spectral sequence
has the following shape:
(
E1pq
=
H q (A• ), for p = 0,
0,
for p =/ 0.
pq
The spectral sequence degenerates at E1 , i.e. E1pq = E∞
for all p, q.
2. For the canonical filtration τ • A• , the associated spectral sequence has the
following shape:
(
E1pq
=
H −p (A• ), for p + q = −p,
0,
for p + q =/ −p.
pq
The spectral sequence degenerates at E1 , i.e. E1pq = E∞
for all p, q.
In both examples, the filtration on the limit terms has only one nontrivial
graded piece, which is not the case in general. Quite often, spectral sequences
arise from double complexes.
(2.2.3) Definition. A double complex A•• is a collection of objects Apq ∈ A,
p, q ∈ ZZ, together with differentials d0pq : Apq → Ap+1,q and d00p,q : Apq → Ap,q+1
such that d0 ◦ d0 = 0 = d00 ◦ d00 and d0 ◦ d00 + d00 ◦ d0 = 0. The associated total
complex
A• = tot (A•• )
is the single complex with An =
is given by the sum of the maps
L
p+q=n
Apq whose differential d : An → An+1
d = d0 + d00 : Apq −→ Ap+1,q ⊕ Ap,q+1 ,
p + q = n.
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104
Chapter II. Some Homological Algebra
Example: Let (C • , dC ) and (D• , dD ) be two complexes of abelian groups.
Their tensor product is the double complex
A•• = C • ⊗ D•
with Apq = C p ⊗ Dq , p, q
∈
ZZ, and with the following differentials:
d0pq = dpC ⊗ idDq ,
d00pq = (−1)p idC p ⊗ dqD .
As there is no danger of confusion, the associated total complex tot(C • ⊗ D• )
is usually also called the tensor product of C • and D• .
Given a double complex A•• , we have the natural filtration F p A• of the total
complex defined by
F p An =
M
Ai,n−i .
i≥p
Example: Let D• be any complex of abelian groups and let C • be the complex
given by
(
ZZ, for i = 0,
i
C =
0, for i =/ 0.
Then tot(C • ⊗D• ) = D• and the induced filtration on D• is the trivial filtration.
Let A•• be a double complex. We will assume in the following that, for
each n, there are only finitely many nonzero Apq on the line p + q = n. Then the
above filtration on A• = tot(A•• ) is biregular and induces a spectral sequence
E1pq ⇒ H p+q (A• )
converging to the cohomology of A• . The initial terms E1pq are obtained by
taking cohomology in direction q:
E1pq = H q (Ap• , d00 ).
The E1 terms give a complex
d0
d0
d0
d0
H q (A•• ) : · · · −→ H q (Ap−1,• ) −→ H q (Ap,• ) −→ H q (Ap+1,• ) −→ · · · ,
whose cohomology yields the E2 -terms:
E2pq = H p (H q (A•• )).
For reasons that will become apparent later, one often forgets the E1 -page and
calls the spectral sequence
E2pq = H p (H q (A•• )) ⇒ H p+q (tot A•• )
the spectral sequence associated to the double complex A•• .
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§2. Filtered Cochain Complexes
105
Remark: For a double complex A•• of abelian groups, the differentials
p+2,q−1
pq
may be described as follows. For each class c ∈ E2pq ,
dpq
2 : E2 → E2
there are elements x ∈ Apq and y ∈ Ap+1,q−1 with the following properties:
1) d00 x = 0, d00 y = −d0 x,
0
2) c is represented by x and dpq
2 c by d y.
(2.2.4) Lemma. Let A•• be a first quadrant double complex, i.e. Apq = 0
if p < 0 or q < 0. Assume that for each q ≥ 0 the horizontal complex
A0q → A1q → A2q → · · · is exact (i.e. trivial cohomology in dimension ≥ 1).
Then E n = H n (B • ), where B • is the complex ker(A0• → A1• ).
Proof: Setting 0Apq = Aqp , we obtain a double complex 0A•• with the same
total complex A• as A•• , and hence a new spectral sequence 0E2pq ⇒ E n with
the same limit terms E n (with different filtrations, however). Now the vertical
pq
sequences 0Ap0 → 0Ap1 → · · · are exact, so that 0E2pq = 0, i.e. 0E∞
= 0 for q > 0
n,0
= 0E2n,0 = H q (H p (A•• ))
and all p ≥ 0. This means that E n = 0F n E n ∼
= 0E∞
for p = 0, q = n.
2
Our first application of spectral sequences is the following
(2.2.5) Proposition. Let R be a commutative ring with unit. Let f : D• → D0 •
be a quasi-isomorphism of complexes of R-modules and let C • be a complex
consisting of flat R-modules. Assume that one of the following conditions is
fulfilled:
(i) C • is bounded above,
(ii) D• and D0 • are bounded below,
(iii) R is a Dedekind domain.
Then the induced homomorphism
idC • ⊗ f : tot (C • ⊗R D• ) −→ tot (C • ⊗R D0 • )
is a quasi-isomorphism.
Proof: We start with the special case D0 • = 0, i.e. D• is exact, and we have
to show that tot (C • ⊗R D• ) is exact.
Let us first assume that D• is bounded above. Then, if (i) or (ii) holds, the
natural filtration on tot (C • ⊗R D• ) is biregular, and for the associated spectral
sequence
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106
Chapter II. Some Homological Algebra
we have
E1pq = H q (C p ⊗ D• ) = 0
as C p is flat for all p. Hence all limit terms H p+q (tot (C • ⊗R D• )) vanish.
If D• is not bounded above, we write D• = lim τ≤n (D• ) and obtain
−→ n
H k (tot (C • ⊗R D• )) = lim
H k (tot (C • ⊗R τ≤n (D• ))) = 0
−→
n
for all k.
Keeping the assumption D0• = 0, we now assume that R is a Dedekind
domain. Then each submodule of a flat R-module is again flat. Hence the
complexes τ≤n (C • ) consist of flat R-modules for all n, and we obtain
H k (tot (C • ⊗R D• )) = lim
H k (tot (τ≤n (C • ) ⊗R D• )) = 0
−→
n
for all k. This settles the case D0• = 0.
In the general case we consider the mapping cone C(f )• of f . It is defined by
C(f )n = Dn+1 ⊕ D0n
with differential d((a, b)) = (−d(a), d0 (b) + f (a)) for a
natural long exact sequence
∈
Dn+1 , b
∈
D0n . The
· · · −→ H n (D• ) −→ H n (D0• ) −→ H n (C(f )• ) −→ · · ·
shows that C(f )• is exact, as f is a quasi-isomorphism. Moreover, the construction of the mapping cone commutes with tensor products, i.e.
C(idC • ⊗ f )• = tot (C • ⊗ C(f )• ).
By the first part of the proof, tot (C • ⊗ C(f )• ) is exact if one of the conditions
(i)–(iii) is satisfied. Hence C(idC • ⊗ f )• is exact and therefore idC • ⊗ f is a
quasi-isomorphism.
2
Exercise 1. Calculate the spectral sequence associated to the stupid filtration
n
A , for n ≥ p,
• n
σ≥p (A ) =
0, for n ≤ p − 1.
Does it degenerate?
Exercise 2. Let p be a prime number and R = ZZ/p2 ZZ. Consider the exact and flat complex
of R-modules
p
p
p
C • = · · · −→ ZZ/p2 ZZ −→ ZZ/p2 ZZ −→ · · · .
Show that H i (tot (C • ⊗R C • )) =/ 0 for all i.
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107
§3. Degeneration of Spectral Sequences
§3. Degeneration of Spectral Sequences
In this section we investigate the degeneration of the spectral sequence
attached to a biregularly filtered cochain complex. The first and easiest case is
degeneration at E1 .
(2.3.1) Theorem. For the spectral sequence
E1pq ⇒ H p+q (A• )
associated to a biregularly filtered complex F • A• the following assertions are
equivalent:
(i) The spectral sequence degenerates at E1 .
(ii) For all n, p we have F p An ∩ d(An−1 ) = d(F p An−1 ).
(iii) For all n, p the natural map H n (F p A• ) → H n (A• ) is injective.
If, moreover, the maps in (iii) are split-injections, we obtain a (non-canonical)
splitting
M
E1pq .
H n (A• ) ∼
=
p+q=n
Proof: Without loss of generality, we may work in the category of modules over a ring. Assume that (i) holds. For sufficiently large p, we have
F p An−1 = 0, hence assertion (ii) holds for p 0. We fix n ∈ ZZ and proceed
by decreasing induction on p. Let x ∈ F p An ∩ d(An−1 ). As the filtration is
biregular, x ∈ d(F m An−1 ) for some m, which we choose as large as possible
(m = ∞ allowed). We assume that m = p−r for some r ≥ 1, and show that this
yields a contradiction. Let y ∈ F p−r An−1 be a pre-image of x. By construction,
y ∈ Zrp−r,n−1−p+r . By assumption, the differential dr : Erp−r,n−1−p+r → Erp,n−p
p,n−p
p+1,n−p−1
is the zero map. Hence x = dy ∈ Br−1
+ Zr−1
, i.e. we may write x in
the form
x = dy 0 + x0 ,
y 0 ∈ F p−r+1 An−1 , x0 ∈ F p+1 An .
As x and dy 0 are coboundaries, the same holds for x0 . By our inductive assumption, we have x0 ∈ d(F p+1 An−1 ), which implies x ∈ F p An ∩ d(F p−r+1 An−1 ),
contradicting the maximality of m = p − r. Hence m ≥ p, showing that
x ∈ d(F p An−1 ). This proves (i)⇒(ii). The implication (ii)⇒(i), as well as the
equivalence (ii)⇔(iii) are elementary.
Finally, assume that (i)–(iii) hold. By definition, F p H n (A• ) is the image
of the natural map H n (F p A• ) → H n (A• ), which is a split-injection. As the
filtration on H n (A• ) is finite, we recursively obtain a splitting
M
H n (A• ) ∼
F p H n (A• )/F p+1 H n (A• )
=
p
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Chapter II. Some Homological Algebra
into a finite direct sum. By (i), we have
p,n−p
= grp H n (A• ) = F p H n (A• )/F p+1 H n (A• ),
E1p,n−p = E∞
2
showing the last assertion of the proposition.
By a formal reindexing procedure, we can displace a spectral sequence in the
pq
following sense: Assume we are given an Em -spectral sequence Em
⇒ E n.
Putting
2p+q,−p
for r
Ẽ n = E n , F p Ẽ n = F p+n E n and Ẽrpq = Er+1
≥
m − 1,
we obtain an Em−1 -spectral sequence converging to the same limit terms, but
with a shifted filtration, which we call the displaced spectral sequence. It is a
remarkable fact that, if the spectral sequence E arises from a biregular filtered
cochain complex, then the displaced spectral sequence Ẽ arises from another
filtration on the same complex, the displaced filtration. This will be useful in
showing that a spectral sequence degenerates at E2 , just by showing that the
displaced spectral sequence Ẽ satisfies the conditions of (2.3.1).
(2.3.2) Definition. Let F • A• be a biregular filtered cochain complex. The
filtration
Dis(F )p An = Z1p+n,−p = {a ∈ F p+n An | da ∈ F p+n+1 An+1 }
is called the displaced filtration. We denote the complex A• , together with
the filtration Dis(F ), by Dis(A• ).
One easily verifies that Dis(A• ) is a filtered complex:
p+n+1,−p
d(Z1p+n,−p ) ⊆ F p+1+n An+1 ∩ ker(d) ⊆ Z∞
⊆
Z1p+n+1,−p ,
and the filtration Dis(F ) is obviously biregular.
Example: Displacing the trivial filtration tr on A• , we obtain the canonical
filtration τ :
Dis(tr)p An = Z1p+n,−p = {a ∈ trp+n An | da ∈ trp+n+1 An+1 }
An ,
for n ≤ −p − 1,
n
n+1
ker(A → A ), for n = −p,
=


0,
for n ≥ −p + 1,



= τ p An .
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109
§3. Degeneration of Spectral Sequences
(2.3.3) Proposition. For all r
≥
1, there are natural isomorphism
∼ E 2p+q,−p (A• )
Erpq (Dis(A• )) −→
r+1
commuting with the corresponding differentials. The displaced spectral sequence is the spectral sequence associated to the displaced filtration.
Proof: For r
≥
0 we have
pq
DisF Zr
=
n
x ∈ Dis(F )p Ap+q | dx ∈ Dis(F )p+r Ap+q+1
o
= {x ∈ F 2p+q Ap+q | dx ∈ F 2p+q+r+1 Ap+q+1 }
=
2p+q,−p
.
F Zr+1
Analogously,
pq
DisF Br
= d(Dis(F )p−r Ap+q−1 ) ∩ Dis(F )p Ap+q
= d(F 2p+q−r−1 Ap+q−1 ) ∩ F 2p+q Ap+q
=
2p+q,−p
.
F Br+1
pq
p+1,q−1
+ Zr−1
). Hence the natural identificaBy definition, Erpq = Zrpq /(Br−1
∼
2p+q,−p
pq
(A• ) for r ≥ 1.
tions above induce isomorphisms Er (Dis(A• )) → Er+1
pq
Therefore, the Er -terms of the displaced spectral sequence and of the spectral
sequence associated to the displaced filtration are canonically isomorphic. The
same holds for the limit terms, which can easily be seen from their definitions.
2
As an application, we obtain the following degeneration result. It should not
be mistaken for the well-known Künneth-formula, which arises from another
filtration on the tensor product (see the exercise below).
(2.3.4) Theorem. Let R be a Dedekind domain and let C • and D• be complexes of R-modules such that the natural filtration on their tensor product
is biregular.∗) If C • consists of flat (i.e. torsion-free) R-modules, then the
spectral sequence of the double complex A•• = C • ⊗R D• degenerates at E2 .
Furthermore, we have a non-canonical splitting
H n (tot (C • ⊗R D• )) ∼
=
M
E2pq .
p+q=n
∗) e.g. both complexes are bounded above, or both complexes are bounded below, or one of
the complexes is bounded.
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Chapter II. Some Homological Algebra
Proof: Let A• = tot(C • ⊗R D• ), and let F p An = i≥p C i ⊗R Dn−i be the
natural filtration. We want to show that the spectral sequence of the double
complex degenerates at E2 . By (2.3.3), it suffices to show that the spectral
sequence associated to the displaced filtration degenerates at E1 . Thus, by
(2.3.1), we have to show that the natural maps
L
H n (Dis(F )p A• ) −→ H n (A• )
are split-injections for all p and n. Using the flatness of C • , we obtain
Dis(F )p An = ker F p+n An → F p+n An+1 /F p+n+1 An+1 ,
=
L
i>p+n
C i ⊗R Dn−i
⊕ C p+n ⊗R ker(d−p
D )
n
= tot C • ⊗R τ≤−p (D• ) .
As the filtration on H n (A• ) is finite, it therefore remains to show that for all
n, m ∈ ZZ the natural map
H n tot(C • ⊗R τ≤m (D• )) −→ H n tot(C • ⊗R τ≤m+1 (D• ))
is a split-injection. Let X • be a complex consisting of projective R-modules
together with a quasi-isomorphism X • → τ≤m+1 (D• ). Using the flatness of C • ,
we obtain a commutative diagram
tot(C • ⊗RHIJG τ≤m (X • ))
tot(C • ⊗R X • )
quasi-iso
quasi-iso
tot(C • ⊗R τ≤m (D• ))
tot(C • ⊗R τ≤m+1 (D• )) .
By (2.2.5)(iii) the horizontal maps are quasi-isomorphisms. Since X • is a
complex consisting of projective R-modules, the inclusion of complexes
τ≤m (X • ) ,→ X •
has a section for each m ∈ ZZ . Indeed, by our assumption on R, d(X m ) ⊆ X m+1
is a projective R-module, and the short exact sequence
0 −→ ker(dm ) −→ X m −→ d(X m ) −→ 0
splits. Using any section s: X m → ker(dm ) in dimension m, and the obvious
maps in the other dimensions, we obtain a splitting of τ≤m (X • ) ,→ X • .
Therefore the left vertical complex homomorphism in the diagram above
has a section, which finishes the proof.
2
The technique of displacing, as well as the results (2.3.1) and (2.3.3), are
due to P. DELIGNE, cf. [34]. The statement of (2.3.4) is implicitly contained
in the article [98] by U. JANNSEN, and we adopted his idea of proof.
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111
§4. The Hochschild-Serre Spectral Sequence
Exercise (Künneth-formula): Let R be a Dedekind domain and let C • and D• be complexes
of R-modules. Assume that C • consists of flat R-modules. Consider the biregular filtration
on A• = tot(C • ⊗R D• ) defined by
 L
i
j

for p ≤ −1,

 Li+j=n C ⊗R D ,
p
n
i
•
j
F (A ) =
i+j=n Z (C ) ⊗R D , for p = 0,



0,
for p ≥ 1,
and show the following assertions regarding the associated spectral sequence:
L
pq
i
•
j
•
/ 0, −1.
(i) E2p,q = i+j=q TorR
−p (H (C ), H (D )); in particular E2 = 0 for p =
(ii) For each n we obtain a short exact sequence
M
M
0→
H i (C • ) ⊗R H j (D• ) → H n (A• ) →
TorR1 (H i (C • ), H j (D• )) → 0.
i+j=n
i+j=n+1
Moreover, these sequences split (non-canonically).
§4. The Hochschild-Serre Spectral Sequence
Relevant for the cohomology of profinite groups is the following
(2.4.1) Theorem. Let G be a profinite group, H a closed normal subgroup
and A a G-module. Then there exists a first quadrant spectral sequence
E2pq = H p (G/H, H q (H, A)) ⇒ H p+q (G, A).
It is called the Hochschild-Serre spectral sequence.
.
Proof: To the standard resolution 0 → A → X of the G-module A, we apply
the functor H 0 (H, −), and get the complex
H 0 (H, X 0 ) −→ H 0 (H, X 1 ) −→ H 0 (H, X 2 ) −→ · · ·
of G/H-modules. For each H 0 (H, X q ), we consider the cochain complex
.
H 0 (H, X q )G/H −→ C (G/H, H 0 (H, X q ))
and we put
C pq = C p (G/H, H 0 (H, X q )) = X p (G/H, X q (G, A)H )G/H ,
p, q
≥
0.
We make C •• into a (anti-commutative) double complex by using the following
differentials: We let
d0pq : C pq −→ C p+1,q
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Chapter II. Some Homological Algebra
.
be the differential of the complex X (G/H, X q (G, A)H )G/H at the place p.
Further, we define
d00pq : C pq −→ C p,q+1
.
as (−1)p times the differential of the complex X p (G/H, X (G, A)H )G/H at the
place q. Then (C •• , d0 , d00 ) is a double complex and we define the HochschildSerre spectral sequence as the associated spectral sequence
E2pq ⇒ E n .
We compute the terms E2pq and E n . By definition, E2pq = H p (H q (C •• )). We
have H q (H 0 (H, X )) = H q (H, A) (see p.34). The functor C p (G/H, −) is
exact (I §3, ex.1). Therefore
H q (C p ) = H q (C p (G/H, H 0 (H, X ))) = C p (G/H, H q (H 0 (H, X )))
= C p (G/H, H q (H, A)),
.
.
hence
.
.
.
E2pq = H p (C (G/H, H q (H, A))) = H p (G/H, H q (H, A)).
.
As for the limit terms, we note that for every q ≥ 0 the complexes C q
= C (G/H, H 0 (H, X q )) are exact. In fact, every X q is an induced G-module,
hence H 0 (H, X q ) is an induced, and thus acyclic, G/H-module by (1.3.6)
and (1.3.7), i.e. H p (C q ) = H p (G/H, H 0 (H, X q )) = 0 for p > 0. By lemma
(2.2.4), we obtain E n = H n (B ), where B is the complex
.
.
.
.
.
.
.
.
.
B = ker(C 0 (G/H, (X )H ) → C 1 (G/H, (X )H )) = ((X )H )G/H = (X )G .
Therefore
.
E n = H n ((X )G ) = H n (G, A).
2
From (2.1.4) follows the
(2.4.2) Corollary. If H q (H, A) = 0 for q > 0, then
H n (G/H, AH ) ∼
= H n (G, A).
Another consequence is the five term exact sequence
0 −→ H 1 (G/H, AH ) −→ H 1 (G, A) −→ H 1 (H, A)G/H
−→ H 2 (G/H, AH ) −→ H 2 (G, A),
which we proved in I §6 in an elementary way, but with some difficulty. It still
requires, however, careful checking to show that the maps are the inflation,
restriction and transgression respectively.
For inf and res, this identification is given in the available literature (e.g.
[128]). For the transgression, we give the proof here.
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§4. The Hochschild-Serre Spectral Sequence
(2.4.3) Theorem. The differential
1
G/H
d0,1
−→ H 2 (G/H, AH )
2 : H (H, A)
is the transgression tg as defined in (1.6.6).
Proof (TH. MOSER, J. STIX): In order to calculate H 1 (H, A) we use the
acyclic resolution A → X (G, A) of the G-module A. Thus an element
z ∈ H 1 (H, A)G/H is represented by an H-invariant 1-cocycle x : G × G → A.
The invariance of z under G/H implies that for ρ, σ ∈ G the cocycles ρx and
σx differ by a 1-coboundary, i.e. there is a map
.
b : G × G −→ X 0 (G, A)H ,
such that
(σ, ρ) 7→ bσ,ρ ,
d(bσ,ρ ) = ρx − σx ,
.
where d denotes the differential of X (G, A). We obtain
bσ,ρ (τ1 ) − bσ,ρ (τ0 ) = (ρx)τ0 ,τ1 − (σx)τ0 ,τ1
(∗)
for all σ, ρ, τ0 , τ1 ∈ G. Therefore we may assume that b is G-invariant and
b1,1 = 0. Furthermore, since τ x = x for τ ∈ H, we may also assume that b
factors through G/H × G/H. Then, for all σ, ρ, γ ∈ G/H,
bρ,γ (τ ) − bσ,γ (τ ) + bσ,ρ (τ )
is an element of the module AH independent of τ
∂b: (G/H)3 → X 0 (G, A)H , given by
∈
G, and so the 2-cocycle
∂bσ,ρ,γ = bρ,γ − bσ,γ + bσ,ρ ,
is constant with value in AH . By the remark on page 105 it represents the
image of z under d0,1
2 . The associated inhomogeneous 2-cocycle, which also
0,1
represents d2 (z), is
a : G/H × G/H −→ AH ,
(σ, ρ) 7→ aσ,ρ ,
where
aσ,ρ = ∂b1,σ,σρ (ζ) = bσ,σρ (ζ) − b1,σρ (ζ) + b1,σ (ζ)
(∗∗)
with ζ
∈
G arbitrary.
We now represent tg(z) as follows. We restrict x to H × H, and pass to
the associated inhomogeneous 1-cocycle x0 : H → A, x0 (τ ) = x1,τ , which also
represents z. Consider then the function
y : G −→ A,
σ 7→ yσ = x1,σ + b1,σ (σ).
We will show that y satisfies the properties (i), (ii), (iii) of the proof of proposition (1.6.6),
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Chapter II. Some Homological Algebra
(i)
y|H = x0 ,
(ii)
yστ = yσ + σyτ
for σ
∈
G, τ
∈
H,
(iii)
yτ σ = yτ + τ yσ for σ ∈ G, τ ∈ H,
hence tg(z) = [∂y]. Since b1,τ = b1,1 = 0 for τ ∈ H, property (i) follows. Let
σ, ρ ∈ G. Then, by (∗) and (∗∗) and since x is a cocycle, we obtain
(∂y)σ,ρ = (dx)1,σ,σρ + (σx)σ,σρ − xσ,σρ
+b1,σ (σ) − b1,σ (σρ) + (∂b)1,σ,σρ (σρ)
= (∂b)1,σ,σρ (σρ) = aσ,ρ .
If σ ∈ H or ρ ∈ H, then the expression above is zero and so (ii) and (iii) follow.
Now, for arbitrary σ, ρ ∈ G, the equality above shows that tg(z) = [∂y] = [a] =
d01
2
2 . This proves the theorem.
A subtle and useful relation of the Hochschild-Serre spectral sequence to
the cup-product is obtained as follows. Let G be a profinite group, H an open
normal subgroup of G and H 0 the closure of the commutator subgroup of H.
Let A be a G-module on which H acts trivially. We then have, for p > 0, two
canonical homomorphisms
(∗)
d2 , u∪ : H p−1 (G/H, H 1 (H, A)) −→ H p+1 (G/H, A)
which are defined as follows. The first map d2 is the differential d2p−1,1 of the
Hochschild-Serre spectral sequence
E2pq = H p (G/H, H q (H, A)) ⇒ H p+q (G, A).
On the other hand, the group extension
1 −→ H ab −→ G/H 0 −→ G/H −→ 1
defines a cohomology class u ∈ H 2 (G/H, H ab ). Using the equality
H 1 (H, A) = Hom(H ab , A), we obtain a canonical pairing
H ab × H 1 (H, A) −→ A ,
which induces a cup-product
∪
H 2 (G/H, H ab ) × H p−1 (G/H, H 1 (H, A)) −→ H p+1 (G/H, A).
The second map u∪ is given by x 7→ u ∪ x.
(2.4.4) Theorem. Let A be a G-module and H an open normal subgroup of
G which acts trivially on A. Then, for p > 0, the maps
d2 , u∪ : H p−1 (G/H, H 1 (H, A)) −→ H p+1 (G/H, A)
are the same up to sign, i.e. d2 (x) = −u ∪ x.
In particular, the transgression tg : H 1 (H, A)G/H → H 2 (G/H, A) is given
by tg(x) = −u ∪ x.
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§4. The Hochschild-Serre Spectral Sequence
Remark: The statement of (2.4.4) remains true for an arbitrary closed normal
subgroup H of G, but then one has to use the continuous cohomology class
2
u ∈ Hcts
(G/H, H ab ) representing the group extension 1 → H ab → G/H 0 →
G/H → 1 (cf. (2.7.7)).
Proof: Suppose first that G is a finite group. Then the projection H → H ab
may be regarded as a G/H-invariant 1-cocycle of H, i.e. as an element ε ∈
H 0 (G/H, H 1 (H, H ab )). From the spectral sequence
E2pq = H p (G/H, H q (H, H ab )) ⇒ H p+q (G, H ab ) ,
we obtain the differential
1
ab G/H
−→ H 2 (G/H, H ab ),
d0,1
2 : H (H, H )
which by (2.4.3) is the transgression tg as defined in (1.6.6). We prove
tg(ε) = −u (in the additive notation of H 2 (G/H, H ab )).
Let s : G/H → G be a section of the projection G → G/H, σ 7→ σ̄. Define
the 1-cochain y : G → H ab by
y(σ) = σ(sσ̄)−1 mod H 0 .
Then y|H = ε and
(∂y)(σ1 , σ2 ) ≡ y(σ1 σ2 )−1σ1y(σ2 )y(σ1 )
≡ s(σ1 σ2 )σ2−1 σ1−1 σ1 σ2 (sσ2 )−1 σ1−1 σ1 (sσ1 )−1
≡ [(sσ1 )(sσ2 )s(σ1 σ2 )−1 ]−1 mod H 0 ,
showing that ∂y depends only on the classes σ1 , σ2 ∈ G/H, and hence tg(ε) =
[∂y] by definition of tg. But the function in square brackets is a 2-cocycle
which represents the class u, hence tg(ε) = −u.
The differential
p
1
p+2
dp,1
(G/H, AH )
2 : H (G/H, H (H, A)) −→ H
is obtained from the part
C p (G/H, XLKM 1 (G, A)H )
∂0
C p+1 (G/H, X 1 (G, A)H )
∂ 00
C p+1 (G/H, X 0 (G, A)H )
∂0
C p+2 (G/H, X 0 (G, A)H )
of the double complex C pq (A) = C p (G/H, X q (G, A)H ) as follows. Let z ∈
H p (G/H, H 1 (H, A)). z is given by an element α ∈ C p,1 (A) such that ∂ 00 α = 0
and that the induced function ᾱ : (G/H)p+1 → H 1 (X (G, A)H ) = H 1 (H, A) is
a p-cocycle representing z. This means that there exists a β ∈ C p+1,0 such that
0
p+2,0
∂ 0 α = ∂ 00 β. The image dp,1
.
2 (z) is then represented by the cocycle ∂ β ∈ C
.
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Chapter II. Some Homological Algebra
This process may also be interpreted as follows. From the complex
0 → AH → X 0 (G, A)H → X 1 (G, A)H → X 2 (G, A)H , we obtain the exact sequence of G/H-modules
(1)
0 −→ AH −→ X 0 (G, A)H −→ Z −→ H 1 (H, A) −→ 0
.
with Z = Z 1 (X (G, A)H ). Splitting it up into two short exact sequences
(2)
0 −→ I(A) −→ Z −→ H 1 (H, A) −→ 0,
(3)
0 −→ AH −→ X 0 (G, A)H −→ I(A) −→ 0,
we obtain two δ-homomorphisms
δ
δ
H p (G/H, H 1 (H, A)) −→ H p+1 (G/H, I(A)) −→ H p+2 (G/H, AH ),
with dp,1
= δ ◦ δ. In fact, the element α above is a lifting α : (G/H)p+1
2
→ Z 1 (G, A)H of the p-cocycle ᾱ representing z, hence δz is represented by
the (p + 1)-cocycle ∂ 0 α. Since ∂ 0 α = ∂ 00 β, β : (G/H)p+2 → X 0 (G, A)H is a
cochain which lifts ∂ 0 α, hence ∂ 0 β represents δ([∂ 0 α]) = δδ(z), showing that
δδ(z) = dp,1
2 (z).
Let us abbreviate the sequences (2) and (3) by
0 −→ S 0 (A) −→ S(A) −→ S 00 (A) −→ 0,
0 −→ T 0 (A) −→ T (A) −→ T 00 (A) −→ 0,
with S 0 = T 00 . Replacing A by the G-module H ab , we obtain exact sequences
of G/H-modules
0 −→ S 0 (H ab ) −→ S(H ab ) −→ S 00 (H ab ) −→ 0,
0 −→ T 0 (H ab ) −→ T (H ab ) −→ T 00 (H ab ) −→ 0.
Setting B = Hom(H ab , A), we have the pairings
S(H ab ) × B −→ S(A),
T (H ab ) × B −→ T (A),
which induce pairings S 0 (H ab ) × B → S 0 (A), S 00 (H ab ) × B → S 00 (A), and
similarly for T . We now assume that H acts trivially on A, i.e. Hom(H ab , A) =
H 1 (H, A). We apply proposition (1.4.3) twice and obtain the commutative
diagram
H 0 (G/H,NPQOVTUSR S 00 (H ab )) × H p (G/H, B)
∪
δ
H 1 (G/H, S 0 (H ab )) × H p (G/H, B)
δ
∪
δ
H 2 (G/H, T 0 (H ab )) × H p (G/H, B)
H p (G/H, S 00 (A))
H p+1 (G/H, S 0 (A))
δ
∪
H p+2 (G/H, T 0 (A)).
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§4. The Hochschild-Serre Spectral Sequence
In the upper pairing, we have S 00 (H ab ) = H 1 (H, H ab ), S 00 (A) = H 1 (H, A) and
ε ∪ z = z (recall that z ∈ H p (G/H, Hom(H ab , A)) = H p (G/H, H 1 (H, A))).
Since δδε = d0,1
2 (ε) = tg(ε) = −u, it follows that
dp,1
2 (z) = δδ(z) = δδ(ε ∪ z) = δδε ∪ z = −u ∪ z.
This proves the theorem for a finite group G.
Now let G be an arbitrary profinite group. We denote by G0 the commutator
subgroup of G. We let W run through the open normal subgroups of G which
are contained in H. Setting G = G/W , H = H/W , we have a commutative
exact diagram
1WXYZ[\]^_`a
H ab
G/H 0
G/H
1
1
H ab
G/H 0
G/H
1.
The lower group extension defines an element uW ∈ H 2 (G/H,H ab ) which is
π
the image of u under H 2 (G/H, H ab ) → H 2 (G/H,H ab ) (see (3.6.1)). On the
other hand, we have a commutative diagram
edcb 1 (H, A))
H p (G/H, H
d2
H p+2 (G/H, A)
d2
H p+2 (G/H, A).
inf
H p (G/H, H 1 (H, A))
We have shown that d2 xW = −uW ∪ xW for xW
diagram
ab
hgfij
H 2 (G/H,H
) × H p (G/H, H 1 (H, A))
π
∈
H p (G/H, H 1 (H̄, A)). The
∪
H p+2 (G/H, A)
∪
H p+2 (G/H, A)
inf
H 2 (G/H, H ab ) × H p (G/H, H 1 (H, A))
is commutative because of the commutativity of the cup-product with the
inflation. So, d2 inf xW = d2 xW = −uW ∪ xW = −πu ∪ xW = −u ∪ inf xW .
Since H 1 (H, A) = lim
H 1 (H/W, A), each x ∈ H p (G/H, H 1 (H, A)) is of
−→ W
the form x = inf xW for some W . This completes the reduction to the case of
a finite group G.
2
If 1 → H → G → G/H → 1 is a split group extension and A is a Gmodule on which H acts trivially, then (2.4.4) shows that the differentials d∗,1
2
are trivial. However, the Hochschild-Serre spectral sequence does, in general,
not degenerate at E2 , even if A is a trivial G-module, see [46], example on
page 83. But, we have the following
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Chapter II. Some Homological Algebra
(2.4.5) Proposition. Let 1 → H → G → G/H → 1 be a split exact sequence
of profinite groups, and let A be a discrete G-module on which H acts trivially.
Then
inf : H ∗ (G/H, A) −→ H ∗ (G, A)
is a monomorphism onto a direct summand, and all differentials into the
horizontal edge of the Hochschild-Serre spectral sequence for A vanish, i.e.
dr∗,r−1 = 0 for all r ≥ 2.
Proof: Let ι: G/H ,→ G be a splitting of the projection π: G G/H, i.e.
πι = id. Since AH = A, the edge homomorphisms
∗,0 ⊆
H ∗ (G/H, A) = E2∗,0 −→ E∞
H ∗ (G, A)
∗,0
∗,0
=
for r ≥ 2. Since Er+1
are split injections, so E2∗,0 = Er∗,0 = E∞
∗,0
∗,r−1
∗,r−1
Er /imdr , we obtain dr
= 0.
2
In case of a direct product, the Hochschild-Serre spectral sequence degenerates at E2 . We follow the proof given by U. JANNSEN, see [98].
(2.4.6) Theorem. Let G and H be profinite groups and let A be a discrete
H-module, regarded as a (G × H)-module via trivial action of the group G.
Then the Hochschild-Serre spectral sequence
E2pq = H p (G, H q (H, A)) ⇒ H n (G × H, A)
degenerates at E2 . Furthermore, it splits in the sense that there is a decomposition
M
H p (G, H q (H, A)).
H n (G × H, A) ∼
=
p+q=n
Proof: For a trivial G-module B, we have a natural isomorphism of complexes
C • (G, B) ∼
= C • (G, ZZ) ⊗ B.
This is easily verified if G is finite, and the result for general G follows by a
straightforward limit process. Therefore, under the assumptions of the theorem,
the Hochschild-Serre spectral sequence is the spectral sequence associated to
the double complex
C • (G, ZZ) ⊗ X • (G × H, A)H .
As the complex C • (G, ZZ) consists of flat ZZ-modules, the result follows from
(2.3.4).
2
Remark: The decomposition in the theorem is non-canonical: it cannot be
made functorial in A. However, for a fixed H-module A, it can be made
functorial in G. For a proof of these assertions see [98].
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§4. The Hochschild-Serre Spectral Sequence
Exercise 1. Show that in the Hochschild-Serre spectral sequence, the edge morphisms
H n (G/H, AH ) −→ H n (G, A) −→ H n (H, A)G/H
are the inflation and the restriction.
Exercise 2. From the exact sequence 0 → A → IndG (A) → A1 → 0, one obtains the four term
exact sequence
1
0 −→ AH −→ IndG (A)H −→ AH
1 −→ H (H, A) −→ 0
of G/H-modules, and hence a homomorphism
δ 2 : H p (G/H, H 1 (H, A)) −→ H p+2 (G/H, AH ).
p,1
p+2,0
Show that δ 2 is the differential dp,1
.
2 : E2 → E2
Exercise 3. Let E(G, H, A) denote the Hochschild-Serre spectral sequence
E2pq = H p (G/H, H q (H, A)) ⇒ H n (G, A).
If G0 is an open subgroup of G and H 0 = H ∩ G0 , then we have two morphisms of spectral
res
sequences E(G,lk H, A) cor E(G0 , H 0 , A).
Exercise 4. (i) Assume that H q (H, A) = 0 for q > 1. Then we have an exact sequence
0mnopqrstuv
H 1 (G/H, H 0 (H, A))
H 1 (G, A)
H 0 (G/H, H 1 (H, A))
H 2 (G/H, H 0 (H, A))
H 2 (G, A)
H 1 (G/H, H 1 (H, A))
H 3 (G/H, H 0 (H, A))
H 3 (G, A)
H 2 (G/H, H 1 (H, A))
(ii) Assume that cd G/H
≤
.
1. Then we have exact sequences
0 −→ H (G/H, H n−1 (H, A)) −→ H n (G, A) −→ H n (H, A)G/H −→ 0
1
for all n ≥ 1 and all discrete G-modules A.
Exercise 5. If A×B → C is a pairing of G-modules, then for the terms Erpq of the HochschildSerre spectral sequence, we have a cup-product
0 0
∪
0
0
Erpq (A) × Erp q (B) −→ Erp+p ,q+q (C)
such that
dr (α ∪ β) = (dr α) ∪ β + (−1)p+q α ∪ dr (β).
Exercise 6. (Künneth-formula) Let G and H be profinite groups and let A be a discrete
H-module, regarded as a (G × H)-module via trivial action of the group G. Show that there
exist natural short exact sequences for all n
M
M
0→
H i (G, ZZ) ⊗ H j (H, A) → H n (G × H, A) →
Tor Z1Z (H i (G, ZZ), H j (H, A)) → 0.
i+j=n
i+j=n+1
Moreover, these sequences split.
Hint: Consider the double complex C • (G, ZZ) ⊗ X • (G × H, A)H , which, by the proof of
(2.4.6), calculates the cohomology H ∗ (G × H, A). Then use the result of the exercise at the
end of §3.
Exercise 7. Let G and H be profinite groups and let F be a field, considered as a trivial
module. Show that
M
∼
H n (G × H, F ) =
H i (G, F ) ⊗F H j (H, F ).
i+j=n
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Chapter II. Some Homological Algebra
§5. The Tate Spectral Sequence
Another spectral sequence, due to J. TATE, which in a sense is dual to the
Hochschild-Serre spectral sequence, is obtained as follows. Let G be a profinite
group. For an abelian group M we set
M ∗ = Hom(M, Q/ZZ),
which is equipped with the natural G-structure g(φ)(m) = g(φ(g −1 (m))), if M
is a G-module. Let A be a G-module. For two open subgroups V ⊆ U of G,
we have the maps
cor ∗ : H n (U, A)∗ −→ H n (V, A)∗ ,
dual to the corestriction, by which the family (H n (U, A)∗ ) becomes a direct
system of abelian groups.
(2.5.1) Definition. Let G be a profinite group, H a closed subgroup and A a
G-module. Then, for every n ≥ 0, we set
Dn (H, A) = lim H n (U, A)∗ ,
−→
U ⊇H
where U runs through the open subgroups of G containing H.
If H is open, then Dn (H, A) = H n (H, A)∗ . If H is a normal closed subgroup
of G, then it suffices to let U run through the normal open subgroups of G
containing H. Dn (H, A) is then a G/H-module. As the functors X 7→ X ∗
and lim
are exact, we see that the family (Dn (H, −))n≥0 is a contravariant ∗)
−→
δ-functor on Mod(G). We set
Dn (A) = Dn ({1}, A).
(2.5.2) Definition. For a G-module A and an integer n ≥ 0, we write
cd(G, A) ≤ n
if H q (H, A) = 0 for all q > n and all closed subgroups H of G. (The letters
“ cd” stand for “cohomological dimension”.)
We call the following spectral sequence the Tate spectral sequence.
∗) This means that the arrows in the usual exact cohomology sequence are reversed.
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§5. The Tate Spectral Sequence
(2.5.3) Theorem. If cd(G, A) ≤ n, then for every closed normal subgroup H,
there exists a first quadrant spectral sequence
E2pq = H p (G/H, Dn−q (H, A)) ⇒ H n−(p+q) (G, A)∗ .
In particular, for H = 1, we have a spectral sequence
E2pq = H p (G, Dn−q (A)) ⇒ H n−(p+q) (G, A)∗ .
.
Proof: We consider the standard resolution A → X of the G-module A and
set Z i = ker(X i → X i+1 ). Splitting up the complex
.
X̃ : 0 → X 0 → X 1 → X 2 → · · · → X n−1 → Z n → 0
into short exact sequences and, recalling that the G-modules X i are cohomologically trivial, we obtain for r > 0
H r (H, Z n ) ∼
= H r+1 (H, Z n−1 ) ∼
= ··· ∼
= H r+n (H, A).
In particular, H r (H, Z n ) = 0 for r > 0, since cd(G, A) ≤ n, i.e. Z n is a
cohomologically trivial G-module.
Let U be an open normal subgroup of G. We apply to the complex X̃
first the functor H 0 (U, −) and then the functor Hom(−, Q/ZZ), and obtain a
complex
0 −→ Y 0 −→ Y 1 −→ Y 2 −→ · · · −→ Y n −→ 0,
.
where Y q = H 0 (U, X̃ n−q )∗ for q ≥ 0, and in particular, Y 0 = H 0 (U, Z n )∗ . This
is a complex of G/U -modules, which by (1.8.2) and (1.8.5) are cohomologically trivial, since X n−q and Z n are cohomologically trivial G-modules. Since
the functor Hom(−, Q/ZZ) is exact, we obtain
.
.
.
H q (Y ) = H q (H 0 (U, X̃ n− )∗ ) = H q (H 0 (U, X̃ n− ))∗ = H n−q (U, A)∗
for all q.
For each Y q , we consider the cochain complex C (G/U, Y q ) and obtain
a double complex C pq = C p (G/U, Y q ), p, q ≥ 0. In order to make it anticommutative, we multiply the differentials (p, q) → (p, q + 1) by (−1)p , cf.
the proof of (2.4.1). As described in §2, this double complex yields a spectral
sequence
E2pq ⇒ E p+q .
.
We compute the initial terms E2pq = H p (H q (C •• )). The functor C p (G/U, −) is
exact (I §3, ex.1), so that
H q (C •• ) = H q (C (G/U, Y )) = C (G/U, H q (Y ))
.
.
.
.
.
= C (G/U, H n−q (U, A)∗ ),
hence
.
E2pq = H p (C (G/U, H n−q (U, A)∗ )) = H p (G/U, H n−q (U, A)∗ ).
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Chapter II. Some Homological Algebra
As for the limit terms E p+q , we note that for each q
q
0 the complex
C (G/U, Y ) −→ C (G/U, Y ) −→ C (G/U, Y q ) −→ · · ·
0
q
≥
1
2
is exact; its homology groups are the groups H p (G/U, Y q ), which are zero for
p > 0, since Y q is cohomologically trivial.
Therefore, setting B = ker(C 0 → C 1 ), we have by (2.2.4)
.
E p+q = H p+q
. .
(B . ) = H (H (G/U, Y . )).
p+q
0
Since Y q is cohomologically trivial, we have Ĥ i (G/U, Y q ) = 0 for all i. By
(1.2.6), we obtain
H 0 (G/U, Y q ) = Hom((X̃ n−q )U , Q/ZZ)G/U = Hom(((X̃ n−q )U )G/U , Q/ZZ)
∗
NG/U
n−q U G/U
←−
) ) , Q/ZZ) = H 0 (G, X̃ n−q )∗
∼ Hom(((X̃
and consequently
E p+q = H p+q (H 0 (G, X̃ n− )∗ ) = H p+q (H 0 (G, X̃ n− ))∗
.
.
= H n−(p+q) (G, A)∗ .
We thus obtain a spectral sequence
E2pq = H p (G/U, H n−q (U, A)∗ ) ⇒ H n−(p+q) (G, A)∗ .
If we now let U run through the open subgroups of G containing H and take
direct limits, we get the spectral sequence
E2pq = H p (G/H, Dn−q (H, A)) ⇒ H n−(p+q) (G, A)∗ .
2
Remark: Tate gave a proof of this spectral sequence using the cohomology
groups in negative dimensions (see [230]), which we have avoided here.
The Tate spectral sequence
E(G, H, A) : E2pq = H p (G/H, Dn−q (H, A)) ⇒ H n−(p+q) (G, A)∗
is functorial in G and H in the following sense. Let G0 be an open subgroup
of G and H 0 = H ∩ G0 . We then have two morphisms of spectral sequences
(∗)
xw
E(G, H,
A)
cor ∗ res
cor res ∗
E(G0 , H 0 , A),
such that the maps on the initial terms and the limit terms are given as follows.
The E2yzpq
E20pq are given as the composites of
H p (G/H, D{~}| n−q (H, A))
res
cor
cor ∗
res ∗
H p (G0 /H 0 , Dn−q (H, A))
H p (G0 /H 0 , Dn−q (H 0 , A)),
where cor ∗ and res ∗ are induced by the direct limit of the maps
€ A)∗
H n−q (U,
cor ∗
res ∗
H n−q (U 0 , A)∗ ,
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§5. The Tate Spectral Sequence
U running through the open normal subgroups of G containing H and U 0 =
U ∩ G0 . The maps on the limit terms are
cor ∗
H n−(p+q)‚ (G, A)∗
H n−(p+q) (G0 , A)∗ .
res ∗
In particular, for H = {1}, we obtain for the edge morphisms a commutative
diagram
‰ˆ‡†ƒ„…Š‹Œ n (A))
H p (G, D
res
H n−p (G, A)∗
res ∗
cor ∗
cor
H p (G0 , Dn (A))
H 0 (G, Dn−p (A))
incl
H n−p (G0 , A)∗
NG/G0
H 0 (G0 , Dn−p (A)).
All this results from the following consideration. Assume that H is an open
subgroup of G. Then the spectral sequence E(G, H, A) is obtained from the
double complex
C pq (G, H, A) = C p (G/H, X̃ n−q (G, A)H∗ ),
where X̃ i = X i for i = 0, · · · , n − 1 and X̃ n = ker(X n → X n+1 ) as in the proof
of (2.5.3). We have on the one hand the homomorphisms
Ž (G, A)H∗ )
C p (G/H, X̃ n−q
res
cor
0
C p (G0 /H 0 , X̃ n−q (G0 , A)H ∗ ).
On the other hand, the duals of the maps

X̃ n−q (G,
A)
res
cor
X̃ n−q (G0 , A)
yield homomorphisms
cor ∗
X̃
n−q
H∗
(G, A)
0
−→
←−
X̃ n−q (G0 , A)H ∗ .
∗
∗)
res
After composing, we obtain two morphisms of double complexes
C pq (G,‘’ H, A)
cor ∗ res
cor res ∗
C pq (G0 , H 0 , A),
and these induce the above morphisms (∗) of spectral sequences. The effects
on the cohomology groups mentioned are obtained in a straightforward manner
by recalling our identifications
E pq ∼
= H p (G/H, H n−q (H, A)∗ ), E p+q ∼
= H n (G, A)∗ .
2
We finish this section on spectral sequences by explicitly determining the
edge morphisms E q → E20,q and E2p,0 → E p of the Tate spectral sequence.
∗) The corestriction cor is defined on all of X̃ n−q (G0 , A) after choosing a sec0
tion s: G/G0 → G of G → G/G0 . It induces a homomorphism cor : X̃ n−q (G0 , A)H →
X̃ n−q (G, A)H if this choice is taken in such a way that s(c)σ = τσ s(cσ) for all c ∈ G0 \G,
σ ∈ H and some τσ ∈ H 0 .
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124
Chapter II. Some Homological Algebra
(2.5.4) Theorem. The edge morphism
H n−q (G, A)∗ −→
H n−q (U, A)∗
lim
−→
G/H
U ⊇H
in the Tate spectral sequence is the direct limit of the maps cor ∗ , dual to the
corestriction maps
cor : H n−q (U, A) −→ H n−q (G, A).
Proof: We use the notation of the proof of (2.5.3). We may assume that H is
open. The spectral sequence is obtained from the double complex
C pq = C p (G/H, Y q ),
where Y q = H 0 (H, X̃ n−q )∗ and X̃ i = X i for i < n and X̃ n = Z n . Let
C = tot (C •• ), B = ker(C 0 → C 1 ) and K = C 0 . Then B is a
subcomplex of C and H q (B ) = H q (C ) = E q by (2.2.4). Therefore the
edge morphism E q → E20,q is the map
(1)
H q (B ) −→ H q (K )
π
induced by the composite B → C → K , which is the inclusion. Identifying
C 0 = C 0 (G/H, Y ) with Y , this is the inclusion
(2)
(Y )G/H ,→ Y .
The image of (1) is contained in
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
∂
.
E20,q = ker(H q (C 0 ) −→ H q (C 1 )) = H q (Y )G/H ,
and the edge morphism becomes the map
edge : H q ((Y )G/H ) −→ H q (Y )G/H ,
induced by the inclusion (2). From what we have seen in the proof of (2.5.3),
this map is identified with a map
H q (G, A)∗ −→ [H q (H, A)∗ ]G/H
as follows. We have the canonical isomorphism
Hom((X̃ n− )G , Q/ZZ) ∼
= H 0 (G/H, Y ) ,
∗
which is the same as the dual NG/H
of
.
.
.
.
(X̃ n−“ )H
We obtain a commutative diagram
.
.
NG/H
.
”–•—˜™š
H q (H 0 (G/H,
Y ))
(X̃ n− )G .
edge
.
H q (Y )G/H
∗
NG/H
.
H q (H 0 (G, X̃ n− )∗ )
H n−q (G, A)∗
∗
NG/H
cor ∗
.
H q (H 0 (H, X̃ n− )∗ )G/H
[H n−q (H, A)∗ ]G/H ,
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125
§5. The Tate Spectral Sequence
which identifies the edge homomorphism with the dual of the corestriction.
2
We now consider the edge morphism E2p,0 → E p in the Tate spectral sequence for the case H = {1}. It is a homomorphism
› n (A))
H p (G, D
H n−p (G, A)∗ .
In particular, for p = n and A = ZZ, we have a canonical homomorphism, called
the trace map,
tr : H n (G, Dn (ZZ)) −→ Q/ZZ,
provided cd(G, ZZ) ≤ n. On the other hand, for every pair V ⊆ U of open
normal subgroups of G and each i ≥ 0, we have the canonical pairing
H i (V, A)∗ × AU −→ H i (V, ZZ)∗ , (χ, a) 7−→ f (x) = χ(ax).
Taking first the direct limit over V and then over U , we obtain a canonical
bilinear map Di (A) × A −→ Di (ZZ), which gives us a cup-product
∪
H p (G, Di (A)) × H n−p (G, A) −→ H n (G, Di (ZZ)).
For i = n this yields, together with the map tr, a homomorphism
cup
H p (G, Dn (A)) −→ H n−p (G, A)∗ .
edge
(2.5.5) Theorem. Suppose that cd(G, ZZ) ≤ n and let A ∈ Mod(G) be finitely
generated as a ZZ-module. If cd(G, A) ≤ n, then the two maps
œ n (A))
H p (G, D
edge
cup
H n−p (G, A)∗
coincide for all p ∈ ZZ.
Proof: The Tate spectral sequence arises from the double complex
C pq (A) = C p (G/U, H 0 (U, X̃ n−q (G, A))∗ )
and the application of lim . We have a canonical pairing
−→ U
ϕ : (X̃ (G, A) ) × A −→(X̃ n−q (G, ZZ)U )∗ , (χ, a) 7−→ ϕ(χ, a) = f,
where f : X̃ n−q (G, ZZ)U −→ Q/ZZ is defined by f (x) = χ(ax).
If z(σ0 , . . . , σp−j ) is a (p − j)-cochain with coefficients in (X̃ n−q (G, A)U )∗
and t(σ0 , . . . , σj ) is an j-cocycle in Z j (G, A), then
(z ∪ t)(σ0 , . . . , σp ) = ϕ(z(σ0 , . . . , σp−j ), t(σp−j , . . . , σp ))
is a p-cochain with coefficients in (X̃ n−q (G, ZZ)U )∗ . Thus we get a map
C p−j,q (A) × Z j (G, A) −→ C p,q (ZZ),
i.e. for fixed t ∈ Z j (G, A), we have a morphism of double complexes
∪t : C −j, (A) −→ C , (ZZ)
n−q
U ∗
. .
..
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126
Chapter II. Some Homological Algebra
of degree (j, 0) and hence a transformation of the associated edge morphisms.
Applying lim , we obtain a map
−→ U
∪t : H n−j (G, Dn (A)) −→ H n (G, Dn (ZZ)),
which only depends on the cohomology class of the cocycle t. As the map tr
is defined via the edge morphism, we obtain a commutative diagram
H n−j (G,ž¢ ¡Ÿ Dn (A)) × H j (G, A)
∪
H n (G, Dn (ZZ))
edge
tr
H j (G, A)∗
× H j (G, A)
Q/ZZ,
where the upper arrow is the cup-product with respect to the pairing
Dn (A) × A −→ Dn (ZZ),
and the lower arrow is the evaluation map (χ, t) 7→ χ(t). From this diagram
we get the commutative diagram
H n−j (G,¥¦¤£ Dn (A))
cup
Hom(H j (G, A), H n (G, Dn (ZZ)))
edge
tr
H j (G, A)∗
Hom(H j (G, A), Q/ZZ),
which shows that the edge morphism coincides with the composite tr ◦ cup.
Setting j = n − p, we obtain the assertion of the theorem.
2
We conclude this section with a vanishing criterion for the terms Di (A).
(2.5.6) Lemma. Suppose that Di (ZZ) = 0. Assume that A ∈ Mod(G) is finitely
generated as a ZZ-module and has torsion divisible only by prime numbers `
for which Di+1 (ZZ) is `-divisible. Then Di (A) = 0.
Proof: If A is finitely generated as a ZZ-module, then A = AU for some open
subgroup U of G. Since in the definition of the Di the group G may be replaced
by U , we may assume that A is a trivial G-module. If A is torsion-free, then
A∼
= ZZn as a G-module, hence Di (A) ∼
= Di (ZZ)n = 0. It remains to consider
the case A = ZZ/mZZ. From the exact sequence 0 → ZZ → ZZ → ZZ/mZZ → 0,
we obtain the exact sequence
m
Di+1 (ZZ) −→ Di+1 (ZZ) −→ Di (ZZ/mZZ) −→ Di (ZZ) = 0,
hence our assumptions imply Di (ZZ/mZZ) = 0.
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127
§6. Derived Functors
Exercise 1. Compute the Tate spectral sequence for the case G ∼
= ẐZ.
Exercise 2. If 0 < cd (G, A) ≤ n, then, for all open normal subgroups U of G, we have
Dn (A)U = H n (U, A)∗ .
Hint: Use (3.3.11).
Exercise 3. Let G be a profinite group and suppose we are given a direct limit A = lim Aα of
−→
G-modules. Show that there exist natural homomorphisms
Di (G, A) −→ lim Di (Aα )
←−
for all i. What are the images of these maps?
§6. Derived Functors
We have constructed the cohomology groups H n (G, A) in a direct and natural
way from the diagram
−→
−→
· · · −→
−→ G × G −→
−→
−→ G.
−→ G × G × G −→
The advantage of this definition is that it is concrete, elementary and down
to earth. Its disadvantage is that it is difficult to generalize and to get deeper
insights. There is another much more general definition of cohomology which
we describe now.
We have explained in I §3 the notion of δ-functor H = (H n )n≥0 between
abelian categories A and A0 . It is a family of additive functors H n : A → A0 ,
which turns a short exact sequence
0 −→ A −→ B −→ C −→ 0
in A functorially into a long exact sequence
δ
· · · −→ H n (A) −→ H n (B) −→ H n (C) −→ H n+1 (A) −→ · · ·
in A0 . A morphism between two δ-functors H and H 0 from A to A0 is a system
f = (f n )n≥0 of functorial morphisms
f n : H n −→ H 0n ,
which commute with δ. That is, for any exact sequence
0→A→B →C →0
in A, the diagram
δ
ª©¨§
H n (C)
H n+1 (A)
f n (C)
H 0n (C)
f n+1 (A)
δ
H 0n+1 (A)
commutes.
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Chapter II. Some Homological Algebra
(2.6.1) Definition. A δ-functor H = (H n )n≥0 from A to A0 is called universal
if, for every other δ-functor H 0 = (H 0n )n≥0 from A to A0 , each morphism
0
f 0 : H 0 → H 0 of functors extends uniquely to a morphism f : H → H 0 of
δ-functors.
We have the following criterion for the universality of a δ-functor. An
additive functor F : A → A0 is called effaceable if, for each object A in A,
there is a monomorphism u : A → I in A such that F (u) = 0.
(2.6.2) Theorem. A δ-functor H = (H n )n≥0 from A to A0 is universal if the
functors H n are effaceable for n > 0.
For the proof we refer to [66], chap.I. The idea is the following. Let
H 0 = (H 0n )n≥0 be an arbitrary δ-functor from A to A0 and let f 0 : H 0 → H 00
be a morphism of functors. Assume that we have shown that there exists a
uniquely determined morphism of functors f i : H i → H 0i , i = 1, . . . , n, which
u
commute with δ. Let A ∈ A and let 0 → A → I → J → 0 be an exact sequence
such that H n+1 (u) = 0. Then we obtain a uniquely determined morphism
f n+1 : H n+1 (A) → H 0n+1 (A) using the exact commutative diagram
H n²±°¯«¬­® (I)
fn
H 0n (I)
δ
H n (J)
fn
H n+1 (A)
0
f n+1
δ
H 0n (J)
H 0n+1 (A).
It remains to show that f n+1 is functorial and commutes with δ.
If G is a profinite group, then the functors H n (G, −), n > 0, are effaceable,
since every G-module A embeds into the induced G-module IndG (A), which
is acyclic, i.e. has trivial cohomology. We therefore have the
(2.6.3) Theorem. The δ-functor (H n (G, −))n≥0 is universal.
With this theorem many proofs of isomorphism, uniqueness etc. are obtained
automatically.
Example 1. Let H be a closed normal subgroup. We then have the following
proof of Shapiro’s lemma (1.6.4)
sh : H n (G, IndH (A)) ∼
= H n (H, A).
G
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129
§6. Derived Functors
H
Noting that IndH
G is exact and that IndG (IndH (B)) = IndG (B), we see that
H
n
n
(H (G, IndG (−))) and (H (H, −)) are effaceable δ-functors on Mod(H), and
are thus universal. They are functorially isomorphic in dimension n = 0.
Hence they are isomorphic as δ-functors. By the uniqueness assertion, the
isomorphism is the composite
π
res
∗
n
H
n
H n (G, IndH
G (A)) −→ H (H, IndG (A)) −→ H (H, A),
where π∗ is induced by π : IndH
G (A) → A, f 7→ f (1). In fact, this composite is
a morphism of δ-functors, which in dimension n = 0 coincides with the initial
isomorphism.
Example 2. For every G-module A, we have the commutative diagram
µ´³
H n (G, IndH
G (A))
ν∗
sh
H n (H, A)
cor
H n (G, A)
as claimed in (1.6.5). In fact, ν∗ and cor ◦ sh are morphisms of universal
δ-functors, which coincide in dimension n = 0 by the definition of sh, cor, ν∗ .
Hence they coincide for all n ≥ 0 by the uniqueness assertion of (2.6.1).
If F : A → A0 is an additive functor, there exists up to a canonical isomorphism at most one universal δ-functor H from A to A0 with H 0 = F .
This δ-functor, if it exists, is then called the right derived functor of F and
is denoted by R F = (Rn F )n≥0 . Obviously, it is defined up to canonical
isomorphism. The question is, when does it exist?
By theorem (2.6.3), the universal δ-functor H (G, −) on Mod(G) is the
right derived functor
H (G, −) = R Γ
.
.
.
.
of the functor
Γ (−) = H 0 (G, −) : Mod(G) −→ Ab,
A 7−→ AG .
Suppose M is a full abelian subcategory of Mod(G) which has the property that
for a discrete G-module M , the induced module IndG (M ) is also in M. Then
the same reasoning as above shows that the restriction of H (G, −) to M is
the right derived functor R Γ of the functor Γ (−) : M −→ Ab, A 7→ AG .
Examples of such subcategories M are the category Modt (G) of discrete
G-modules which are torsion groups, or the category Mod(p) (G) of discrete
G-modules which are p-torsion groups, where p is a prime number.
.
.
Recall that an additive functor F : A → A0 is called left exact if for each
exact sequence 0 → A → B → C the sequence
0 −→ F (A) −→ F (B) −→ F (C)
is also exact.
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130
Chapter II. Some Homological Algebra
The left exactness of F is clearly a necessary condition for the existence of
the right derived functor R F , since H 0 is left exact. This condition is already
sufficient if A has sufficiently many injectives:
.
An object A in A is injective if for every monomorphism B → C in A the
map Hom(C, A) → Hom(B, A) is surjective. A is said to have sufficiently
many injectives if, for any object A, there exists a monomorphism A → I into
an injective object.
(2.6.4) Theorem. Let A have sufficiently many injectives. Then for each left
exact additive functor F : A → A0 , the right derived functor R F = (Rn F )n≥0
exists.
.
For the proof we refer to [21], chap. V, §3, but we explain the idea of it.
Since A has sufficiently many injectives, each object A ∈ A has an injective
resolution, i.e. there is an exact complex
0 −→ A −→ I 0 −→ I 1 −→ I 2 −→ · · ·
with injective objects I n in A. We apply the functor F and get a complex
0 −→ F (A) −→ F (I 0 ) −→ F (I 1 ) −→ F (I 2 ) −→ · · · .
We define
.
Rn F (A) = H n (F (I )),
n ≥ 0;
in particular, R0 F (A) = ker(F (I 0 ) → F (I 1 )) = F (A).
The independence of this definition from the injective resolution chosen is
seen as follows. If A → I and A0 → I 0 are injective resolutions of A and A0 ,
then, because of the injectivity property of the I n , every morphism u : A → A0
extends to a morphism of complexes
A¸¹¶·
I
.
.
.
u
u
.
A0
I0 ,
and every two such extensions are homotopic (cf. I §3, exercise 6). This means
that the induced maps from F (I ) to F (I 0 ) are homotopic, hence induce
the same homomorphism H n (F (I )) → H n (F (I 0 )) on the homology. In
particular, if A = A0 , we find extensions u : I → I 0 , v : I 0 → I , such that
u ◦ v and v ◦ u are homotopic to the identity, hence induce mutually inverse
isomorphisms H n (Fº» (I ))
H n (F (I 0 )). This shows the independence.
The property of being a δ-functor is seen as follows. Any exact sequence
0 → A → B → C → 0 in A may be extended to an exact sequence
.
.
.
.
. .
.
.
.
.
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131
§6. Derived Functors
0¼½¾¿ÀÁÂÃÄÅÆ
A
B
C
0
0
IA
.
IB
.
IC
.
0
of injective resolutions. Since IAn is injective, all exact sequences
0 −→ IAn −→ IBn −→ ICn −→ 0
split and therefore
0 −→ F (IAn ) −→ F (IBn ) −→ F (ICn ) −→ 0
remains exact. The exact sequence
0 −→ F (IA ) −→ F (IB ) −→ F (IC ) −→ 0
of complexes yields, in the same way as in I §2, a long exact sequence
.
.
.
δ
· · · −→ Rn F (A) −→ Rn F (B) −→ Rn F (C) −→ Rn+1 F (A) −→ · · · .
We have obtained a δ-functor R F = (Rn F )n≥0 . For an injective object I, we
id
have Rn F (I) = 0 for n > 0 since 0 → I → I → 0 is an injective resolution of
I. Since A has sufficiently many injectives, the Rn F , n > 0, are effaceable,
hence R F is universal.
.
.
(2.6.5) Lemma. If G is a profinite group, then the category Mod(G) of discrete
G-modules has sufficiently many injectives.
Proof: To every abstract G-module M we can associate the submodule
[
MU,
M δ :=
U ⊆G
where U runs through the open subgroups of G. If we endow M with the
discrete topology, then M δ is the maximal submodule on which G acts continuously (compare with the remark after (1.1.8)). One easily verifies that
every G-homomorphism from a discrete G-module N to M factors through
M δ . In particular, we see that the discrete module I δ is an injective object
in Mod(G) provided the (abstract) G-module I is injective. The category of
abstract G-modules has sufficiently many injective objects (see [79], chap. IV:
it is canonically equivalent to the category of modules over the group ring
ZZ[G]). Therefore we can embed a given discrete G-module M into an injective abstract module I and then M is automatically contained in the injective
discrete module I δ .
2
The Hochschild-Serre spectral sequence (2.4.1) becomes a special case of
the following general result.
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Chapter II. Some Homological Algebra
(2.6.6) Theorem. Let A and A0 be abelian categories with sufficiently many
injectives and let A00 be another abelian category. Let
F
F0
A −→ A0 −→ A00
be left exact additive functors. Assume that F maps injective objects from A
to F 0 -acyclic objects, i.e. those annihilated by Rn F 0 for n > 0. Then there is
a cohomological spectral sequence
E2pq = Rp F 0 (Rq F (A)) ⇒ Rp+q (F 0 ◦ F )(A) ,
which is called the Grothendieck spectral sequence.
This spectral sequence is obtained as follows. There exists a homomorphism
of the complex F (I ) into a double complex of A0 -injective objects I •• which
induces injective resolutions of all groups F (I q ) and also for all cocycle,
coboundary and cohomology groups of the complex F (I ) (a so-called CartanEilenberg resolution, cf. [21], chap.XVII). Applying to the double complex
I •• = (I pq )p,q≥0 the functor F 0 , we obtain a double complex
(Apq )p,q≥0 = (F 0 (I pq ))p,q≥0 .
The spectral sequence E pq ⇒ E n associated with this double complex is the
maintained spectral sequence
E2pq = Rp F 0 (Rq F (A)) ⇒ Rn (F 0 ◦ F )(A).
For the proof we refer to [66] and [21].
.
.
If G is a profinite group and H a closed subgroup, then we have the additive
left exact functors
F = H 0 (H, −) : Mod(G)
→ Mod(G/H),
A 7→ AH ,
F 0 = H 0 (G/H, −) : Mod(G/H) → Ab,
B 7→ B G/H ,
F 0 ◦ F = H 0 (G, −) : Mod(G)
→ Ab,
A 7→ AG .
In this case, the Grothendieck spectral sequence
E2pq = Rp F 0 (Rq F (A)) ⇒ Rp+q (F 0 ◦ F )(A)
coincides with the Hochschild-Serre spectral sequence. We easily see that the
E2 -terms and the limit terms are the same, since
Rq F = H q (H, −), Rp F 0 = H p (G/H, −), Rn (F 0 ◦ F ) = H n (G, −).
That both spectral sequences actually coincide follows from the fact that the
functor ‘homogeneous cochain complex’ is a ‘resolving functor’, see [66].
So far we have dealt with the right derivation of a left exact, covariant
functor. In the later applications it will be often useful to work with certain
modifications of this concept.
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§6. Derived Functors
Assume that we are given abelian categories B and B 0 . We say that a family
H = (Hn )n∈ ZZ of functors Hn : B → B 0 is a homological δ-functor if the
family K = (K n )n∈ ZZ , defined by K n := H−n , is a (cohomological) δ-functor
as defined before. The following notions and statements are dual to those
given before for cohomological δ-functors and we leave their verification to
the reader. We also note that, up to the obvious modifications, one can also
work with contravariant functors.
We say that a homological δ-functor H = (Hn )n≥0 is universal if, for every
other homological δ-functor H 0 = (Hn0 )n≥0 , each morphism f0 : H00 → H0
of functors extends uniquely to a morphism f : H 0 → H of homological
δ-functors. A functor G : B → B 0 is called coeffaceable if, for every object
B ∈ B, there is an epimorphism φ : P → B with G(φ) = 0. A homological δfunctor H = (Hn )n≥0 : B → B 0 is universal if the functors Hn are coeffaceable
for n > 0. If G : B → B 0 is an additive functor, there exists up to canonical
isomorphism at most one universal homological δ-functor H from B to B 0 with
H0 = G. This δ-functor, if it exists, is then called the left derived functor of
G and is denoted by L G = (Ln G)n≥0 .
An object P of B is called projective if for every epimorphism A → B
in B the map Hom(P, A) −→ Hom(P, B) is surjective. We say that B has
sufficiently many projectives if for every object B there exists an epimorphism
P → B with projective P .
The left derived functor of G : B → B 0 exists if G is right exact and B has
sufficiently many projectives.
.
Now we introduce the homology of profinite groups. The homology groups
are compact abelian groups and they have compact G-modules as coefficients.
In order to prevent confusion, we use the notation D = D(G) for the category of
discrete G-modules, which so far has been denoted by Mod(G). The category
of compact G-modules will be denoted by C = C (G).
(2.6.7) Definition. Let G be a profinite group and let A ∈ C be a compact
G-module. The cofixed module (or module of coinvariants) AG of A is the
largest Hausdorff quotient of A on which G acts trivially, i.e. AG is the quotient
of A by the closed subgroup generated by the elements (ga − a), g ∈ G, a ∈ A.
We denote the category of compact abelian groups by Abc and in order
to stress the difference we write Abd for the category of (discrete) abelian
groups. One easily verifies that (−)G is a right exact functor from C to Abc .
Furthermore, the category C is dual to D by Pontryagin duality and therefore
it has sufficiently many projectives by (2.6.5).
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Chapter II. Some Homological Algebra
(2.6.8) Definition. For a profinite group G and a compact module A, the
homology groups are defined as the left derivatives of the cofixed-module
functor
Hn (G, A) := Ln (−)G (A).
In particular, we have H0 (G, A) = AG , and if 0 → A → B → C → 0 is an
exact sequence in C , then we get a long exact homology sequence
· · · → Hn+1 (G, C) → Hn (G, A) → Hn (G, B) → Hn (G, C) → · · · .
The homology theory for profinite groups is dual to the cohomology theory:
every cohomological result has its homological analogue. Fortunately we do
not have to prove everything twice because of the following
(2.6.9) Theorem. Let G be a profinite group and A be a compact G-module.
Then there are functorial isomorphisms for all i ≥ 0
Hi (G, A)∨ ∼
= H i (G, A∨ ),
where
∨
denotes the Pontryagin dual.
Proof: The theorem is true for i = 0 by the definition of the fixed and the
cofixed module. Now the following diagram of categories and functors is
commutative
∨
CÊÉÈÇ
D
(−)G
(−)G
∨
Abc
Abd .
Furthermore, Pontryagin duality is an exact, contravariant functor that transfers
C -projectives to D-injectives. The statement of the theorem now follows from
the universal property for the derived functors.
2
We see from the last theorem that, in principle, one can avoid the use
of homology groups, working only with cohomology. Indeed, the decision
whether to work with cohomology or homology, is more or less a question of
personal taste.
We finish this section with a spectral sequence for Ext-groups. Let R be a
ring (with unit). Then the functor HomR (−, −) is a bifunctor from the category
of R-modules to abelian groups, which is contravariant in the first and covariant
in the second variable. Its derivations
ExtiR (−, −)
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§6. Derived Functors
may likewise be computed using projective resolutions of the first, or using
injective resolutions of the second, variable. (See [79] or any textbook about
homological algebra for this basic fact.) The Hom-acyclic objects in the first
resp. second variable are projective resp. injective R-modules. The following
spectral sequence connects the Ext-groups for modules over a group ring R[G]
with that over R.
(2.6.10) Theorem. Let R be a commutative ring with unit, let G be a finite
group and let M and N be R[G]-modules. Then there exists a natural spectral
sequence
E2pq = H p (G, ExtqR (M, N )) ⇒ Extp+q
R[G] (M, N ) .
Proof: First we observe that
HomR[G] (M, N ) ∼
= HomR (M, N )G ,
thus the left exact functor HomR[G] (M, −) is the composition of the left
exact functors HomR (M, −) and H 0 (G, −). Now assume that N is injective. Then N is a direct summand of IndG N . By (1.3.6)(iii), the G-module
HomR (M, IndG N ) is induced. Thus HomR (M, N ) is cohomologically trivial
because it is a direct summand of an induced module. Therefore theorem
(2.6.6) gives us the desired spectral sequence.
2
(2.6.11) Corollary. Let G be a finite group whose order is invertible in the
commutative ring R. Then an R[G]-module M is projective if and only if it is
R-projective.
Proof: A free R[G]-module is free as an R-module. If M is a projective
R[G]-module, then it is a direct summand of a free R[G]-module and therefore
also projective as an R-module. In order to show the other implication, assume
that M is an R[G]-module which is R-projective. The cohomology groups
H i (G, M ) are R-modules and annihilated by #G for i ≥ 1 by (1.6.1). Hence
they are trivial for i ≥ 1 and for an arbitrary R[G]-module N , the spectral
sequence (2.6.10) degenerates to a sequence of isomorphisms
ExtiR[G] (M, N ) −→ ExtiR (M, N )G = 0
for i ≥ 1. Hence M is R[G]-projective.
We obtain the following result, which is known as Maschke’s Theorem.
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Chapter II. Some Homological Algebra
(2.6.12) Corollary (MASCHKE). Let G be a finite group and let K be a field
whose characteristic does not divide the order of G. Then the category of
K[G]-modules is semi-simple.
Exercise 1. Define for an abstract group G the homology with values in a G-module as the
left derivation of the cofixed module functor on the category of abstract G-modules. Assume
that G is a finite group and let A be a finite G-module. Then we can view G as an abstract
group and A as an abstract module or we can view G as a profinite group and A as a compact
module. Show that the corresponding homology groups are the same and that both coincide
with the homology groups introduced in I §9.
Exercise 2. Let R be a commutative ring with unit, let G be a finite group and let M and N
be R[G]-modules such that M is R-projective and N is cohomologically trivial. Show that
Ext1R[G] (M, N ) = 0 .
§7. Continuous Cochain Cohomology
In chapter I, we started considering the simplicial diagram
−→
· · · −→
−→
−→ G × G × G −→
−→ G,
−→ G × G −→
−→
which gave rise to a standard complex C (G, A), and defined the cohomology
of the profinite group G with values in the discrete module A. In the last
section we learned that this cohomology can be characterized by a universal
property which, in particular, explains its good functorial behaviour. This
functorial approach via universal constructions is extremely useful and clarifies
the principles behind classical homological notions.
.
Sometimes, however, the simplicial approach seems to reach further. For
instance, the categories of finite, compact, locally compact or of all topological
G-modules lack the existence of sufficiently many “good” objects, and the
existence of derived functors on these categories is not guaranteed. But we
can still define cohomology using the standard complex.
Let G be a profinite group and let A be any topological G-module (see
I §1). We form the continuous homogeneous cochain complex of G with
coefficients in A , denoted by Ccts (G, A), by taking the G-invariants of the
continuous standard resolution Xcts (G, A), which is defined in exactly the
same way as in I §2 for discrete modules.
.
.
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137
§7. Continuous Cochain Cohomology
(2.7.1) Definition. We call the n-th cohomology group of the complex
Ccts (G, A) the n-th continuous cochain cohomology group of G with con
efficients in A. We denote this group by Hcts
(G, A).
.
n
If A is discrete, then clearly Hcts
(G, A) = H n (G, A), but for an arbitrary
topological G-module A, the right-hand side of the equation is not defined.
The same arguments as used in I §3 show the following
(2.7.2) Lemma. Let
β
α
0 −→ A −→ B −→ C −→ 0
be a short exact sequence of topological G-modules such that the topology
of A is induced by that of B and such that β has a continuous section (just a
continuous map, not necessarily a homomorphism). Then there exist canonical
boundary homomorphisms
n
n+1
δ: Hcts
(G, C) −→ Hcts
(G, A)
and we obtain an exact sequence
δ
n
n
n
n+1
· · · → Hcts
(G, A) → Hcts
(G, B) → Hcts
(G, C) → Hcts
(G, A) → · · ·
δ
1
which begins 0 → AG → B G → C G → Hcts
(G, A).
Note that we can apply this lemma in the particular case that A is an open
submodule of B and C = B/A is the quotient module with the quotient
topology, which is discrete.
Suppose
B: A × B −→ C
is a continuous G-pairing, i.e. a continuous biadditive map such that σ(a · b) =
(σa · σb) for a ∈ A, b ∈ B, σ ∈ G, where a · b denotes B(a, b). Then B induces
biadditive maps
p
q
p+q
Ccts
(G, A) × Ccts
(G, B) −→ Ccts
(G, C)
via the formula
(f ∪ g)(σ0 , . . . , σp+q ) = f (σ0 , . . . , σp ) · g(σp , . . . , σp+q ).
This cup-product of cochains satisfies the identity
∂(f ∪ g) = (∂f ) ∪ g + (−1)p f ∪ (∂g)
and consequently induces pairings
p
q
p+q
Hcts
(G, A) × Hcts
(G, B) −→ Hcts
(G, C) ,
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Chapter II. Some Homological Algebra
which enjoy the same properties as the cup-products considered in I §4.
The maps inf, res, cor with respect to a change of groups are defined and have
the same properties as the maps introduced in I §5. (Note that we exclusively
made calculations there on the chain level!)
In some important special situations, we can still relate the continuous
cochain cohomology to derived functors. For this we need some preparations.
Let A be an abelian category and let AIN be the category of inverse systems
in A indexed by the set IN of natural numbers with the natural order. Thus
objects in AIN are inverse systems
· ·ËÌÍÎÏ ·
An+1 dn An
···
in A and morphisms are commutative diagrams
· ·ÝÜÛÐÑÒÓÔÕÖ×ØÙÚ ·
An+1
An
fn+1
A1
···
A1
d0
A0
A0
f1
fn
f0
···
Bn+1
Bn
···
B1
B0 .
The category A is abelian, with kernels and cokernels taken componentwise.
One can show that AIN has sufficiently many injectives, provided that this is
the case for A. Moreover, an object (An , dn ) in AIN is injective if and only if
all An are injective and all dn are split surjections (see [96], prop. 1.1). A left
exact functor
F : A −→ B
into another abelian category induces a functor F IN : AIN −→ B IN in the obvious
way. If A has sufficiently many injectives, then the right derivatives Ri F IN
exist and we have (loc.cit. prop. 1.2)
Ri F IN = (Ri F )IN for i ≥ 0, i.e. Ri F IN (An , dn ) = (Ri F An , Ri F (dn )).
If inverse limits over IN exist in B, we can define the functor
F : AIN −→ B,
lim
←−
n
which is by definition the composition of the functor F IN with the limit functor
lim
: B IN −→ B.
←−
Assume that A and B have sufficiently many injectives, F is left exact and
IN
F sends injectives to lim
-acyclic objects. Then, by (2.6.6), we have a spectral
←−
sequence
E2pq = lim p (Rq F An ) ⇒ Rp+q (lim F )(An , dn ) ,
.
←−
n
←−
n
where lim denotes the right derived functor of lim .
←−
←−
We say that a system (An , dn ) satisfies the Mittag-Leffler property if for
each n the images of the transition maps An+m → An are the same for
all sufficiently large m. Observe that every inverse system of finite abelian
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§7. Continuous Cochain Cohomology
groups automatically has the Mittag-Leffler property. We call a system (An , dn )
ML-zero if for each n there is an m = m(n) such that the transition map
An+m → An is zero.
(2.7.3) Proposition. Let (An , dn ) be a ML-zero system in A. Then
Rp (lim F )(An , dn ) = 0
←−
n
for all p ≥ 0.
Proof: Given a system A = (An , dn ), we define a new system A[1] by
can
A[1] = (An+1 , dn+1 ). There is a canonical map A[1] −→ A given by the
composition of the transition maps. If A ,→ I is an injective resolution, then
A[1] ,→ I[1] is too, and we obtain a commutative diagram
.
.
áÞßà
A[1]
I[1]
can
.
can
.
A
I .
∼ lim I m for all m
Since lim I[1]m −→
←−
←−
for all p ≥ 0
≥
0, we obtain natural isomorphisms
∼ Rp (lim F )A .
Rp (lim F )A[1] −→
←−
n
←−
n
If all transition maps of A are zero, then this canonical isomorphism is the zero
map, and hence Rp (lim F )A = 0 for all p ≥ 0 in this case.
←−
Let (An ) be a ML-zero system and let J = {n1 , n2 , . . .} ⊂ IN be a cofinal
subset such that all transition maps in the inverse system (Anj )j ∈IN are zero.
Now lim F is also the composition of the exact forgetful functor
←−
VJ : AIN −→ AIN ,
(An ) 7→ (Anj ),
which preserves injectives, with lim F . The spectral sequence implies
←−
p
R (lim
F )(An ) = Rp (lim
F )(Anj ) = 0
←−
←−
n
j
for all p ≥ 0.
2
The following well-known proposition was formulated in [182], prop. 1,
with too much generality (see [150] for a counter-example). Therefore we
give a complete proof here in the case that B is the category of modules over a
ring.
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Chapter II. Some Homological Algebra
(2.7.4) Proposition. Let B be the category of R-modules for some ring R.
Then lim p = 0 for p ≥ 2, and there is a canonical exact sequence
←−
0 −→ lim (Bn ) −→
←−
n
If B
∈
Y
id−dn
Y
Bn −−−→
n
Bn −→ lim 1 (Bn ) −→ 0.
←−
n
n
1
B = 0.
Ob(B IN ) satisfies the Mittag-Leffler property, then lim
←−
Proof: By the definition of lim
, we have an exact sequence
←−
0 −→ lim
(Bn ) −→
←−
n
Y
id−dn
Bn −−−→
n
Y
Bn ,
n
which is also exact on the right, if all transition maps of (Bn ) are surjective (for
Q
any element (bn ) ∈ n Bn a pre-image under (id − dn ) is easily constructed
recursively). Let
0 −→(An ) −→(Bn ) −→(Cn ) −→ 0
be an exact sequence in B IN such that all transition maps of the systems (An )
and (Bn ) are surjective. Then the transition maps of (Cn ) are also surjective
and the snake lemma shows that the sequence of inverse limits
(Cn ) −→ 0
(Bn ) −→ lim
(An ) −→ lim
0 −→ lim
←−
←−
←−
n
n
n
IN
is exact. Since injective objects in B have surjective transition maps, we
obtain lim p (Bn ) = 0 for all p ≥ 1 and any system (Bn ) with surjective transition
←−
maps.
If (Bn ) is an arbitrary inverse system, we consider the inverse systems (PBn )
Q
Q
and (QPn ) given by PBn = ni=1 Bi and QPn = n−1
i=1 Bi , and with the natural
Qn+1
Qn
projections pn : i=1 Bi → i=1 Bi as transition maps. We obtain an exact
sequence of inverse systems
0 −→(Bn ) −→(PBn ) −→(QPn ) −→ 0,
given by the commutative exact diagrams
0èåæçäâãëéêì
Bn+1
(id,dn ,dn dn−1 ,...)
n+1
Y
Bi
pn −(dn prn+1 ,dn−1 prn ,...)
i=1
Bn
(id,dn−1 ,dn−2 dn−1 ,...)
n
Y
i=1
Bi
0
i=1
pn−1
pn
dn
0
n
Y
Bi
pn−1 −(dn−1 prn ,dn−2 prn−1 ,...)
n−1
Y
Bi
0,
i=1
p
p
where pri denotes the projection onto Bi . Since lim
(PBn ) = 0 = lim
(QBn )
←−
←−
p
≥
≥
for p 1, we obtain lim (Bn ) = 0 for p 2 and the exact sequence stated in
←−
the proposition.
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§7. Continuous Cochain Cohomology
Finally, let (Bn ) be a Mittag-Leffler system. We consider the subsystem (Bn0 )
of (Bn ), where, for each n, Bn0 is the image of the transition map Bn+m → Bn ,
for sufficiently large m, which is independent of m. Then the transition maps of
(Bn0 ) are surjective and the cokernel (Bn00 ) of the natural inclusion (Bn0 ) ,→ (Bn )
is ML-zero. Therefore we obtain isomorphisms
p
∼ lim p (B )
0 = lim
(Bn0 ) −→
n
←−
←−
n
n
2
for all p ≥ 1.
Suppose we are given a profinite group G and a compact topological Gmodule A whose underlying topology is profinite. It is not difficult to see
that such a module is the inverse limit of finite G-modules and we call such
modules profinite modules for short. The continuous cochain cohomology of
profinite modules can also be defined by means of derived functors, namely as
an Ext-group in the category of profinite G-modules.∗) We will not make use
of this in the following, but the reader should notice that this is the conceptual
reason behind the next theorem.
(2.7.5) Theorem. Suppose that the compact G-module A has a presentation
A = lim An
←−
n∈IN
as a countable inverse limit of finite, discrete G-modules. Then there exists a
natural exact sequence
i
0 −→ lim 1 H i−1 (G, An ) −→ Hcts
(G, A) −→ lim H i (G, An ) −→ 0.
←−
n
←−
n
Proof: Consider the categories (G-Mod)IN and AbIN of inverse systems of
discrete G-modules and of abelian groups, respectively. Both categories are
abelian and have sufficiently many injective objects. Furthermore, the functor
H 0 (G, −) sends injective G-modules to injective (i.e. divisible) abelian groups:
indeed, this is obvious for modules of the form IndG M for a divisible module
M . If I is injective, then it is divisible and it is a direct summand of IndG I.
Let us consider the functor lim
H 0 (G, −). By (2.7.4), applied to B = Ab,
←−
the spectral sequence for the composition of derived functors degenerates to a
series of short exact sequences
0 → lim 1 H i−1 (G, An ) → Ri (lim H 0 (G, −)) (An ) → lim H i (G, An ) → 0.
←−
n
←−
n
←−
n
∗) H i (G, A) =
∼ Exti
cts
profinite G-modules (ẐZ, A), where G acts trivially on ẐZ. This can easily
be deduced from (5.2.14).
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Chapter II. Some Homological Algebra
It therefore remains to show that there are isomorphisms for all i:
i
Ri (lim
H 0 (G, −)) (An ) ∼
(G, A).
= Hcts
←−
n
To prove this, let us first replace the system (An ) by its subsystem (A0n ) with
A0n =
\
im(An+m )
m
(since the Ai are finite, all intersections are finite too). The system A0n has
surjective transition maps and A ∼
A0n as a topological G-module. The
= lim
←−
cokernel of the natural inclusion map (A0n ) ,→ (An ) is ML-zero. By (2.7.3),
we have isomorphisms
∼ Ri (lim H 0 (G, −)) (A )
Ri (lim H 0 (G, −)) (A0n ) →
n
←−
←−
(A0n ),
for all i 0. Therefore, replacing (An ) by
we may assume that all trani
(G, A) is the
sition maps of the system (An ) are surjective. By definition, Hcts
i-th cohomology of the continuous homogeneous cochain complex Ccts (G, A),
which can be identified with the inverse limit over n of C (G, An ). The complexes C (G, An ) are the G-invariants of the complexes X (G, An ), which
are acyclic resolutions of An for all n, see I §3. Moreover, these complexes are
functorial, hence we get a H 0 (G, −)IN -acyclic resolution
(An ) ,→ (X (G, An )),
i
from which Hcts (G, A) is obtained by applying lim H 0 (G, −) and taking
←−
cohomology. Hence it remains to show that the systems (X (G, An )) are
lim
H 0 (G, −)-acyclic, which follows easily from the fact that the systems of
←−
abelian groups (C i (G, An )) = (X i (G, An )G ) have surjective transition maps
for all i (cf. I §3, ex.1).
2
≥
.
.
.
.
.
.
(2.7.6) Corollary. Let A be a compact G-module having a presentation
A = lim An
←−
n∈IN
as a countable inverse limit of finite, discrete G-modules. If H i (G, An ) is finite
for all n, then
i+1
Hcts
(G, A) = lim
H i+1 (G, An ).
←−
n
If A = lim
←− n∈IN
An , where An is finite, then the corollary applies, for example,
1
2
to Hcts
(G, A) and, if the group G is finitely generated, also to Hcts
(G, A).
As we did in I §2, p.17, in the case of a finite A, we consider for a profinite
G-module A the set EXT(G, A) of equivalence classes of exact sequences of
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§7. Continuous Cochain Cohomology
profinite groups
1 −→ A −→ Ĝ −→ G −→ 1
such that the action of G on A is given by
a = σ̂aσ̂ −1 ,
σ
where σ̂ ∈ Ĝ is a pre-image of σ
also in this case the
∈
G. The same proof as that of (1.2.4) shows
(2.7.7) Theorem. We have a canonical bijection of pointed sets
2
Hcts
(G, A) ∼
= EXT(G, A).
Let ` be a prime number and T a topological G-module which, as a topological group, is a finitely generated ZZ` -module with the natural topology, and
on which G acts ZZ` -linearly.
n
(G, T ).
(2.7.8) Proposition. Let Y be a finitely generated ZZ` -submodule of Hcts
n
Then the quotient group Hcts (G, T )/Y contains no nontrivial `-divisible subgroup.
n
Proof: (cf. [232], prop. 2.1) Suppose xi ∈ Hcts
(G, T ), 0 ≤ i < ∞, such that
xi ≡ `xi+1 mod Y for all i. We must show x0 ∈ Y . Let yj , 1 ≤ j ≤ m, be a
finite set of generators for Y . For each i, let fi be an n-cocycle representing
xi , and for each j, let gj be an n-cocycle representing yj . Then there are
(n − 1)-cochains hi and elements aij ∈ ZZ` such that
fi = `fi+1 +
m
X
aij gj + ∂hi .
j=1
Hence
f0 =
m
X
aj gj + ∂h
j=1
with aj = i≥0 `i aij and h = i≥0 `i hi . The use of infinite sums here is
formally justified by the fact that T is the inverse limit of its quotients T /`i T
and consequently
P
P
.
.
Ccts (G, T ) = lim C (G, T /`i T )
←−
i
is a projective limit of modules, each of which is killed by a fixed power of `.
This completes the proof.
2
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Chapter II. Some Homological Algebra
n
(2.7.9) Corollary. The ZZ` -module Hcts
(G, T ) is finitely generated if and only
n
n
if Hcts (G, T )/`Hcts (G, T ) is finite.
Proof: In order to show the nontrivial assertion, assume that y1 , . . . , ym
n
generate Hcts
(G, T ) modulo `. Putting Y = hy1 , . . . , ym i, we conclude that the
n
group Hcts (G, T )/Y is `-divisible, hence trivial by the last proposition.
2
(2.7.10) Corollary. Assume that the cohomology groups of G with coefficients
n
(G, T ) is a finitely generated
in finite `-primary modules are finite. Then Hcts
ZZ` -module for all n and the canonical map
n
(G, T ) ⊗ Q` /ZZ` −→ H n (G, T ⊗ Q` /ZZ` )
Hcts
is an isogeny, i.e. has finite kernel and finite cokernel.
Proof: Replacing, if necessary, T by an open submodule, we may assume
`m
that T is torsion-free. Then the exact sequence 0 → T → T → T /`m → 0
implies the exact sequence
n
n
0 −→ Hcts
(G, T )/`m −→ H n (G, T /`m ) −→ `m Hcts
(G, T ) −→ 0 .
Now the statements follow from (2.7.9).
2
Suppose now that T is torsion-free. Tensoring it, over ZZ` , by the exact
sequence 0 → ZZ` → Q` → Q` /ZZ` → 0 gives an exact sequence
(∗)
0 −→ T −→ V −→ W −→ 0 ,
in which V is a finite dimensional Q` -vector space, T is an open compact
subgroup and W is a discrete divisible `-primary torsion group.
(2.7.11) Proposition. There are isomorphisms for all n
n
n
Hcts
(G, V ) ∼
(G, T ) ⊗ ZZ` Q` .
= Hcts
In the exact cohomology sequence associated with (∗), the kernel of the boundary homomorphism
n
δ: H n−1 (G, W ) −→ Hcts
(G, T )
is the maximal divisible subgroup of H n−1 (G, W ), and its image is the torsion
n
subgroup of Hcts
(G, T ).
n
Proof: Since V is a vector space over Q` , so is Hcts
(G, V ) for all n. Furthern
more, H (G, W ) is an `-torsion group for all n. By (2.7.2), we have a long
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§7. Continuous Cochain Cohomology
145
exact sequence associated to (∗). Tensoring over ZZ` with Q` implies the first
statement. Clearly,
n
(G, T )) = im(H n−1 (G, V ) → H n−1 (G, W ))
ker(H n−1 (G, W ) → Hcts
is `-divisible. On the other hand, by (2.7.8), each divisible subgroup of
H n−1 (G, W ) must be contained in the kernel. Since W is torsion, the group
n
im(H n−1 (G, W ) → Hcts
(G, T )) is a torsion group. On the other hand, it is
n
n
equal to the kernel of the map Hcts
(G, T ) → Hcts
(G, V ) and therefore must
n
2
contain all torsion elements of Hcts (G, T ).
(2.7.12) Corollary. Assume that the cohomology groups of G with coefficients
in finite `-primary modules are finite. Then
n
n
(G, T ) = dimQ` Hcts
(G, V )
rank ZZ` Hcts
= corank ZZ` H n (G, T ⊗ Q` /ZZ` )
= rank ZZ` Hn (G, Hom(T, ZZ` )) .
Proof: The first two equalities follow from (2.7.11) and (2.7.10). Since T is
finitely generated, we have
Hom(T ⊗ Q` /ZZ` , Q` /ZZ` ) = Hom(T, Hom(Q` /ZZ` , Q` /ZZ` )) = Hom(T, ZZ` ) ,
thus (2.6.9) implies the third equality.
2
Continuous cochain cohomology was introduced by J. TATE in [232]. In
addition, we have taken several arguments from the paper [96] of U. JANNSEN.
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Chapter III
Duality Properties of Profinite Groups
§1. Duality for Class Formations
Let G be a finite group. If A and B are two G-modules, the cup-product
associated with the canonical pairing
Hom(A, B) × A −→ B,
(f, a) 7→ f (a),
yields a pairing
∪
Ĥ i (G, Hom(A, B)) × Ĥ n−i (G, A) −→ Ĥ n (G, B) ,
which we call again the cup-product. When B = Q/ZZ, we set
A∗ = Hom(A, Q/ZZ).
(3.1.1) Proposition. Let G be a finite group and let A be a G-module. Then
for all i ∈ ZZ, the pairing
∪
Ĥ i (G, A∗ ) × Ĥ −i−1 (G, A) −→ Ĥ −1 (G, Q/ZZ) =
1
ZZ/ZZ ⊆
#G
Q/ZZ
induces an isomorphism
∼
Ĥ i (G, A∗ ) −→ Ĥ −i−1 (G, A)∗ .
Remark: Via the identification (see (1.9.2)) Hi = Ĥ −i−1 , i > 0, we recover
the isomorphism H i (G, A∗ ) ∼
= Hi (G, A)∗ from (1.9.1).
Proof: First let i = 0. A homomorphism f : A → Q/ZZ is a G-homomorphism, i.e. f ∈ H 0 (G, A∗ ), if and only if f (IG A) = 0. Therefore the map
H 0 (G, A∗ ) −→ H0 (G, A)∗ ,
which associates to a G-homomorphism f : A → Q/ZZ the induced homomorphism g : A/IG A → Q/ZZ, is an isomorphism. If f ∈ NG A∗ , i.e.
P
P
f = σ∈G σh for h ∈ A∗ , then for a ∈ NG A we have f (a) = (σh)(a)
P
= h(σ −1 a) = h(NG a) = 0. This shows that we have a well-defined map
A∗G /NG A∗ −→ (NG A/IG A)∗ .
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148
Chapter III. Duality Properties of Profinite Groups
For the surjectivity, let g : NG A → Q/ZZ be a homomorphism that vanishes on
IG A. Since Q/ZZ is an injective ZZ-module, it can be extended to a homomorphism g : A → Q/ZZ, which is a G-homomorphism because g(IG A) = 0. This
shows the surjectivity.
For the injectivity, let f ∈ A∗G be such that f (NG A) = 0. As
NG : A/NG A −→ NG A
is an isomorphism, there exists a g ∈ (NG A)∗ such that f (a) = g(NG a) for
all a ∈ A. We may extend g to a homomorphism g : A → Q/ZZ and then
f = NG g, since
X
(NG g)(a) =
g(σ −1 a) = g(NG a) = f (a).
σ ∈G
This proves the injectivity.
Now let i ∈ ZZ be arbitrary. We use dimension shifting (see I §3, p.32) in
order to reduce to the case i = 0. From (1.4.6) we obtain the commutative
diagram
ïîíñð
Ĥ i (G, Hom(A,
Q/ZZ)) × Ĥ −i−1 (G, A)
∪
(−1)i(i+1)/2
δi
δi
Ĥ −1 (G, Q/ZZ)
Ĥ 0 (G, Hom(A, Q/ZZ)i ) × Ĥ −1 (G, A−i )
∪
Ĥ −1 (G, Q/ZZ).
Since Hom(A, Q/ZZ)i ∼
= Hom(A−i , Q/ZZ), the desired result follows.
2
Next, we consider the case B = ZZ and get the following
(3.1.2) Proposition. If G is a finite group and A is a ZZ-free G-module, then
for all integers i ∈ ZZ, the pairing
∪
Ĥ i (G, Hom(A, ZZ)) × Ĥ −i (G, A) −→ Ĥ 0 (G, ZZ) = ZZ/#GZZ
yields an isomorphism
Ĥ i (G, Hom(A, ZZ)) ∼
= Ĥ −i (G, A)∗ .
Proof: Since A is ZZ-free, the sequence
0 −→ Hom(A, ZZ) −→ Hom(A, Q) −→ Hom(A, Q/ZZ) −→ 0
is exact. Multiplication by #G on Hom(A, Q) is an isomorphism, hence also
on the group Ĥ n (G, Hom(A, Q)), which is therefore zero by (1.6.1). For this
reason, we get a commutative diagram
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149
§1. Duality for Class Formations
ôóòöõ ZZ)) × Ĥ −i (G, A)
Ĥ i−1 (G, Hom(A, Q/
δ
∪
Ĥ −1 (G, Q/ZZ)
δ
id
Ĥ i (G, Hom(A, ZZ)) × Ĥ −i (G, A)
∪
Ĥ 0 (G, ZZ),
in which the vertical arrows are isomorphisms. Hence the proposition follows
from (3.1.1).
2
(3.1.3) Definition. Let G be a finite group. We call a G-module C a class
module if for all subgroups H of G
(i) H 1 (H, C) = 0 and
(ii) H 2 (H, C) is cyclic of order #H.
A generator γ of H 2 (G, C) is called a fundamental class.
Obviously, C is also a class module for every subgroup H of G. Note that
2
if γ is a generator of H 2 (G, C), then γH = res G
H γ is a generator of H (H, C):
since cor ◦ res γ = (G : H)γ, the order of γH is divisible by #H. If U ⊆ G is
normal, then C U is not necessarily a class module for G/U .
If G is cyclic, then ZZ is a class module because of the isomorphisms
(cf. (1.7.1)) Ĥ 1 (H, ZZ) ∼
= Ĥ −1 (H, ZZ) = 0 and H 2 (H, ZZ) ∼
= Ĥ 0 (H, ZZ) =
ZZ/(#H)ZZ. We have seen in (1.7.1) that the cup-product with a fundamental
class is a periodicity operator for the cohomology of G, i.e.
∼ Ĥ n+2 (G, A)
γ∪ : Ĥ n (G, A) −→
for every G-module A and for all n ∈ ZZ.
We relate the property of being a class module to the property of cohomological triviality. To each G-module C and each class γ ∈ H 2 (G, C) we associate
L
a G-module C(γ) as follows. Let B = σ=/1 ZZbσ be the free abelian group
with basis bσ , indexed by the elements σ ∈ G, σ =/ 1. We set
C(γ) = C ⊕ B,
and we let G act on C(γ) by means of an inhomogeneous cocycle c(σ, τ )
representing γ as follows: we set b1 = c(1, 1) and define
σbτ = bστ − bσ + c(σ, τ ).
This is really a G-action, i.e. (ρσ)bτ = ρ(σbτ ) and 1bτ = bτ , because of the
cocycle relation ρc(σ, τ ) − c(ρσ, τ ) + c(ρ, στ ) − c(ρ, σ) = 0 ∗) . The G-module
C(γ) is constructed in such a way that the cocycle c(σ, τ ) becomes necessarily
∗) The reader may check that, up to isomorphism, C(γ) depends only on the class γ, not on
the cocycle c(σ, τ ).
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Chapter III. Duality Properties of Profinite Groups
in C(γ) a coboundary c(σ, τ ) = σb(τ ) − b(στ ) + b(σ), where the 1-cochain b(σ)
is defined by b(σ) = bσ . Thus H 2 (G, C) → H 2 (G, C(γ)) maps γ to zero. C(γ)
is therefore called the splitting module of γ. ∗)
We obtain a four term exact sequence
ϕ
i
0 −→ C −→ C(γ) −→ ZZ[G] −→ ZZ −→ 0
of G-modules if we define ϕ by
ϕ(c) = 0 for c ∈ C and ϕ(bσ ) = σ − 1 for σ =/ 1.
Splitting up this sequence into two short exact sequences, we obtain, for each
n ∈ ZZ and each subgroup H ⊆ G, a homomorphism
δ 2 : Ĥ n (H, ZZ) −→ Ĥ n+2 (H, C).
(3.1.4) Theorem. Let G be a finite group. For each n ∈ ZZ and each subgroup
H ⊆ G, the homomorphism
δ 2 : Ĥ n (H, ZZ) → Ĥ n+2 (H, C)
is given by the cup-product β 7→ γH ∪ β, where γH = res G
H (γ). The following
conditions are equivalent.
(i) C(γ) is a cohomologically trivial G-module,
(ii) C is a class module with fundamental class γ,
(iii) δ 2 is an isomorphism for all n ∈ ZZ and all H.
Remark: If C is a class module for G, then, by the above theorem, we have
isomorphisms
1
1
∼
ZZ/ZZ, γH 7→ #H
mod ZZ,
(δ 2 )−1 : H 2 (H, C) −→
#H
where γ ∈ H 2 (G, C) is a chosen fundamental class. These are called invariant
maps and denoted by inv.
Proof: The map δ 2 arises from the two exact sequences
(1)
0 −→ IG −→ ZZ[G] −→ ZZ −→ 0,
(2)
0 −→ C −→ C(γ) −→ IG −→ 0
and is the composite of the maps
δ
δ
1
2
n+1
(3)
Ĥ n (H, ZZ) −→
(H, IG ) −→
Ĥ n+2 (H, C),
∼ Ĥ
where δ1 is always an isomorphism. For n = 0 we have the maps
(4)
δ
δ
1
2
H 1 (H, IG ) −→
H 2 (H, C).
ZZ/#H ZZ = Ĥ 0 (H, ZZ) −→
∗) The module C(γ) also arises as the extension corresponding to the class γ
∼ Ext2
∼ Ext1
via the isomorphisms H 2 (G, C) =
Z, C) =
ZZ [G] (Z
ZZ [G] (IG , C).
∈
H 2 (G, C)
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151
§1. Duality for Class Formations
For the generator 1̄ = 1 mod #H of Ĥ 0 (H, ZZ), we have
δ2 δ1 1̄ = γH = res G
H γ.
(5)
In fact, a pre-image of the 0-cocycle 1 ∈ Z 0 (H, ZZ) in C 0 (H, ZZ[G]) is 1 ∈ ZZ[G]
and δ1 1̄ ∈ H 1 (H, IG ) is represented by the 1-cocycle (∂1)(σ) = σ − 1. A lift of
∂1 in C 1 (H, C(γ)) is given by x(σ) = bσ , and δ2 δ1 1̄ is represented by
(∂x)(σ, τ ) = σbτ − bστ + bσ = c(σ, τ ) .
This proves (5). Now let β ∈ Ĥ n (H, ZZ), where n is arbitrary. Applying
proposition (1.4.3) to B = ZZ and to the two exact sequences (1) and (2), we
obtain
δ 2 β = δ2 δ1 (1̄ ∪ β) = δ2 (δ1 1̄ ∪ β) = δ2 δ1 1̄ ∪ β = γH ∪ β.
Noting that Ĥ i (H, IG ) = Ĥ i−1 (H, ZZ) = 0 for i = 0 and 2, we obtain from (2)
the exact sequence
0 −→ H 1 (H, C) −→ H 1 (H, C(γ)) −→ H 1 (H, IG )
δ
−→ H 2 (H, C) −→ H 2 (H, C(γ)) −→ 0.
If C(γ) is cohomologically trivial, then H 1 (H, C) = 0 and the composite
δ
δ
1
2
ZZ/#H ZZ = Ĥ 0 (H, ZZ) −→
∼ H (H, IG ) −→ H (H, C)
is an isomorphism which maps 1̄ onto γH . Therefore C is a class module
with fundamental class γ. Conversely, if this is true, then H 1 (H, C) = 0 and
δ : H 1 (H, IG ) → H 2 (H, C) is an isomorphism; hence H i (H, C(γ)) = 0 for
i = 1, 2, and therefore for all i by (1.8.4).
The equivalence (i) ⇔ (iii) follows from (3) and from the exact sequence
. . . −→ Ĥ n (H, C) −→ Ĥ n (H, C(γ)) −→ Ĥ n (H, IG )
δ
2
−→ Ĥ n+1 (H, C) −→ Ĥ n+1 (H, C(γ)) −→ . . .
For ZZ-free G-modules A we can now prove the following duality theorem.
(3.1.5) Theorem (NAKAYAMA-TATE). Let G be a finite group, let C be a class
module for G and let γ ∈ H 2 (G, C) be a fundamental class. Then, for all
integers i ∈ ZZ, the cup-product
∪
Ĥ i (G, Hom(A, C)) × Ĥ 2−i (G, A) −→ H 2 (G, C) ∼
=
where H 2 (G, C) ∼
=
phism
1
ZZ/ZZ
#G
is given by γ 7→
1
#G
1
ZZ/ZZ,
#G
mod ZZ, induces an isomor-
Ĥ i (G, Hom(A, C)) ∼
= Ĥ 2−i (G, A)∗ ,
provided that A is ZZ-free. If, in addition, A is finitely generated, then this is
an isomorphism of finite abelian groups.
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Chapter III. Duality Properties of Profinite Groups
Proof: Let 0 −→ C −→ C(γ) −→ IG −→ 0 be an exact sequence as in the
proof of (3.1.4). As A is ZZ-free, the sequences
0 −→ Hom(A, C) −→ Hom(A, C(γ)) −→ Hom(A, IG )−→ 0
0 −→ Hom(A, IG )−→ Hom(A, ZZ[G]) −→ Hom(A, ZZ) −→ 0
are exact and the G-modules in the middle are cohomologically trivial by
(1.8.5), (3.1.4), and (1.3.7). We now have for i ∈ ZZ a commutative diagram
úûùø÷ÿþýü
Ĥ i−2 (G, Hom(A,
ZZ)) × Ĥ 2−i (G, A)
δ
∪
Ĥ 0 (G, ZZ)
δ
id
Ĥ i−1 (G, Hom(A, IG )) × Ĥ 2−i (G, A)
δ
∪
Ĥ 1 (G, IG )
δ
id
∪
Ĥ i (G, Hom(A, C)) × Ĥ 2−i (G, A)
Ĥ 2 (G, C),
where the vertical arrows are isomorphisms. Hence the asserted isomorphism
follows from (3.1.2). If A is finitely generated, then the cohomology groups
Ĥ i (G, A) are finite, and so are their duals, by (1.6.1).
2
For a subgroup H of G we have commutative diagrams
, Ĥ i (H, Hom(A,
C))
Ĥ 2−i (H, A)∗
res
cor ∗
cor
res ∗
Ĥ i (G, Hom(A, C))
Ĥ 2−i (G, A)∗
where cor ∗ and res ∗ are the maps dual to the corestriction and restriction, and
the upper map relies on the choice of the fundamental class γH ∈ Ĥ 2 (H, C).
The commutativity of the diagram follows at once from cor (γ ∪ res β) =
(cor γ) ∪ β (see (1.5.3)(iv)) and inv ◦ cor = inv.
Applying the duality theorem to the case A = ZZ, i = 0, and recalling
H 2 (G, ZZ)∗ ∼
= H 1 (G, Q/ZZ)∗ = Hom(Gab , Q/ZZ)∗ = Gab ,
we obtain
(3.1.6) Theorem. If C is a class module for the finite group G, then we have
an isomorphism
∼
ρ = ρG : Gab −→ C G /NG C,
called the Nakayama map. It depends on the choice of a fundamental class
γ ∈ H 2 (G, C) and satisfies (by definition) the formula
χ(σ) = inv(ρ(σ) ∪ δχ)
δ
2
for all characters χ ∈ H 1 (G, Q/ZZ) −→
Z).
∼ H (G, Z
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153
§1. Duality for Class Formations
The inverse map
∼
ρ−1 : C G /NG C −→ Gab
is often called the reciprocity isomorphism. We call its composite with the
natural projection C G C G /NG C, i.e. the map
rec = rec G : C G
Gab ,
the reciprocity homomorphism.
The reciprocity homomorphism is also called the norm residue symbol and,
if α ∈ C G , we also write
(α, G) := rec(α).
The name norm residue symbol reflects the fact that this symbol detects whether
α ∈ C G is the norm of an element in C. Indeed, (α, G) = 0 if and only if
α ∈ NG C.
The isomorphism Gab ∼
= C G /NG C is an abstract version of class field
theory. If L|K is a finite Galois extension of local or global fields, then the
multiplicative group L× is a class module in the local case, and the idèle class
group CL in the global case, as we shall see. Combined with the above theorem
this gives the main theorem of local and global class field theory.
The following theorem gives an explicit description of the Nakayama map.
(3.1.7) Theorem. If C is a class module for the finite group G and γ
H 2 (G, C) is a fundamental class, then the Nakayama map
∼
ρ : Gab −→ C G /NG C
is explicitly given by
σ mod [G, G] 7−→
X
c(τ, σ) mod NG C,
τ
where c is a cocycle representing γ.
Proof: We will prove the theorem by means of the diagram
G
ab
(∗)
ν
1
ZZ/ZZ
#G
× H 1 (G, Q/ZZ)
ρ
Ĥ 0 (G, C) ×
inv
δ
H 2 (G, ZZ)
∪
H 2 (G, C),
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154
Chapter III. Duality Properties of Profinite Groups
in which the upper horizontal arrow is the evaluation map (σ, χ) 7→ χ(σ),
1
mod ZZ, where ρ is the Nakayama isomorphism and ν is the map
inv(γ) = #G
given by
X
σ̄ 7−→ c(τ, σ) mod N C , ∗)
G
τ
where σ̄ = σ mod [G, G]. It suffices to show that the diagram is commutative
with both maps ν and ρ, since then theorem (3.1.5) identifies ν and ρ with the
dual δ ∗ of δ. The commutativity of the diagram with the map ρ follows from
the definition of ρ and is an equivalent version of the formula
χ(σ) = inv(ρ(σ) ∪ δχ) ,
see (3.1.5). It remains to show the commutativity for ν. For this we consider
the larger diagram
Gab
Ĥ −1 (G, Q/ZZ) =
× H 1 (G, Q/ZZ)
δ1
δ3
id
Ĥ −1 (G, IG ) × H 1 (G, Q/ZZ)
id
∪
Ĥ 0 (G, IG ⊗ Q/ZZ)
−δ4
δ
Ĥ −1 (G, IG ) ×
δ2
∪
H 2 (G, ZZ)
H 2 (G, ZZ)
H 1 (G, IG )
δ5
id
Ĥ 0 (G, C) ×
1
ZZ/ZZ
#G
∪
H 2 (G, C),
in which δ1 : Gab → Ĥ −1 (G, IG ) = IG /IG2 is defined by σ̄ 7→ σ − 1 mod IG2
and the other δi ’s arise from the exact sequences
(1)
0 −→ IG −→ ZZ[G] −→ ZZ −→ 0,
(2)
0 −→ IG ⊗ Q/ZZ −→ ZZ[G] ⊗ Q/ZZ −→ Q/ZZ −→ 0,
(3)
0 −→ IG −→ IG ⊗ Q −→ IG ⊗ Q/ZZ −→ 0,
(4)
0 −→ C −→ C(γ) −→ IG −→ 0.
We make the homomorphisms δ explicit by choosing in (1) - (4) sections of
the last arrows, by which the cocycles with coefficients in the last groups are
sent to cochains with coefficients in the middle groups, to which we apply the
coboundary operator ∂. For example, the map
C(γ) = C ⊕
M
ZZbσ −→ IG
/1
σ=
∗) Note that the sum is contained in C G because of the cocycle relation ρc(τ, σ) = c(ρτ, σ) +
c(ρ, τ ) − c(ρ, τ σ) and the fact that ρτ and τ σ run through G if τ does.
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155
§1. Duality for Class Formations
is defined by C → 0 and bσ 7→ σ − 1. It has the section σ − 1 7→ bσ ,
and b : G → C(γ), σ 7→ bσ , is an inhomogeneous cochain with coboundary
(∂ 2 b)(σ, τ ) = c(σ, τ ), which lifts the 1-cocycle x : G → IG , x(σ) = σ − 1.
We set g = #G, ā = a mod ZZ for a ∈ Q with ga ∈ ZZ. Noting that the elements
P
in (IG ⊗ Q/ZZ)G are of the form τ (τ − 1) ⊗ ā = NG ⊗ ā, the δ’s are given as
follows
δ1 : σ̄
7−→ σ − 1 mod IG2 ,
δ2 : σ − 1
7−→
X
c(τ, σ) mod NG C,
τ
7−→
δ3 : ā
X
(τ − 1) ⊗ ā,
τ
−δ4 : NG ⊗ ā 7−→ gax,
7−→ c.
δ5 : x
The first assertion holds by definition and the others rely on the relations
∂ 0 bσ =
X
τ
τ bσ =
X
bτ σ −
τ
∂ 0 (1 ⊗ ā) = NG (1 ⊗ ā) =
X
∂ 1(
X
bτ +
τ
X
X
c(τ, σ) =
c(τ, σ),
τ
τ
X
X
τ
τ
τ ⊗ ā =
(τ − 1) ⊗ ā,
(τ − 1) ⊗ a)(ρ) = (ρ − 1)[(NG ⊗ a) − ga1] = −ga(ρ − 1),
τ
(∂ 2 b)(σ, τ ) = σbτ − bστ + bσ = c(σ, τ ).
This result shows that the composite −δ5 δ4 δ3 is inverse to inv, since it maps
1
mod ZZ to γ.
g
The upper partial diagram is commutative, because δ1 (σ) ∪ χ is represented
by
X
X
X
τ (σ − 1) ⊗ χ(τ −1 ) = τ σ ⊗ χ(τ −1 ) − τ ⊗ χ(τ −1 )
τ
τ
τ
=
X
=
X
τ ⊗ χ(στ −1 ) −
τ
τ
X
τ ⊗ χ(τ −1 )
τ
τ ⊗ χ(σ) =
X
(τ − 1) ⊗ χ(σ)
τ
(see (1.4.8)), hence by the same element as δ3 (χ(σ)). The middle and the lower
partial diagrams are commutative by (1.4.3) and the remark following (1.4.6).
This proves that the diagram (∗) is commutative.
2
From now on let G be a profinite group. We extend the notion of “class
module”, which we needed for the duality theorem (3.1.5), to profinite groups
as follows. We denote open subgroups of G by the letters U, V, W .
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Chapter III. Duality Properties of Profinite Groups
(3.1.8) Definition. Let G be a profinite group. A formation module for G is
a discrete G-module C together with a system of isomorphisms
∼
invU/V : H 2 (U/V, C V ) −→
1
ZZ/ZZ
(U :V )
for every pair V ⊆ U of open subgroups, V normal in U , such that the following
conditions hold.
(i) H 1 (U/V, C V ) = 0.
(ii) For open normal subgroups W
C V )
H 2 (U/V,
inf
inv
1
ZZ/ZZ
(U :V )
⊆
V of an open subgroup U , the diagram
res
H 2 (U/W, C W )
H 2 (V /W, C W )
inv
incl
inv
(U :V )
1
ZZ/ZZ
(U :W )
1
ZZ/ZZ
(V :W )
is commutative.
The pair (G, C) is called a class formation.
Remarks: 1. For finite G the notion of a formation module is stronger than that
of a class module because there is no compatibility condition for the passage
to quotients for the latter.
2. From (ii) it follows that the diagram
$#"! C W )
H 2 (V /W,
inv
1
ZZ/ZZ
(V :W )
cor
H 2 (U/W, C W )
incl
inv
1
ZZ/ZZ
(U :W )
is commutative because cor VU ◦ res UV = (U : V ).
3. The isomorphisms
∼
inv : H 2 (G/V, C V ) −→
1
ZZ/ZZ
(G:V )
form a direct system. Passing to the direct limit, we obtain a homomorphism
inv : H 2 (G, C) −→ Q/ZZ,
which is called the invariant map. It is injective and its image is
1
ZZ/ZZ.
lim (G:V
)
1
ZZ/ZZ
#G
=
−→ V
4. In terms of G-modulations (see I §5), a formation module for G is a discrete
G-module C together with an isomorphism
inv : H 2 (C) → Ĥ 0 (ZZ)∗
of G-modulations, which has the additional property that H 1 (C) = 0.
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157
§1. Duality for Class Formations
5. For every open normal subgroup U ⊆ G, the module of invariants C U is a
1
Z) ∈ H 2 (G/U, C U )
class module for G/U . The elements inv−1
G/U ( (G:U ) mod Z
form a compatible system of fundamental classes for varying U . We therefore obtain reciprocity homomorphisms recG/U : C G (G/U )ab which are
compatible for open normal subgroups V ⊆ U in the sense that recG/U is the
composition of recG/V with the natural projection (G/V )ab (G/U )ab . Passing to the projective limit, we therefore obtain the reciprocity homomorphism
rec = recG : C G −−−→ Gab ,
which is also called the norm residue symbol. It has dense image and kernel
NG C.
Our aim is to prove a version of theorem (3.1.5) for profinite groups. We
consider a discrete G-module A which is finitely generated as a ZZ-module.
Then, for any discrete G-module B, Hom(A, B) is a discrete G-module and
the pairing
Hom(A, B) × A −→ B, (f, a) 7−→ f (a),
induces a cup-product
Ĥ i (G, Hom(A, B)) × Ĥ 2−i (G, A) −→ H 2 (G, B)
for all i ∈ ZZ: for i = 1 this is the usual cup-product and for i =/ 1 it is the
cup-product constructed in chap. I, §9, see (1.9.7).
We recall the definition of the topology on the Tate cohomology groups of
A from chap. I, §9. By (1.6.1), the groups Ĥ i (G/U, AU ) are finite for all i and
all open normal subgroups U of G. Therefore the groups
Ĥ i (G, A) = lim Ĥ i (G/U, AU ),
←−
U, def
i ≤ 0,
are naturally equipped with a profinite topology. For i > 0 we give Ĥ i (G, A)
the discrete topology. Now assume that C is a formation module for G and
that A is free as a ZZ-module. For sufficiently small U we have AU = A, and
hence Hom(A, C U ) = Hom(A, C)U . Therefore the groups
Ĥ i (G, Hom(A, C)) = lim Ĥ i (G/U, Hom(A, C)U ),
←−
U, def
i ≤ 0,
are naturally equipped with a profinite topology, as the groups on the right hand
side are finite by (3.1.5). For i ≥ 1, we give Ĥ i (G, Hom(A, C)) the discrete
topology.
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Chapter III. Duality Properties of Profinite Groups
(3.1.9) Theorem. Let G be a profinite group and let C be a formation module
for G. Let A be a discrete G-module which is finitely generated and free as a
ZZ-module. Then the cup-product
&% × Ĥ 2−i (G, A)
Ĥ i (G, Hom(A, C))
∪
H 2 (G, C)
inv
1
ZZ/ZZ ,
#G
induces a topological isomorphism
∼
Ĥ i (G, Hom(A, C)) −→ Ĥ 2−i (G, A)∨ .
for all i ∈ ZZ.
Proof: For sufficiently small U we have AU = A and therefore Hom(A, C U ) =
Hom(A, C)U . Thus, for i =/ 1, the theorem follows easily from (3.1.5) and
(1.9.6) by passing to the limit over G/U , where U runs through the open
normal subgroups in G. For i = 1, let U ⊆ G be an open normal subgroup
which acts trivially on A. Since A is ZZ-free, H 1 (U/V, AV ) = 0 for every open
normal subgroup V ⊆ U and therefore
∼ H 1 (G/V, AV ).
H 1 (G/U, AU ) →
Using the fact that C is a formation module for G, we obtain
H 1 (U/V, Hom(A, C)V ) = H 1 (U/V, Hom(A, C V )) ∼
= H 1 (U/V, C V )rank ZZ A = 0.
This implies that
∼ H 1 (G/V, Hom(A, C)V ).
H 1 (G/U, Hom(A, C)U ) →
A stationary limit process shows that (3.1.5) implies the desired result also in
dimension 1.
2
Recall from (1.9.10) that a level-compact G-module is a discrete G-module C
with a natural compact topology on C U for each open subgroup U ⊆ G.
(3.1.10) Corollary. Let G be a profinite group and let C be a level-compact
formation module for G. Then the reciprocity homomorphism
rec : C G −→ Gab
is surjective.
Proof: We apply (3.1.9) for i = 0 and A = ZZ. The reciprocity map rec is the
composite
∼
C G −→ Ĥ 0 (G, C) −→ H 2 (G, ZZ)∨ = Gab .
If C is level-compact, then, by (1.9.11), the first arrow is surjective.
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§1. Duality for Class Formations
159
The central result of this section is the following duality theorem for levelcompact class formations, which generalizes (3.1.9) to the case when A is not
necessarily free. In what follows the reader should note the following
Topological Remarks: Let A be a discrete G-module which is finitely generated as a ZZ-module and let C be a level-compact G-module.
For sufficiently small V we have AV = A and therefore
Hom(A, C)V = Hom(A, C V ).
Considering A as a group with the discrete topology, Hom(A, C V ) endowed
with the compact open topology is compact. For an arbitrary open subgroup U ⊆ G, the group Hom(A, C)U is a closed subgroup of the compact
group Hom(A, C)V , where V is chosen sufficiently small. We conclude
that Hom(A, C) is level-compact in a natural way. In particular, the groups
Ĥ i (G, Hom(A, C)) carry a natural profinite topology for i ≤ 0. If, as happens
in the situation of theorem (3.1.9), the groups Ĥ i (G/U, Hom(A, C)U ) are finite for all U , then the knowledge of the topology of the groups Hom(A, C)U
is redundant for an understanding of the profinite topology of the groups
Ĥ i (G, Hom(A, C)), i ≤ 0. Finally, recall that Ĥ i (G, Hom(A, C)) carries the
discrete topology for i ≥ 1 by convention.
In addition, assume that C is a formation module. Let U ⊆ G be an open
normal subgroup such that AU = A. Then, for i ∈ ZZ, the cup-product
1
Ĥ i (G/U, Hom(A, C)U'( ) × Ĥ 2−i (G/U, AU ) ∪ H 2 (G/U, C U ) inv (G:U
ZZ/ZZ
)
is continuous with respect to the topologies induced on the cohomology groups
by the complete standard complex. As the group H 2 (G/U, C U ) is compact
and finite, it is discrete. The groups Ĥ 2−i (G/U, AU ) are discrete and finite by
(1.6.1). Thus, for each i ≤ 0, we obtain a continuous homomorphism
Ĥ i (G/U, Hom(A, C)U ) −→ Ĥ 2−i (G/U, AU )∨
from a compact group to a finite discrete group. Passing to the projective limit
over all U , we obtain a continuous homomorphism
Ĥ i (G, Hom(A, C)) −→ H 2−i (G, A)∨
between compact abelian groups. For i ≥ 2 and U as above, the cup-product
defines a homomorphism
H i (G/U, Hom(A, C)U ) −→ Ĥ 2−i (G/U, A)∨ .
Passing to the direct limit, we obtain the (continuous) homomorphism
H i (G, Hom(A, C)) −→ Ĥ 2−i (G, A)∨
between discrete groups. Finally, for i = 1, the usual cup product induces a
(continuous) homomorphism
H 1 (G, Hom(A, C)) −→ H 1 (G, A)∨
from a discrete to a compact group.
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Chapter III. Duality Properties of Profinite Groups
(3.1.11) Duality Theorem. Let G be a profinite group and let C be a levelcompact formation module for G such that the group of universal norms NU C
is divisible for all open subgroups U of G. Let A be a discrete G-module
which is finitely generated as a ZZ-module. Consider the cup-product
1
)* × Ĥ 2−i (G, A) ∪ H 2 (G, C)
Ĥ i (G, Hom(A, C))
Z/ZZ .
inv #G Z
The following assertions are true:
(i) Let i ≤ 0. Then the pairing above induces a topological isomorphism
∼
Ĥ i (G, Hom(A, C)) −→ Ĥ 2−i (G, A)∨ .
This holds true for all i ∈ ZZ if A is ZZ-free.
(ii) Let i ∈ ZZ and let p be a prime number with p∞ |#G. Assume that C is
p-divisible. Then the pairing above induces an isomorphism
∼
Ĥ i (G, Hom(A, C))(p) −→ Ĥ 2−i (G, A)(p)∨ .
Remark: Part (ii) is only interesting if A has p-torsion because otherwise the
statement follows easily from (i). The isomorphism in (ii) is a topological
isomorphism for i =/ 1. For i = 1 it is an algebraic isomorphism from a discrete
group onto a compact group. It is a topological isomorphism if and only if
H 1 (G, A)(p) is finite. This is true in our main application, when G is the Galois
group of the maximal extension of a global field which is unramified outside a
given finite set of places.
Proof: If A is ZZ-free, assertion (i) follows from (3.1.9). For arbitrary A,
there exists an exact sequence
0 −→ R −→ F −→ A −→ 0
of G-modules, where R and F are finitely generated and ZZ-free. Applying the
functor Hom(−, C), we obtain an exact sequence of level-compact modules
0 −→ Hom(A, C) −→ Hom(F, C) −→ Hom(R, C) .
Let U run through the open normal subgroups such that F U = F . Then
NU Hom(F, C) = Hom(F, NU C), NU Hom(R, C) = Hom(R, NU C)
and the map
NU Hom(F, C) −→ NU Hom(R, C)
is surjective since NU C is divisible. For i
diagram in which we write ∼ for Hom(−, C):
≤
0 we obtain a commutative
+,-./01234567 F̃ )
Ĥ i−1 (G,
Ĥ i−1 (G, R̃)
Ĥ i (G, Ã)
Ĥ i (G, F̃ )
Ĥ i (G, R̃)
H 3−i (G, F )∨
H 3−i (G, R)∨
H 2−i (G, A)∨
H 2−i (G, F )∨
H 2−i (G, R)∨ .
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161
§1. Duality for Class Formations
The upper row is exact by (1.9.13) and the lower row is exact, too. The
vertical arrows, except the middle one, are isomorphisms of compact abelian
groups by (3.1.9). Furthermore, using (1.4.6), it is easy to see that all squares
commute up to sign. Hence, by the five lemma (in the abelian category of
compact abelian groups), the middle one is also a topological isomorphism.
This proves assertion (i).
In order to prove (ii), we may, without loss of generality, assume that the
torsion part of A consists only of p-torsion. Since C is p-divisible, we now
obtain the exact sequence
0 −→ Hom(A, C) −→ Hom(F, C) −→ Hom(R, C) −→ 0,
and, using (1.9.14), the long exact cohomology sequence which is unbounded
in both directions
· · · −→ Ĥ i (G, Ã)(p) −→ Ĥ i (G, F̃ )(p) −→ Ĥ i (G, R̃)(p) −→ · · ·
in which we wrote ∼ for Hom(−, C). Furthermore, (1.9.9) gives us a corresponding long exact cohomology sequence associated to the short exact
sequence 0 → R → F → A → 0 in which all groups are compact or discrete
and all maps are continuous. Hence this sequence remains exact after taking
Pontryagin duals. Thus the duality map of (3.1.11) defines a map between two
long exact sequences. This map commutes up to signs with the differentials.
For the square
Ĥ 0 (G,89:; R̃)(p)
H 1 (G, Ã)(p)
H 2 (G, R)∨ (p)
H 1 (G, A)∨ (p)
this follows from the surjectivity of the map H 0 (G, R̃) → Ĥ 0 (G, R̃) by
(1.9.11). For the square
H 1 (G,<=>? R̃)(p)
H 2 (G, Ã)(p)
H 1 (G, R)∨ (p)
Ĥ 0 (G, A)∨ (p)
it follows from the isomorphism Ĥ 0 (G, A)(p) ∼
= H 0 (G, A) ⊗ ZZp by (1.9.8).
That all other squares commute up to sign is obvious by the definition of
the maps occurring and by the compatibilities proved in I §4. Therefore the
statement (ii) for A follows from that for F and R and from the five lemma.
2
Remark: Theorem (3.1.11)(i), together with the idea of the proof, was formulated by G. POITOU in 1966 (see [168]) in order to prove an important duality
theorem over local and global fields that was announced without proof by
J. TATE in 1962 (see [229]). In 1969, a proof based on the same idea was
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162
Chapter III. Duality Properties of Profinite Groups
independently given by K. UCHIDA (see [236]). A similar proof was presented
by K. HABERLAND in 1978 (see [72]), but this, however, is not without mistakes. UCHIDA’s proof is correct, but rather terse, and we have given a detailed
account of his arguments. These papers, however, all failed to prove the central fact that the cup-product for the Tate cohomology of profinite groups is
well-defined (see I §9). This gap was filled in [196], where also the extension
of the theorem by assertion (ii) was proven.
We mention a further feature of a class formation (G, C), which plays an
important role in “non-abelian class field theory”. Assume that C has trivial
T
universal norms, i.e. NU C = V ⊆U NU/V C V = 1 for every open subgroup U
of G, and that C is a topological G-module such that C V /C U is compact for
every pair V ⊆ U of open subgroups. For every triple W ⊆ V ⊆ U of open
subgroups, W, V normal in U , we may consider the diagram
1GHIJFE@ABCD
CW
NV /W
(1)
1
W(U/W )
U/W
ϕV /W
CV
1
π
W(U/V )
U/V
1,
where the horizontal sequences are the group extensions defined by the fundamental classes uU/W ∈ H 2 (U/W, C W ) and uU/V ∈ H 2 (U/V, C V ). One can
show that the diagram of solid arrows can be commutatively completed by an
arrow ϕV /W . Because of the compactness assumption, one can moreover show
that there exists a transitive family of arrows ϕV /W , i.e. ϕW/W 0 ◦ϕV /W = ϕV /W 0
for U ⊇ V ⊇ W ⊇ W 0 . It is therefore possible to take the (left exact) projective
limit, which yields an exact sequence
f
g
1 −→ C × −→ W −→ G
(2)
with C × = lim C V ; the right arrow has a dense image. The group W is
←− V
called the Weil group of the class formation (G, C). W is a topological group
and has the following properties.
(i) Let W(U ) := g −1 (U ) and let W(U )c be the closure of the commutator
subgroup of W(U ). If V is open and normal in U , then
W(U/V ) = W(U )/W(V )c ,
and we have a commutative exact diagram
1QRSTPOKLMN
C×
N
1
CV
W(U )
ϕ
W(U/V )
U
π
U/V
1,
∼ W(U )ab .
and, in particular, a canonical isomorphism ρU : C U →
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§1. Duality for Class Formations
(ii) For every pair of open subgroups V
CXWVU V
ρV
CU
W(U )ab
ρU
∈
W, the diagrams
W(U )ab
σ∗
Ver
ρU
U and every σ
C\[ZY U
W(V )ab
incl
⊆
,
cσ
CU
σ
ρU σ
W(U σ )ab
are commutative, where U σ = σU σ −1 , σ ∗ is the action of g(σ)
cσ (x) = σxσ −1 .
∈
G on C and
The Weil group W is determined by these properties up to isomorphism in the
following sense. If W 0 is another topological group with these properties, then
there is an isomorphism W → W 0 , compatible with the above structures, and
this isomorphism is uniquely determined up to an inner automorphism of W 0
by an element of C × .
The proofs of these assertions are rather deep. Since we will not make use of
the Weil group in this book, we refer for the details to [6], chap.14 and [124],
chap.IX, 3.
Exercise 1. If G is a finite group and C a class module, then H 3 (G, C) = 1 and H 4 (G, C) ∼
=
Hom (G, Q/ZZ).
Exercise 2. Let G be a finite group, C a class module and A a ZZ-free G-module. Show that
every fundamental class γ ∈ H 2 (G, C) defines an isomorphism
γH ∪ : Ĥ n (H, A) → Ĥ n+2 (H, A ⊗ C)
for every n ∈ ZZ and every subgroup H.
Exercise 3. Show that the duality theorem (3.1.11)(i) may be interpreted as an isomorphism
of G-modulations
Ĥ i (Hom(A, C)) ∼
= Ĥ 2−i (A)∨ .
Exercise 4. Let G be a finite group, A a G-module and α ∈ H 2 (G, A). Apply to the four term
exact sequence
(1)
0 −→ A −→ A(α) −→ ZZ[G] −→ ZZ −→ 0
the exact functor Hom ( , Q/ZZ) and obtain the exact sequence
(2)
0 −→ Q/ZZ −→ ZZ[G]∗ −→ A(α)∗ −→ A∗ −→ 0.
(i) Show that the homomorphism
δ 2 : Ĥ n (G, A∗ ) −→ Ĥ n+2 (G, Q/ZZ),
arising from this sequence, coincides with the cup-product β 7→ −α ∪ β which is induced by
the pairing A × A∗ → Q/ZZ.
(ii) Show that δ 2 = −α∪ is an isomorphism for all n
fundamental class.
∈
ZZ if A is a class module and α is a
Hint: (i) Apply proposition (1.4.6) to the two pairs of exact sequences
0 −→ IG −→ ZZ[G] −→ ZZ −→ 0, 0 −→ A −→ A(α) −→ IG −→ 0,
∗
0 −→ ZZ∗ −→ ZZ[G]∗ −→ IG
−→ 0,
∗
0 −→ IG
−→ A(α)∗ −→ A∗ −→ 0.
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Chapter III. Duality Properties of Profinite Groups
Let β ∈ Ĥ n (G, A∗ ) and consider the element 1 ∈ H 0 (G, ZZ). The homomorphism δ 2 :
H 0 (G, ZZ) → H 2 (G, A) arising from (1) maps 1 to α (see the proof of (3.1.5)). Therefore
δ 2 β = δ(δβ) ∪ 1 = (−1)n+2 (δβ ∪ δ1) = (−1)n+2+n+1 (β ∪ δ 2 1)
= −(β ∪ α) = −α ∪ β.
(ii) If A is a class module, then A(α) is cohomologically trivial by (3.1.4), hence also A(α)∗
is by (1.8.5).
§2. An Alternative Description of the
Reciprocity Homomorphism
Let G be a profinite group and let C be a G-module. If C is a formation
module for G, then we have the reciprocity homomorphism
recG : C G −→ Gab ,
which was obtained from the Nakayama-Tate duality theorem (3.1.5) by passing to the projective limit. If the pair (G, C) satisfies certain conditions described below, we get a simple criterion for C being a formation module and
an alternative, non-cohomological, description of the reciprocity homomorphism as presented in [160], chap.IV. This method applies, for instance, to the
absolute Galois group of a local or global field and becomes important if one
wants to understand the reciprocity homomorphism more explicitly. However,
we will not use the results below in the following, so the reader might skip this
section on the first reading.
Assume we are given a G-module C and a pair of homomorphisms
d
v
(G −→ ẐZ, C G −→ ẐZ) ,
where d is surjective and Z = v(C G ) has the properties
ZZ ⊆ Z
Z/nZ ∼
= ZZ/nZZ for all n ∈ IN ,
and
in particular, the cokernel ẐZ/Z of v is uniquely divisible. Furthermore, we
assume that, for every open subgroup U of G,
v(NG/U C U ) = fU Z
with fU = (ẐZ : d(U )). Then we have the surjective homomorphisms
dU =
1
d
fU
: U −→ ẐZ
,
vU =
1
v
fU
◦ NG/U : C U −→ Z.
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§2. An Alternative Description of the Reciprocity Homomorphism
For every pair V ⊆ U of open subgroups, we set fU/V = fV /fU , eU/V =
(U : V )/fU/V , and we get the commutative diagrams
V a`_^]b ab
dV
fU/V
i Ver
Chgfedc V
ẐZ
eU/V
NU/V
vV
ẐZ
fU/V
incl
eU/V
d
vU
U
U ab
ẐZ,
CU
ẐZ.
The situation is most briefly, and best, formulated in the language of G-modulations (see I §5). We have on the one hand the fundamental G-modulation
π ab : U 7−→ U ab ,
where, for two open subgroups V
⊆
U , the maps
res U
V
U jiab
V ab
ind V
U
are the transfer Ver : U ab → V ab and the map induced by the inclusion V ,→ U .
On the other hand, we consider C as the G-modulation
C : U 7−→ C U ,
where res UV and ind VU are the inclusion and the norm. C is endowed with the
submodulation N C : U 7→ NU C. The submodulation N π ab : U 7→ NU U ab of
π ab is trivial, because we have a surjection
1 = lim
V lim
V [U, U ]/[U, U ] = NU U ab .
←−
←−
V ⊆U
V ⊆U
Finally, we consider the G-modulation U 7→ ẐZ, where
res UV = eU/V
and
ind VU = fU/V .
We denote this G-modulation by ẐZ (observe that it depends on d : G → ẐZ
since the numbers fU/V and eU/V do). Then, in the above situation, we have
two morphisms
d : π ab −→ ẐZ,
v : C −→ ẐZ
of G-modulations. The main result of abstract class field theory may now be
formulated as follows.
(3.2.1) Theorem. Assume that for every pair V
with V normal in U and U/V cyclic, we have
(
i
V
#Ĥ (U/V, C ) =
(U : V )
⊆
U of open subgroups of G,
for
i = 0,
1
for i = 1,
the class field axiom. Then there is a unique morphism
r : C −→ π ab
of G-modulations such that v = d ◦ r.
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166
Chapter III. Duality Properties of Profinite Groups
Proof: The theorem is just a reformulation of the results in [160], chap.IV.
For every pair V ⊆ U of open subgroups of G, V normal in U , we have by
[160], chap.IV, (6.3) a canonical isomorphism
V
∼ C U /N
r̂U/V : (U/V )ab −→
U/V C .
We briefly recall the definition of r̂U/V (see [160], chap.IV, (5.6), where, in
contrast to our notation, this map was called the reciprocity homomorphism).
Let I = ker d and IU = I ∩ U . The semigroup
Frob(U/IV ) = {σ̃
∈
U/IV | dU (σ̃) ∈ IN} ,
whose elements we call Frobenius lifts, maps surjectively onto U/V
Frob(U/IV ) → U/V,
cf. [160], chap.IV, (4.4). If σ
r̂U/V (σ̄) is defined by
∈
σ̃ 7→ σ = σ̃ mod V ,
U/V and σ̄ is its image in (U/V )ab , then
r̂U/V (σ̄) = NU/S (πS ) mod NU/V C V ,
where S = hσ̃iIV ⊆ U for some Frobenius lift σ̃ of σ and πS ∈ C S is any
element with vS (πS ) = 1. If the class field axiom holds, it was shown in [160],
chap.IV that r̂U/V is a well-defined homomorphism which, in particular, does
not depend on the various choices and which moreover is an isomorphism.
Taking the projective limit over V of the surjections
rU/V kl : C U
C U /NU/V C V
yields a family of homomorphisms
(r̂U/V )−1
(U/V )ab
∼ U ab = π ab (U )
rU : C(U ) = C U C U /NU C ,→ lim C U /NU/V C V −→
←−
V
with dense image. By [160], chap.IV, (6.4) and (6.5), this family defines a
morphism r : C −→ π ab of G-modulations such that the diagram
Cnmo
r
v
π ab
d
ẐZ
is commutative.
In order to prove uniqueness, first observe that any morphism r : C → π ab
factors through C/N C, because π ab has trivial universal norms, and the same
is true for v. Let U be an open subgroup of G. For an open normal subgroup
V of U , we consider the group U/IV . Obviously, it is enough to show that in
the commutative diagram
rU/IV
Crpq U
vU
(U/IV )ab
dU
ẐZ
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167
§2. An Alternative Description of the Reciprocity Homomorphism
the homomorphism rU/IV = lim
←− IV ⊆W ⊆U
rU/W (W open and normal in U ) is
uniquely determined by dU and vU , since then we can pass to the projective
limit over V in order to obtain the desired result. Let S be the finite set of
splittings of the surjection dU : U/IV ẐZ. We denote the images by Ws /IV
for s ∈ S, where Ws is an open subgroup of U . Since
hWs /IV , s ∈ Si = U/IV ,
we see that hWs , s ∈ Si = U . By [160], chap.IV, (6.7), we have the equality
0
0
NU |W C W · NU |W 0 C W = NU |W W 0 C W W for two open subgroups W and W 0 of
U . Since S is finite, we obtain
hNU |Ws C Ws , s ∈ Si = C U .
Now the commutative diagram
rU/IV
Cstuvwx U
(U/IV )ab
NU |Ws
rWs /IV
C Ws
vWs
Ws /IV
dWs
ẐZ
shows that rU/IV is uniquely determined by dU and vU since this is the case for
all maps rWs /IV .
2
The morphism r : C → π ab yields, in particular, a homomorphism
rG : C G −→ Gab ,
which we now want to compare with the homomorphism at the beginning of
this section.
d
U
For every open subgroup U of G, we again set IU = ker(U −→
ẐZ) and ΓU =
U/IU . We have a commutative diagram
C Iy{|z U
ṽU
Z
v
U
CU
Z
where ṽU = lim
vU 0 . Recalling that Z/ZZ is uniquely divisible, we get
−→ IU ⊆U 0 ⊆U
the sequence of canonical homomorphisms
(ṽ )
U ∗
∼ Q/Z
∼ H 1 (Γ , Q/Z
Z) −→
Z,
H 2 (ΓU , C IU ) −→
H 2 (ΓU , Z) = H 2 (ΓU , ZZ) −→
U
and we denote its composite by
invU : H 2 (ΓU , C IU ) −→ Q/ZZ.
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168
Chapter III. Duality Properties of Profinite Groups
For a pair V ⊆ U of open subgroups, we have the compatible pair of homomorphisms ΓV → ΓU , C IU ,→ C IV , which yields a canonical homomorphism
res UV : H 2 (ΓU , C IU ) −→ H 2 (ΓV , C IV ),
and we have the commutative diagram
H 2 (ΓV„…†ƒ‚}~€ , C IV )
res
H 2 (ΓU , C IU )
δ −1
H 2 (ΓV , ZZ)
H 1 (ΓV , Q/ZZ)
eU/V ·res
H 2 (ΓU , ZZ)
eU/V ·res
δ −1
H 1 (ΓU , Q/ZZ)
Q/ZZ
(U :V )
Q/ZZ,
noting that eU/V · fU/V = (U : V ) and that the canonical generator ϕV of ΓV is
mapped onto the fU/V -th power of the canonical generator ϕU of ΓU .
(3.2.2) Proposition. If C satisfies the class field axiom (3.2.1), then, for every
open subgroup U of G, the homomorphism
invU : H 2 (ΓU , C IU ) −→ Q/ZZ
is an isomorphism.
(ṽ )
U ∗
Proof: We have to show that H 2 (ΓU , C IU ) −→
H 2 (ΓU , Z) is an isomorphism.
−1
Un
Let Un := dU (nẐZ) and Dn := ker(vUn : C → Z). From [160], chap.IV,
(6.2) it follows that the class field axiom (3.2.1) implies Ĥ i (U/Un , Dn ) = 0 for
i = 0, −1, hence for all i. Since
[
Dn ,
D := ker(ṽU : C IU → Z) = Dn = lim
−→
n
n
we get for i ≥ 1
H i (ΓU , D) = lim H i (U/Un , Dn ) = 0 .
−→
n
ṽ
U
Z −→ 0
Taking the cohomology of the exact sequence 0 −→ D −→ C IU −→
of ΓU -modules, we get the exact sequence
(ṽ )
δ
U ∗
0 = H 2 (ΓU , D) −→ H 2 (ΓU , C IU ) −→
H 2 (ΓU , Z) −→ H 3 (ΓU , D) = 0.
Therefore (ṽU )∗ is an isomorphism.
2
(3.2.3) Proposition. Under the above assumptions, the following assertions
are equivalent.
(i) C satisfies the class field axiom (3.2.1).
(ii) H 2 (ΓU , C IU ) = H 2 (U, C) for all open subgroups U of G and C is a
formation module with respect to the isomorphisms
∼
invU : H 2 (U, C) −→
Q/ZZ.
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§2. An Alternative Description of the Reciprocity Homomorphism
169
Proof: If C is a formation module, then for every open normal subgroup V
of U ,
(
ZZ/(U : V )ZZ
if i = 2 ,
i
V ∼
H (U/V, C ) =
0
if i = 1 ,
and H 2 (U/V, C V ) ∼
= Ĥ 0 (U/V, C V ) if U/V is cyclic, so that the class field
axiom holds.
Conversely, assume that C satisfies the class field axiom. We claim that
#H 2 (U/V, C V ) (U : V )
for every pair V / U . In fact, this is true if U/V is cyclic because H 2 ∼
= Ĥ 0 .
If U/V is a p-group, it follows inductively from the exact sequence
0 −→ H 2 (U/W, C W ) −→ H 2 (U/V, C V ) −→ H 2 (W/V, C V ),
where W/V is a normal subgroup of U/V of order p . In the general case, let
(U/V )p be a p-Sylow subgroup of U/V and Up the pre-image of (U/V )p in U .
Since the restriction map
res : H 2 (U/V, C V ) ,→
M
H 2 ((U/V )p , C Up )
p
is injective by (1.6.10), we obtain
Y
#H 2 (U/V, C V ) #H 2 ((U/V )p , C Up )
p
Y
#(U/V )p = #(U/V ) .
p
For every open normal subgroup V of U , we have the exact sequence
inf
res
0 −→ H 2 (U/V, C V ) −→ H 2 (U, C) −→ H 2 (V, C)
and we identify H 2 (U/V, C V ) with its image in H 2 (U, C). Let n = (U : V )
and let Un = d−1
Z). Then
U (nẐ
H 2 (U/V, C V ) = H 2 (U/Un , C Un ).
In fact, because
#H 2 (U/V, C V ) (U : V ) = (U : Un ) = #H 2 (U/Un , C Un ) ,
it suffices to show the inclusion "⊇". But this follows from the exact commutative diagram
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170
Chapter III. Duality Properties of Profinite Groups
0Ž‡ˆ‰Š‹Œ
H 2 (U/V, C V )
H 2 (U, C)
res
H 2 (V, C)
H 2 (ΓU , C IU )
res
H 2 (ΓV , C IV )
invU
invV
(U :V )
Q/ZZ
Q/ZZ,
in which invU and invV are isomorphisms by (3.2.2). Since H 2 (U/Un , C Un ) ⊆
H 2 (ΓU , C IU ) has order n = (U : V ), it is mapped by the middle arrow res , and
thus by the upper arrow res , to zero, hence
H 2 (U/Un , C Un ) ⊆ H 2 (U/V, C V ).
We therefore obtain
H 2 (U, C) =
[
H 2 (U/V, C V ) =
[
H 2 (U/Un , C Un ) = H 2 (ΓU , C IU ).
n
V
For V open and normal in U , we have the commutative diagram
“’‘ C)
H 2 (V,
invV
∼
Q/ZZ
res
(U :V )
H 2 (U, C)
invU
∼
Q/ZZ,
and the induced isomorphisms
invU/V : H 2 (U/V, C V ) −→ (U1:V ) ZZ/ZZ
define on C the structure of a formation module in the sense of (3.1.8).
2
Let us assume that C satisfies the class field axiom and thus is, in particular,
a formation module with respect to the isomorphisms
∼ Q/Z
invU : H 2 (U, C) −→
Z.
Noting that H 2 (U, ZZ) ∼
= H 1 (U, Q/ZZ) = Hom(U ab , Q/ZZ), the cup-product
inv
lim
Ĥ 0 (U/V, C V ) × H 2 (U, ZZ) −→ H 2 (U, C) −→
Z
∼ Q/Z
←−
V
yields by (3.1.5) homomorphisms
∼ U ab .
recU : C U C U /NU C ,→ lim C U /NU/V C V −→
←−
V
These homomorphisms commute with conjugation, restriction and corestriction by the rules (1.5.3) and (1.5.7), i.e. they form a morphism
rec : C −→ π ab
of G-modulations.
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171
§3. Cohomological Dimension
(3.2.4) Theorem. The morphisms
r, rec : C −→ π ab
of G-modulations coincide.
Proof: Because of the uniqueness assertion of (3.2.1), it suffices to show that
the diagram
rec
C”–•
π ab
v
d
ẐZ
commutes. Let U be an open subgroup of G. As before, we put IU =
dU
ker(U −→
ẐZ) and ΓU = U/IU . We have the commutative diagram
›™š¡œ˜—¢ Ÿž C)
H 0 (U,
∪
× H 2 (U, ZZ)
H 2 (U, C)
H 0 (ΓU , C IU ) × H 2 (ΓU , ZZ)
H 0 (ẐZ, Z)
Q/ZZ
inf
inf
∪
H 2 (ΓU , C IU )
(dU )∗
vU
inv
× H 2 (ẐZ, ZZ)
(dU )∗ (ṽU )∗
∪
H 2 (ẐZ, Z)
inv
Q/ZZ ,
which induces the commutative diagram
C¤¥¦£ U
vU
Z
recU
U ab
dU
ẐZ .
This proves the theorem.
2
§3. Cohomological Dimension
Let G be a profinite group, Mod(G) the category of (discrete) G-modules, and
Modt (G), Modp (G), Modf (G) the category of G-modules which are torsion,
p-torsion, finite respectively as abelian groups. A fundamental numerical
invariant of G is the cohomological dimension.
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Chapter III. Duality Properties of Profinite Groups
(3.3.1) Definition. The cohomological dimension cd G (resp. strict cohomological dimension scd G) of G is the smallest integer n such that
H q (G, A) = 0 for all
q>n
and all A ∈ Modt (G) (resp. A ∈ Mod(G)), and is ∞ if no such integer exists.
Let p be a prime number. The cohomological p-dimension cdp G (resp.
strict cohomological p-dimension scdp G) is the smallest integer n such that
the p-primary part
H q (G, A)(p) = 0 for all q > n ∗)
and all A ∈ Modt (G) (resp. A ∈ Mod(G)), and is ∞ if no such integer exists.
The profinite group G is of virtual cohomological dimension vcd G = n
resp. virtual strict cohomological dimension vscd G = n if there exists an
open subgroup U of G such that cd U = n resp. scd U = n. The p-versions
vcdp G and vscdp G are defined analogously.
Since
M every abelian torsion group X decomposes into the direct sum
X=
X(p) of its p-primary parts X(p), we have
p
cd G = sup cdp G and
p
scd G = sup scdp G.
p
If G has an element of order p − in particular, if G is finite and p | #G −
then cdp G = ∞. In fact, if H is a subgroup of order p, then by (1.6.4) and
(1.7.1),
H 2n (G, IndH
Z/pZZ)) ∼
= H 2n (H, ZZ/pZZ) ∼
= Ĥ 0 (H, ZZ/pZZ) = ZZ/pZZ =/ 0
G (Z
for all n ≥ 0.
(3.3.2) Proposition. The following conditions are equivalent.
(i) cdp G ≤ n,
(ii) H q (G, A) = 0 for all q > n and all A ∈ Modp (G),
(iii) H n+1 (G, A) = 0 for all simple G-modules A with pA = 0.
For a pro-p-group G, we have, in particular,
cd G ≤ n
⇔
H n+1 (G, ZZ/pZZ) = 0.
∗) The p-primary part X(p) of an abelian torsion group X consists of all elements of X
which have p-power order.
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§3. Cohomological Dimension
Proof: (i) ⇔ (ii) follows from A =
M
A(p) if A ∈ Modt (G), and H q (G, A)(p)
p
= H q (G, A(p)). Assume (iii). If A is finite of p-power order, then H n+1 (G, A) =
0 follows by induction on #A by the exact cohomology sequence associated
to the exact sequence 0 → B → A → A/B → 0, where B is a simple non-zero
submodule of A. It then follows for general A ∈ Modp (G) by taking direct
limits. From this, we get (ii) by induction on q, considering the exact sequence 0 → A → IndG (A) → A1 → 0, which by (1.3.8) yields the isomorphism
H q+1 (G, A) ∼
2
= H q (G, A1 ).
(3.3.3) Proposition.
Proof: cdp G
sequences
≤
cdp G ≤ scdp G ≤ cdp G + 1.
scdp G is trivial. Let A
p
0 −→ p A −→ A −→ pA −→ 0,
∈
Mod(G) and consider the exact
0 −→ pA −→ A −→ A/pA −→ 0.
q
Let q > cdp G + 1. Then H (G, p A) = H q−1 (G, A/pA) = 0 since p A and
A/pA ∈ Modp (G). Therefore
H q (G, A) −→ H q (G, pA)
and H q (G, pA) −→ H q (G, A)
are injective. The composite is multiplication by p since the composite of
p
A → pA → A is multiplication by p. It follows that H q (G, A)(p) = 0, showing
scdp G ≤ cdp G + 1.
2
(3.3.4) Corollary. Assume that cdp G = n is finite. Then scdp G = n if and
only if H n+1 (U, ZZ)(p) = 0 for all open subgroups U of G.
Proof: Assume the latter. If the G-module A is finitely generated as a ZZmodule, then there is an open subgroup U of G acting trivially on A, and A
is a quotient B/C of B = IndUG (ZZm ) for some m. Since scdp G ≤ n + 1, we
have H n+1 (G, A)(p) = 0, and this result extends to an arbitrary G-module A
by passing to the direct limit.
2
Example: Let G = ẐZ = lim ZZ/nZZ. Then
←− n
cdp G = 1
and scdp G = 2.
In fact, if A is a finite G-module of p-power order, then every extension
0 −→ A −→ Ĝ −→ G −→ 1
splits (the closed subgroup (σ) of Ĝ, topologically generated by a pre-image
σ ∈ Ĝ of 1 ∈ G, is mapped isomorphically onto G). By (1.2.4), H 2 (G, A) ∼
=
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Chapter III. Duality Properties of Profinite Groups
EXT(G, A), hence H 2 (G, A) = 0 for finite A ∈ Modp (G), and by taking
inductive limits, for all A ∈ Modp (G). Noting that H 1 (G, ZZ/pZZ) = ZZ/pZZ =/ 0,
this shows cdp G = 1. On the other hand, H 2 (G, ZZ) ∼
= H 1 (G, Q/ZZ) ∼
= Q/ZZ,
so that scdp G = 2.
(3.3.5) Proposition. If H is a closed subgroup of G, then
cdp H ≤ cdp G and scdp H ≤ scdp G.
We have equality in each of the following cases:
(i) (G : H) is prime to p,
(ii) H is open and cdp G < ∞.
Proof: The inequalities follow from Shapiro’s lemma (1.6.4)
∼ q
H q (G, IndH
G (A)) = H (H, A),
noting that if A is torsion, then so is IndH
G (A). On the other hand, in case (i),
q
res : H (G, A)(p) −→ H q (H, A)(p)
is injective (see the proof of (1.6.10)). In case (ii), consider for A ∈ Mod(G)
the exact sequence
ν
0 −→ B −→ IndH
G (A) −→ A −→ 0
of G-modules, where ν is given by νx = σ∈G/H σx(σ −1 ). We obtain a
homomorphism
n
H n (H, A)(p) = H n (G, IndH
G (A))(p) −→ H (G, A)(p),
which is surjective if H n+1 (G, B)(p) = 0. This is the case if either n = scdp G
or n = cdp G and A ∈ Modt (G). Thus in either case, (i) or (ii), we have the
implication
H n (H, A)(p) = 0 ⇒ H n (G, A)(p) = 0
P
and this means scdp H
≥
scdp G or cdp H
≥
cdp G respectively.
2
SERRE has shown that a much weaker condition than cdp G < ∞ guarantees
the equality cdp G = cdp H for an open subgroup H. One requires only that G
contains no element of order p (see [211] and [75]).
(3.3.6) Corollary. If Gp is a p-Sylow subgroup of G, then
cdp G = cdp Gp = cd Gp and scdp G = scdp Gp = scd Gp .
(3.3.7) Corollary. Let G be a profinite group. Then
cdp G = 0 if and only if #G is prime to p.
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§3. Cohomological Dimension
Proof: By (3.3.6), we may assume that G is a pro-p-group, and then the
assertion follows from (1.6.14)(ii).
2
(3.3.8) Proposition. If H is a closed normal subgroup of G, then
cdp G ≤ cdp G/H + cdp H.
If cdp G/H < ∞ and cdp H < ∞, and if H n (U, ZZ/pZZ) is finite for n = cdp H
and all open subgroups U of H ∗) , then equality holds.
Proof: We may assume that m = cdp G/H and n = cdp H are finite. Consider
for A ∈ Modp (G) the Hochschild-Serre spectral sequence
E2ij = H i (G/H, H j (H, A)) ⇒ H i+j (G, A) = E i+j .
Let q > m + n. If i + j = q, then either i > m or j > n, hence E2ij = 0. As
ij
of E2ij , we get
H q (G, A) has a filtration, whose quotients are subquotients E∞
H q (G, A) = 0, so that cdp G ≤ m + n.
Now assume that H n (U, ZZ/pZZ) is finite for all open subgroups U of H. Since
cdp H = n, there exists a finite H-module A with pA = 0 and H n (H, A) =/ 0.
Let G0 be an open subgroup of G such that H 0 = G0 ∩ H acts trivially on A.
The canonical surjection
0
ν
IndH
H (A) → A,
x 7→
σx(σ −1 ),
X
σ ∈H/H 0
induces a surjection
0
ν
∗
n
H n (H 0 , A) = H n (H, IndH
H (A)) −→ H (H, A),
again because n = cdp H. Therefore H n (H 0 , A) =/ 0, and hence the group
H n (H 0 , ZZ/pZZ) is finite and non-zero.
Now let G00 /H 0 be a p-Sylow subgroup of G0 /H 0 . Then cdp G00 /H 0 =
cdp G0 /H 0 = cdp G/H = m and cdp H 0 = cdp H = n by (3.3.5). Therefore, by
(2.1.4),
H m+n (G00 , ZZ/pZZ) = H m (G00 /H 0 , H n (H 0 , ZZ/pZZ)).
This group is non-zero. Namely, since G00 /H 0 is a pro-p-group, there is
a surjective homomorphism H n (H 0 , ZZ/pZZ) → ZZ/pZZ of G00 /H 0 -modules,
which induces a surjective homomorphism
H m (G00 /H 0 , H n (H 0 , ZZ/pZZ)) −→ H m (G00 /H 0 , ZZ/pZZ) =/ 0.
We therefore obtain cdp G ≥ cdp G00
≥
m + n.
We take the following theorem from [247], see also [45].
∗) If H is a pro-p-group, this is already true if H n (H, ZZ/pZZ) is finite.
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Chapter III. Duality Properties of Profinite Groups
(3.3.9) Theorem. Let N be a closed normal subgroup of a profinite group G
and let cdp G = n and cdp N = k. Assume that the group H k (N, ZZ/pZZ) is
finite and nonzero. Then
vcdp G/N = n − k .
Remark: The condition H k (N, ZZ/pZZ) =/ 0 is necessary. If N is a pro-pgroup, then cdp N = k implies H k (N, ZZ/pZZ) =/ 0, see (3.3.2). This is not true
for a general profinite group N .
Proof: In the following we write H ∗ (−) for H ∗ (−, ZZ/pZZ). Replacing
G by the pre-image of a p-Sylow subgroup of G/N under the canonical
projection and using (3.3.6), we may assume that G/N is a pro-p-group. Since
H k (N ) = lim 0 H k (N/N 0 ) is finite, there exists an open subgroup M of N ,
−→ N
which is normal in G, such that the inflation map
inf : H k (N/M ) −→ H k (N )
is surjective.
Now there exists an open subgroup U0 of G containing M such that
(∗)
U0 ∩ N = M and U0 N/M ∼
= N/M × U0 /M .
This is a consequence of the following group theoretical
Claim: Let N be a finite normal subgroup of the profinite group G. Then there
exists an open subgroup U of G, such that U is contained in the centralizer of
N in G and U ∩ N = 1. In particular, U N is open in G and U N ∼
= U × N.
Proof of the claim: Let K be the kernel of the canonical homomorphism
G → Aut(N ) which is induced by the conjugation. Then K is open and normal
in G, and K ∩ N is the center Z(N ) of N . Choosing for every element
z ∈ Z(N )\{1} an open normal subgroup Uz of K such that z ∈/ Uz , then the
T
group U = z∈Z(N )\{1} Uz has the desired properties.
Using the claim for the finite subgroup N/M of G/M , we obtain U0 as in
(∗). Since H k (N/M ) is finite, replacing U0 by a suitable open subgroup, we
may further assume that U0 acts trivially on H k (N/M ). Then H k (N ) is a finite
trivial U -module where U = U0 N .
Now we consider the Hochschild-Serre spectral sequences
E2p,q = H p (U/N, H q (N )) ⇒ H p+q (U )
and
p
q
p+q
E p,q
(U/M ) ,
2 = H (U/N, H (N/M )) ⇒ H
which are associated to the group extensions
1 → N → U → U/N → 1 and 1 → N/M → U/M → U/N → 1 .
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177
§3. Cohomological Dimension
By (2.4.6), the second spectral sequence degenerates at E 2 .
Let X • (U, ZZ/pZZ), X • (U/M, ZZ/pZZ) and X • (U/N, ZZ/pZZ) be the standard
resolutions of ZZ/pZZ as a trivial U -module, U/M -module and U/N -module,
respectively. Then the canonical map X • (U/M, ZZ/pZZ) → X • (U, ZZ/pZZ) induces a homomorphism of double complexes
X • (U/N, X • (U/M, ZZ/pZZ)N/M )U/N → X • (U/N, X • (U, ZZ/pZZ)N )U/N ,
and so we obtain a morphism of the associated E2 -spectral sequences
pq
¯
(E pq
r , dr ) −→(Er , dr ) .
Since cdp N = k, and as the second spectral sequence degenerates, we have a
commutative diagram
n+1−k,k
§¨©ª
E∞
E2n+1−k,k
E n+1−k,k
∞
E n+1−k,k
.
2
Next we prove that E2n+1−k,k vanishes. By (3.3.5)(ii), we have cdp U = n,
n+1−k,k
= 0. Therefore it suffices to show that the right-hand
which implies E∞
vertical arrow in the commutative diagram above is surjective. This can be
seen as follows: By construction of M , the map
inf : H k (N/M ) −→ H k (N )
is an epimorphism of trivial U/N -modules, which splits, as both groups are
IFp -vector spaces. Hence the map
E 2n+1−k,k = H n+1−k (U/N, H k (N/M )) −→ E2n+1−k,k = H n+1−k (U/N, H k (N ))
is surjective.
We obtain E2n+1−k,k = H n+1−k (U/N, H k (N )) = 0. By assumption, H k (N ) ∼
=
(ZZ/pZZ)m for some m ≥ 1, and by construction of U , H k (N ) is a trivial U/N module. Hence
0 = H n+1−k (U/N, H k (N )) ∼
= H n+1−k (U/N )m .
This implies H n+1−k (U/N ) = 0, and so cdp (U/N ) ≤ n − k, as U/N is a prop-group, see (3.3.2). On the other hand, (3.3.8) implies cdp (U/N ) ≥ n − k,
which finishes the proof.
2
(3.3.10) Corollary. Let G be a pro-p-group, cdp G = n < ∞, and let N be a
closed normal subgroup of G such that cdp N = cdp G and H n (N, ZZ/pZZ) is
finite. Then N is open in G.
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Chapter III. Duality Properties of Profinite Groups
Proof: By (3.3.9) and the subsequent remark, we obtain vcdp G/N = 0.
Hence G/N is finite by (3.3.7).
2
The cohomological dimension has the following effect on the corestriction.
(3.3.11) Proposition. Let U be an open subgroup of G. If scd G = n (resp.
cdp G = n), then for every G-module A (resp. for every p-torsion G-module),
the corestriction
cor : H n (U, A) −→ H n (G, A)
is surjective. If U is normal in G, then
cor : H n (U, A)G/U −→ H n (G, A)
is an isomorphism.
Proof: The first assertion follows from the second. Indeed, if U0 is an open
normal subgroup of G contained in U , then the composition
cor UG0 : H¬« n (U0 , A)
U
cor U0
cor U
G
H n (U, A)
H n (G, A)
shows that the surjectivity of cor UG0 implies the surjectivity of cor UG . Therefore
we may assume U to be normal in G.
Let A → X be the standard resolution of the G-module A. Then the map
cor : H i (U, A) → H i (G, A) is given by taking cohomology of the map of
complexes N = NG/U : X U → X G . The G-module Y n = ker(X n → X n+1 )
is cohomologically trivial by our assumption on the cohomological dimension.
Therefore
0 −→ A −→ X 0 −→ · · · −→ X n−1 −→ Y n −→ 0
is a finite resolution of A by cohomologically trivial G-modules. By (1.3.9),
we obtain the exact commutative diagram
.
.
³´µ²±­®¯° )U
(X n−1
N
.
(Y n )U
N
H n (U, A)
0
cor
(X n−1 )G
(Y n )G
H n (G, A)
0.
Taking G/U -coinvariants of the upper row, we obtain the exact commutative
diagram
((X n−1¼½¾»º¶·¸¹ )U )G/U
N
((X n−1 )U )G/U
((Y n )U )G/U
N
((Y n )U )G/U
H n (U, A)G/U
0
cor
H n (G, A)
0.
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179
§3. Cohomological Dimension
By (1.8.2), the G/U -modules (X n−1 )U and Y U are cohomologically trivial.
Therefore, by (1.2.6), the maps N are bijective, and hence also cor , by the five
lemma.
2
We will now consider the Euler-Poincaré characteristic of a pro-p-group.
(3.3.12) Definition. Let G be a pro-p-group of finite cohomological dimension.
Assume that the groups H i (G, IFp ) are finite for all i ≥ 0. Then the EulerPoincaré characteristic of G is the alternating sum
χ(G) =
X
(−1)i dimIFp H i (G, IFp ) .
i
If A is a finite p-primary G-module with pA = 0, then we put
χ(G, A) =
(−1)i dimIFp H i (G, A) .
X
i
Let G be as above and let 0 −→ A1 −→ A2 −→ A3 −→ 0 be an exact sequence
of finite IFp [G]-modules. Then obviously
χ(G, A2 ) = χ(G, A1 ) + χ(G, A3 ) ;
in particular, from (1.6.13), it follows for an IFp [G]-module A of order pr that
χ(G, A) = r · χ(G) .
(3.3.13) Proposition. Let G be a pro-p-group of finite cohomological dimension and assume that the groups H i (G, IFp ) are finite for all i ≥ 0. If U is an
open subgroup of G, then
χ(U ) = (G : U )χ(G) .
Proof: Using (1.6.4), we obtain χ(U ) = χ(G, IndUG IFp ) = (G : U )χ(G).
2
If we drop the assumption that the pro-p-group G is of finite cohomological
dimension (but keep the assumption on the finiteness of H i (G, IFp ) for i ≤ n
for some n), then we define the partial Euler-Poincaré characteristic of G.
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Chapter III. Duality Properties of Profinite Groups
(3.3.14) Definition. Let n ≥ 0 and let G be a pro-p-group such that the
groups H i (G, IFp ) are finite for all i with 0 ≤ i ≤ n. Then the n-th partial
Euler-Poincaré characteristic of G is the alternating sum
χn (G) =
n
X
(−1)i dimIFp H i (G, IFp ) .
i=0
For a finite IFp [G]-module A, we define
χn (G, A) =
n
X
(−1)i dimIFp H i (G, A) .
i=0
Induction on the length of a composition series of the IFp [G]-module A
yields the formula:
(3.3.15) Lemma. If G and A are as above, then
(−1)n χn (G, A) ≤ (−1)n dimIFp A · χn (G) .
The following theorem, due to H. KOCH [110], can be considered as a
converse of (3.3.13).
(3.3.16) Theorem. Let G be a pro-p-group such that the groups H i (G, IFp ) are
finite for 0 ≤ i ≤ n. Let U be a cofinal set of open subgroups of G. Then the
following assertions are equivalent:
(i) χn (U ) = (G : U )χn (G) for all U
∈
U.
(ii) cdp G ≤ n.
Proof: The implication (ii) ⇒ (i) is just (3.3.13), so let us assume that (i) holds.
By (3.3.2)(iii), it suffices to show that H n+1 (G, IFp ) = 0. Let ā ∈ H n+1 (G, IFp )
with a ∈ C n+1 (G, IFp ). Then there exists a U ∈ U such that a depends only on
the cosets of G/U . Define the finite IFp [G]-module A by the exact sequence
ϕ
0 −→ IFp −→ IndUG IFp −→ A −→ 0 .
We obtain a commutative diagram
ÁÀ¿ IFp )
H n+1 (G,
ϕ∗
H n+1 (G, IndUG IFp )
res
sh
H
n+1
(U, IFp )
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181
§4. Dualizing Modules
which, by the choice of U , shows that res ā = 0, and so ϕ∗ ā = 0. Furthermore,
we get from the short exact sequence above the exact cohomology sequence
0 −→ H 0 (G, IFp ) −→ · · · −→ H n (G, A) −→ ker(ϕ∗ ) −→ 0 ,
which implies, using (3.3.15), (1.6.4) and the assumption (i),
dimIFp ker(ϕ∗ ) = (−1)n (χn (G) + χn (G, A) − χn (G, IndUG IFp ))
≤ (−1)n (χn (G) + dimIF A · χn (G) − χn (U ))
p
= (−1)n ((G : U )χn (G) − χn (U )) = 0 .
Thus ker(ϕ∗ ) = 0 and therefore ā = 0, which proves the theorem.
2
Remark: One can prove the following generalization of (3.3.16), cf. [194]: If
there exists a number N such that (−1)n χn (U ) + N ≥ (−1)n (G : U )χn (G) for
all U ∈ U, then G is finite or cd G ≤ n.
Exercise 1. If cdp G = 1, then scdp G = 2.
Exercise 2. Let p =/ 2 and let G be the group of affine transformations x 7→ ax + b with b ∈ ZZp
and a ∈ ZZ×
p . Then cd p G = scd p G = 2.
Exercise 3. If cdp G =/ 0, ∞, then the exponent of p in the order of G is infinite.
Exercise 4. cdp G = cdp G/H +cdp H if H is contained in the center of G and cdp G/H < ∞.
Exercise 5. Let H be a closed normal subgroup of G. If cdp G/H =/ 0, then scdp G
cdp G/H + scdp H.
≤
Exercise 6. If H n+1 (U, ZZ) = H n+2 (U, ZZ) = 0 for all open subgroups U of G, then scdp G ≤ n.
p
Hint: Using the exact sequence 0 → ZZ → ZZ → ZZ/pZZ → 0, one sees that H n+1 (Gp , ZZ/pZZ)
= 0 for a p-Sylow subgroup Gp , hence cdp G = cd Gp ≤ n, and scdp G ≤ n by (3.3.4).
§4. Dualizing Modules
Let G be a profinite group and let Mod(G) be the category of discrete
G-modules A. In II §5, we introduced the G-modules
Di (A) = lim H i (U, A)∗ ,
−→
U
where U runs through the open normal subgroups of G and the limit is taken
over the maps cor ∗ dual to the corestriction. It comes equipped with the
canonical homomorphism
H i (G, A)∗ −→ Di (A)
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Chapter III. Duality Properties of Profinite Groups
and the pairing of G-modules
Di (A) × A −→ Di (ZZ)
(cf. p.125). We obtain a canonical homomorphism
ϕA : H i (G, A)∗ −→ HomG (A, Di (ZZ)).
(3.4.1) Theorem. If n = scd G < ∞, then the map
ϕA : H n (G, A)∗ −→ HomG (A, Dn (ZZ))
is an isomorphism for all A ∈ Mod(G). We call the G-module D = Dn (ZZ) the
dualizing module of G.
Remark: By definition, the dualizing module of an open subgroup U of G is
the dualizing module of G regarded as a U -module.
Proof: By a straightforward limit argument using (1.5.1), we are reduced to
the case where A is a finitely generated ZZ-module. Let U be an open normal
subgroup of G acting trivially on A, i.e. A is a ZZ[G/U ]-module.
Suppose the theorem is proven for free ZZ[G/U ]-modules. Then we may
choose an exact sequence
0 ← A ← F0 ← F1
with free ZZ[G/U ]-modules F0 , F1 and obtain the bijectivity of ϕA from the
exact commutative diagram
0ÈÉÊÇÆÂÃÄÅ
H n (G, A)∗
H n (G, F0 )∗
ϕF0
ϕA
0
HomG (A, D)
H n (G, F1 )∗
ϕF1
HomG (F0 , D)
HomG (F1 , D).
So let A = ZZ[G/U ] for an open normal subgroup U of G. Using Shapiro’s
lemma (1.6.4), we obtain the commutative diagram
H n (G, ZZÍÎÌË [G/U ])∗
ϕG, ZZ [G/U ]
HomG (ZZ[G/U ], D)
sh∗
H n (U, ZZ)∗
ϕU, ZZ
HomU (ZZ, D).
Finally, the map ϕU, ZZ is the direct limit over V of the maps
cor ∗ : H n (U, ZZ)∗ −→ HomU (ZZ, H n (V, ZZ)∗ ) = (H n (V, ZZ)U/V )∗ ,
which are isomorphisms by (3.3.11).
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183
§4. Dualizing Modules
If U is an open subgroup of G, then, by construction, the diagram
H i (U,ÒÓÔÕÑÐÏ A)∗
ϕA
HomU (A, Di (ZZ))
Hom(A, Di (ZZ))
ϕA
HomG (A, Di (ZZ))
Hom(A, Di (ZZ))
cor ∗
H i (G, A)∗
is commutative. Passing to the direct limit, we obtain the
(3.4.2) Corollary. For A ∈ Mod(G), we have a functorial homomorphism
Di (A) −→ Hom(A, Di (ZZ)),
which is an isomorphism for i = scd G if A is a finitely generated ZZ-module.
We defined the trace map
tr : H n (G, D) −→ Q/ZZ
as the first edge morphism of the Tate spectral sequence H i (G, Dn−j (ZZ))
⇒ H n−(i+j) (G, ZZ)∗ (see also ex.2). The pairing
Hom(A, D) × A −→ D
gives us a cup-product
∪
H i (G, Hom(A, D)) × H n−i (G, A) −→ H n (G, D) ,
which, together with the map tr, yields a homomorphism
H i (G, Hom(A, D)) −→ H n−i (G, A)∗ .
We can now prove the central result of this section.
(3.4.3) Duality Theorem. Let G be a profinite group of strict cohomological
dimension scd G = n < ∞ such that Dk (ZZ) = 0 for k < n. Let A ∈ Mod(G) be
finitely generated over ZZ and assume that the `-primary part of A is nontrivial
only for prime numbers ` such that D = Dn (ZZ) is `-divisible.
Then for all i ∈ ZZ ∗) the cup-product and the trace map
∪
tr
H i (G, Hom(A, D)) × H n−i (G, A) −→ H n (G, D) −→ Q/ZZ
yield an isomorphism
H i (G, Hom(A, D)) ∼
= H n−i (G, A)∗ .
∗) H i = 0 for i < 0 by definition.
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Chapter III. Duality Properties of Profinite Groups
Proof: Because of (3.4.2), we may replace Hom(A, D) by Dn (A) and consider the map
(∗)
H i (G, Dn (A)) −→ H n−i (G, A)∗
induced by the cup-product with respect to the canonical pairing Dn (A) ×
A → Dn (ZZ). But by (2.5.5) this map is the edge morphism E2i,0 → E i of the
Tate spectral sequence
E2ij = H i (G, Dn−j (A)) ⇒ H n−(i+j) (G, A)∗ = E i+j .
From the assumption in the theorem, it follows that Dk (A) = 0 for k < n by
lemma (2.5.6), hence the spectral sequence degenerates, showing that (∗) is an
isomorphism.
2
Remarks: 1. Setting i = n and A = ZZ, we see that the trace map tr : H n (G, D)
→ Q/ZZ is an isomorphism. It is clear that the duality theorem remains valid if
we replace it by any other isomorphism H n (G, D) ∼
= Q/ZZ. This can be useful
in the applications, where we may have a canonical such isomorphism without
it being clear that it is the edge morphism.
Q/ZZ = 0 is equivalent to the assertion that every
2. The condition D0 = lim
−→ U
prime number p divides the order of G infinitely often, in the sense that all
Sylow subgroups Gp are infinite.
We obtain an important variant of the dualizing module and the duality
theorem as follows. Let P be a nonempty set of prime numbers. Let IN(P )
denote the set of all natural numbers which are divisible only by prime numbers
in P . Let ModP (G) denote the category of all P -torsion G-modules, i.e.
modules A consisting of elements a such that na = 0 for some n ∈ IN(P ).
We define the cohomological P -dimension cdP G as the smallest number
n ≥ 0 such that H i (G, A) = 0 for all i > n and all A ∈ ModP (G). If no such
number exists, we set cdP G = ∞. In other words, cdP G = sup{cdp G}.
p∈P
We now set
Di (ZZP ) = lim Di (ZZ/mZZ) = lim lim H i (U, ZZ/mZZ)∗ ,
−→
m∈IN(P )
−→
−→
m∈IN(P ) U
and, in particular,
Di (ẐZ) = lim Di (ZZ/mZZ).
−→
m∈IN
Again we have for A ∈ ModP (G) a functorial homomorphism
ϕA : H i (G, A)∗ −→ HomG (A, Di (ZZP ))
which is obtained as above from the canonical pairing
H i (G, A)∗ × m AU −→ H i (V, ZZ/mZZ)∗ , (χ, a) 7→ χa (x) = χ(cor (ax)),
and by taking direct limits over V and m ∈ IN(P ), and then over U .
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§4. Dualizing Modules
(3.4.4) Theorem. If cdP G = n < ∞, then for all A ∈ ModP (G), the map
ϕA : H n (G, A)∗ −→ HomG (A, Dn (ZZP ))
is an isomorphism, and, if A is finite, we obtain an isomorphism of G-modules
Dn (A) ∼
= Hom(A, Dn (ZZP )).
The proof is the same as that of (3.4.1) (resp. (3.4.2)); we have just to replace
ZZ by ZZ/mZZ (m ∈ IN(P )) and lim by lim lim .
−→
−→
−→
U
m∈IN(P ) U
The G-module Dn (ZZP ) represents the functor
T : ModP (G) −→ Ab,
A 7−→ T (A) = H n (G, A)∗ ,
and is called the dualizing module of G at P (or of ModP (G)). We denote it
briefly by DP . If P consists of a single prime number p, then we write D(p) .
(3.4.5) Corollary (Serre Criterion). Let p be a prime number and let 1 ≤
n = cdp G < ∞. We have scdp G = n + 1 if and only if there exists an open
U
subgroup U of G such that D(p)
contains a subgroup isomorphic to Qp /ZZp .
U
Proof: The existence of a subgroup of D(p)
isomorphic to Qp /ZZp is equivalent to HomU (Qp /ZZp , D(p) ) =/ 0. Since D(p) is also the dualizing module
of U at p, this means by (3.4.4) that H n (U, Qp /ZZp ) =/ 0. But this last
group is the p-primary component of H n (U, Q/ZZ). The exact sequence
0 → ZZ → Q → Q/ZZ → 0 yields H n (U, Q/ZZ)(p) ∼
= H n+1 (U, ZZ)(p), and the
corollary follows now from (3.3.4) and (3.3.5) (ii).
2
The Tate spectral sequence
H i (G, Dn−j (ZZ/mZZ)) ⇒ H n−(i+j) (G, ZZ/mZZ)∗ ,
m ∈ IN(P ), gives an edge morphism
H n (G, D(ZZ/mZZ)) −→ H 0 (G, ZZ/mZZ)∗ =
1
ZZ/ZZ.
m
Taking limits over m ∈ IN(P ), we obtain a homomorphism
tr : H n (G, DP ) −→ QP /ZZP :=
M
Qp /ZZp ,
p∈P
again called the trace map. We now obtain the following variant of the duality
theorem (3.4.3).
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Chapter III. Duality Properties of Profinite Groups
(3.4.6) Duality Theorem. For a profinite group with cdP G = n < ∞, the
following assertions are equivalent.
(i) Di (ZZ/pZZ) = 0 for all p ∈ P and all i < n.
(ii) For all i ∈ ZZ and all finite G-modules A
and the trace map
∈
ModP (G), the cup-product
∪
tr
H i (G, Hom(A, DP )) × H n−i (G, A) −→ H n (G, DP ) −→ QP /ZZP
yield an isomorphism
H i (G, Hom(A, DP )) ∼
= H n−i (G, A)∗ .
In this case G, is called a duality group at P of dimension n.
Proof: The implication (i) ⇒ (ii) follows by the same argument as in the
proof of (3.4.3). So we have only to show the implication (ii) ⇒ (i). For every
pair V ⊆ U of open normal subgroups of G, we have a commutative diagram
of G-modules
V
νU
IndVG×ÖØ (A)
IndUG (A)
V
νG
where νUV (x)(σ) =
X
U
νG
A
τ x(τ −1 σ). By the lemma of Shapiro (1.6.4) and by
τ ∈U/V
(1.6.5) and the subsequent remark, we obtain from this a commutative diagram
of G-modules
ÞÝÜàßÛÚÙ V (A))
H i (G, Ind
G
sh
ν∗
H i (V, A)
cor
H i (G, A)
ν∗
ν∗
sh
H i (G, IndUG (A))
cor
cor
H i (U, A).
In this diagram the maps sh are isomorphisms, and hence define an isomorphism of projective systems (H n (G, IndUG (A))) ∼
= (H n (U, A)) of G-modules.
We therefore have a canonical isomorphism
Di (A) ∼
H i (G, IndUG (A))∗ .
= lim
−→
U
Applying (ii) and (1.5.1), we obtain for all i < n and p ∈ P ,
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§4. Dualizing Modules
H i (G, IndUG (ZZ/pZZ))∗
Di (ZZ/pZZ) ∼
= lim
−→
U
∼
H n−i (G, Hom(IndUG (ZZ/pZZ), DP ))
= lim
−→
U
∼
H n−i (G, IndUG (Hom(ZZ/pZZ, DP ))
= lim
−→
U
∼
H n−i (U, p DP ) = 0.
= lim
−→
U,res
2
(3.4.7) Corollary. If G is a duality group at P of dimension n, then for every
p ∈ P there exists an exact sequence
p
0 −→ Dn (ZZ/pZZ) −→ DP −→ DP −→ 0.
In particular, DP is p-divisible for all p ∈ P .
Proof: Let p ∈ P , m > 1, and consider the exact diagram
0áâãäåæçèéêë
ZZ/pm ZZ
ZZ/pm+1 ZZ
ZZ/pZZ
0
0
ZZ/pm−1 ZZ
ZZ/pm ZZ
ZZ/pZZ
0.
n
∗
Applying the functor lim
H (U, −) and passing to the limit over m, we
−→ U
obtain the exact sequence
p
0 −→ Dn (ZZ/pZZ) −→ D(p) −→ D(p) −→ 0.
This proves the corollary since D(p) is the p-torsion subgroup of the torsion
group DP .
2
(3.4.8) Corollary. Assume that scd G = n and Dk (ZZ) = 0 for k = 0, . . . , n−1.
Let p be a prime number. Then Dn (ZZ) is p-divisible if and only if G is a duality
group at p of dimension n. In this case
D(p) = Dn (ZZ)(p) .
Proof: Consider the exact sequence
p
0 −→ ZZ −→ ZZ −→ ZZ/pZZ −→ 0
and apply the functor lim
H k (U, −)∗ . We obtain Dk (ZZ/pZZ) = 0 for k
−→
and an exact sequence
≤
n−2
p
0 → Dn (ZZ/pZZ) → Dn (ZZ) → Dn (ZZ) → Dn−1 (ZZ/pZZ) → 0 .
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Chapter III. Duality Properties of Profinite Groups
Hence Dn−1 (ZZ/pZZ) = 0, i.e. G is a duality group at p of dimension n by
(3.4.6) if and only if Dn (ZZ) is p-divisible. Further, in this case,
H n (U, ZZ/pm ZZ)∗
D(p) = lim
−→
U,m
= lim H 0 (U, Hom(ZZ/pm ZZ, Dn (ZZ)))
−→
U,m
= Dn (ZZ)(p) .
2
Remark: The duality theorems (3.4.3) and (3.4.6) are due to J. TATE (see
[230]). Tate gave the duality isomorphism as the edge morphism of the Tate
spectral sequence, which by (2.5.5) is induced by the cup-product. Another
slightly different method, due to J. L. VERDIER, is to prove (3.4.3) and (3.4.6)
by analyzing the group of “local homomorphisms” (see [240]). The concept
of duality groups was also inspired by the work of R. BIERI and B. ECKMANN.
in the case of discrete groups (see [11]).
Exercise 1. Let n = scd G. If the homomorphism
H p (G, Dn (A)) −→ H n−p (G, A)∗
is bijective for all p ∈ ZZ and all A ∈ Mod(G) finitely generated over ZZ, then Di = 0 for i < n
and Dn is divisible.
Exercise 2. Applying (3.4.1) to A = D, we obtain a canonical isomorphism
H n (G, D)∗ ∼
= HomG (D, D).
The trace map tr : H n (G, D) −→ Q/ZZ, defined as the first edge morphism of the Tate spectral
sequence, is an element of the left-hand group, which corresponds to id in the right-hand
group, and HomG (D, D) ∼
= ZZ.
Exercise 3. Let G = ẐZ. Show that scd G = 2, cd G = 1. Calculate D and DP for the set P of
all prime numbers. Show that there are isomorphisms
H i (G, Hom(A, Q/ZZ)) ∼
= H 1−i (G, A)∗
for all i and all finite G-modules A.
Exercise 4. Let Fn = hx1 , . . . , xn i be a free pro-p-group on n generators. Show that the
dualizing module D(p) of Fn is given by the exact sequence
diag Ln
hxi i
0îïìí
Qp /ZZp
Zp
D(p)
0,
i=0 IndFn Qp /Z
−1
−1
where x0 := x−1
n xn−1 · · · x1 and hxi i is the closed procyclic group generated by xi in Fn
(i = 0, . . . , n).
(The characterization of D(p) given in ex.4 on p.V-24 (Cinquième éd. p.76) in [240] is incorrect!)
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§5. Projective pro-c-groups
§5. Projective pro-c-groups
In this section we introduce the concept of embedding problems and consider profinite groups G of cohomological dimension cd G ≤ 1. We start by
considering the abstract embedding problem for a profinite group G.
(3.5.1) Definition. (i) An embedding problem E (G) = E (G, ϕ, α) for a
profinite group G is a diagram
ôñòóð
G
ϕ
1
N
E α G
1
with an exact sequence of profinite groups and a surjective homomorphism ϕ.
(ii) A solution of the embedding problem E (G) is a homomorphism ψ: G → E
such that α ◦ ψ = ϕ. A solution is called proper if ψ is surjective.
(iii) Two solutions ψ and ψ 0 are called equivalent if
ψ 0 (σ) = a−1 ψ(σ)a
for all σ ∈ G with a fixed element a ∈ N . The set of all solutions of E (G) modulo equivalence is denoted by SE (G) and is considered as a discrete topological
space.
An embedding problem for a Galois group G corresponds to a field theoretical problem. Let G = G(k̃|k) be the Galois group of a Galois extension k̃ of a
field k and let K|k be a Galois subextension. If Eõ
G(K|k) is a group extenö G can G(K|k), α)
sion, then a proper solution of the embedding problem E (G,
defines a Galois extension L ⊇ K ⊇ k with Galois group G(L|k) isomorphic
to E such that α is the canonical projection G(L|k) G(K|k). If the solution
is not proper, then one obtains only a Galois algebra with group E instead of
a field L.
(3.5.2) Definition. (i) A class c of finite groups is called a full class if it is
closed under taking subgroups, homomorphic images and group extensions.
(i.e. if 1 → G0 → G → G00 → 1 is an exact sequence of finite groups, then G is
in c if G0 and G00 are).
(ii) If c is a full class of finite groups, a pro-c-group is a projective limit of
groups in c.
(iii) For a full class c of finite groups, S(c) denotes the set of prime numbers p
with ZZ/pZZ ∈ c. For a set S of prime numbers, c(S) denotes the full class of
finite groups whose orders are divisible only by prime numbers in S.
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Chapter III. Duality Properties of Profinite Groups
Remark: We have S(c(S)) = S and an inclusion c
general, strict.
⊆
c(S(c)) which is, in
If c is the class of all finite groups, finite p-groups, or finite solvable
groups, we get the profinite groups, the pro-p-groups, or the prosolvable
groups, respectively. The class of pro-p-groups does not contain a nontrivial, proper full subclass, because every finite p-group G has a series of subgroups 1 = G0 ⊆ G1 ⊆ · · · ⊆ Gn = G such that Gi−1 is normal in Gi and
Gi /Gi−1 ∼
= ZZ/pZZ for i = 1, . . . , n. In particular, the class of pro-c-groups
contains the class of pro-p-groups if and only if ZZ/pZZ ∈ c.
(3.5.3) Definition. Let c be a full class of finite groups.
(i) For a profinite group G, the group
G(c) = lim
G/U,
←−
G/U ∈c
where U runs through the open normal subgroups of G such that G/U
called the maximal pro-c-factor group of G.
∈
c, is
(ii) For an abelian torsion group A, the subgroup
[
B,
A(c) =
B ∈c
where the union is taken over all finite subgroups B
the c-torsion subgroup of A.
⊆
A with B
∈
c, is called
If G is a finite abelian torsion group, then both possible interpretations of
G(c) coincide in the sense that the c-torsion subgroup maps isomorphically
onto the maximal pro-c-factor group.
(3.5.4) Proposition. Let G be a profinite group and let c be a full class of finite
groups.
(i) For the kernel H = ker(G G(c)) we have H(c) = 1.
(ii) Let G0 → G → G00 → 1 be an exact sequence of profinite groups. Then
the sequence
G0 (c) → G(c) → G00 (c) → 1
is exact.
(iii) Let 1 → G0 → G → G00 → 1 be an exact sequence of profinite groups.
If G00 is a pro-c-group, then the sequence
1 → G0 (c) → G(c) → G00 (c) → 1
is exact.
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191
§5. Projective pro-c-groups
Proof: We start by showing (i). Suppose H has a proper open normal
subgroup N with H/N ∈ c. We let G act on H by conjugation. The subgroup
G0 of all σ ∈ G such that σN σ −1 = N is open in G, since it consists of
all elements σ ∈ G which maps a compact set (namely N ) into an open set
(namely N ). Therefore G/G0 is finite and
\
Ñ :=
σN σ −1
σ ∈G/G0
is open in H and normal in G. The injective homomorphism
H/Ñ −→
Y
H/σN σ −1
σ ∈G/G0
shows that H/Ñ ∈ c. The group extension 1 → H/Ñ → G/Ñ → G/H → 1
shows that G/Ñ is a pro-c-group, contradicting the definition of H.
Now let G0 → G → G00 → 1 be exact. The surjectivity of G(c) → G00 (c) is
immediate. Therefore, in order to show (ii), we may replace G0 by its image
in G and suppose that 1 → G0 → G → G00 → 1 is exact. Let H be the kernel
of G G(c). We have an exact sequence of pro-c-groups
1 → G0 /(G0 ∩ H) → G/H → G/G0 H → 1
and a surjection G0 (c) G0 /(G0 ∩ H). Consider the exact sequence
1 → G0 H/G0 → G/G0 → G/G0 H → 1.
The group G0 H/G0 is a quotient of H and has no nontrivial homomorphisms
∼
to c-groups by (i). We therefore obtain an isomorphism G00 (c) = (G/G0 )(c) →
G/G0 H, showing (ii).
If G00 is a pro-c-group, then H ⊆ G0 and, again using (i), the surjection
∼ G0 /H. This shows (iii).
G0 → G0 /H induces an isomorphism G0 (c) →
2
(3.5.5) Definition. Let c be a full class of finite groups.
(i) A profinite group G is called c-projective if every embedding problem for
G where the kernel is a pro-c-group has a solution.
(ii) A profinite group G is projective, if it is f-projective, where f is the class
of all finite groups.
(iii) If S is a set of prime numbers, then a c(S)-projective group is called
S-projective. If S consists only of one prime number p, i.e. c(S) is the class
of finite p-groups, we call such a group p-projective.
Remark: If G is a pro-c-group which is c-projective in the above sense, then
G is a projective object in the category of pro-c-groups.
Our aim is to prove the following result due to K. W. GRUENBERG [71].
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Chapter III. Duality Properties of Profinite Groups
(3.5.6) Theorem. Let c be a full class of finite groups. For a profinite group
G the following assertions are equivalent:
(i) G is c-projective.
(ii) Every group extension 1 → N → E → G → 1 where N is a pro-c-group
splits.
(iii) Every group extension 1 → A → E → G → 1 where A is a finite abelian
p-group with p ∈ S(c) splits.
(iv) cdp G ≤ 1 for all prime numbers p ∈ S(c).
(v) G is S(c)-projective.
The crucial step in the proof of (3.5.6) is the following result.
(3.5.7) Proposition. A profinite group G is c-projective if every embedding
problem
÷øùúû
G
(∗)
1
N
E
for G is solvable, where E is finite and N
subgroup of E.
G
∈
1
c is a minimal abelian normal
Proof: We have to show that every embedding problem (∗) with N a pro-cgroup is solvable. Firstly, we reduce to the case when N is finite. Assume that
all embedding problems (∗) with N a finite c-group are solvable and consider
an embedding problem (∗) with N an arbitrary pro-c-group. Let X be the set
of all pairs (N 0 , ψ 0 ) consisting of a closed subgroup N 0 of N which is normal
in E and a solution ψ 0 : G → E/N 0 of the induced embedding problem
ýþÿü
G
ψ0
1
N/N 0
E/N 0
ϕ
ᾱ
G
1.
We write (N 00 , ψ 00 ) ≥ (N 0 , ψ 0 ) if N 00 ⊆ N 0 and if ψ 0 coincides with the composite
ψ 00
G −→ E/N 00 E/N 0 . Then X is inductively ordered and nonempty. By
Zorn’s lemma, there exists a maximal element (N 0 , ψ 0 ). We have to show that
N 0 = 1. Assume the contrary. Then there exists a proper open subgroup U
of N 0 . As the topology of N 0 is induced by that of E, there exists a normal
open subgroup Ẽ ⊆ E such that N 00 = Ẽ ∩ N 0 is contained in U . The group
N 00 is a proper open subgroup in N 0 and normal in E. By assumption, the
embedding problem with finite kernel
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193
§5. Projective pro-c-groups
G
ψ0
1
N 0 /N 00
E0
im(ψ 0 )
1
1
N 0 /N 00
E/N 00
E/N 0
1,
where E 0 is the pre-image of im(ψ 0 ) in E/N 00 , has a solution. Therefore we
obtain a homomorphism ψ 00 : G → E/N 00 such that the pair (N 00 , ψ 00 ) is strictly
larger than the pair (N 0 , ψ 0 ). This contradicts the maximality of (N 0 , ψ 0 ).
Next we show that we can also assume E to be finite. Assume all embedding
problems (∗) with E finite and N ∈ c are solvable, and consider an arbitrary
embedding problem with finite kernel N ∈ c. Since N is finite, there exists an
open normal subgroup U of E with U ∩ N = 1. We obtain the commutative
and exact diagram
G
ϕ
N
1
E
G
1
G/im(U )
1,
ψ
N
1
E/U
where the broken arrow is obtained by the assumption of this reduction. By
(3.5.8) below, the right hand square is cartesian, i.e. the natural map from E to
the fibre product E/U ×G/im(U ) G is an isomorphism. Therefore the pair (ψ, ϕ)
induces a homomorphism G −→ E, completing the diagram commutatively.
Now we prove by induction on the (finite) order of N ∈ c that every embedding problem (∗) with E finite has a solution. We distinguish between the
following three cases:
(1) N is not a minimal normal subgroup of E. Then we can choose a normal
subgroup M of E such that 1 $ M $ N . By induction,
!
G
ψ
N/M
1
E/M
G
1
has a solution ψ: G → E/M . Again by induction, the embedding problem
#$%&"
G
ψ
1
M
E
E/M
1
has a solution.
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Chapter III. Duality Properties of Profinite Groups
(2) Let N be a minimal normal subgroup of E and assume that N is not
contained in the Frattini subgroup Φ(E) of E (which is the intersection of all
maximal subgroups of E). Then there exists a maximal subgroup E1 of E such
/ E1 , hence N E1 = E, i.e. E1 projects onto G. Then N1 = E1 ∩ N is a
that N ⊆
proper subgroup of N and thus, by induction, the embedding problem
'()*+,-./0123
G
1
N1
E1
G
1
1
N
E
G
1
has a solution.
(3) Let N be a minimal normal subgroup of E and assume that N ⊆ Φ(E).
Since Φ(E) is nilpotent (see [81], III Satz 3.6), the group N is abelian. By
assumption, we get a solution of the embedding problem in this case.
2
(3.5.8) Lemma. Let E be a profinite group and let U and V be closed normal
subgroups of G. Then the diagram of profinite groups
E/U6754 ∩V
pU
E/U
pV
E/V
E/U V
is cartesian, i.e. the projections define an isomorphism
E/U8 ∩V
(pU ,pV )
E/U ×E/U V E/V,
where the group on the right hand side is the fibre product, i.e. the subgroup
of elements (a, b) in the product E/U × E/V such that a and b project to the
same element in E/U V .
Proof: Injectivity is trivial. Let a, b ∈ E be elements with the same image
in E/U V and consider the element (pU (a), pV (b)) ∈ E/U ×E/U V E/V . By
assumption, ab−1 = uv for elements u ∈ U , v ∈ V . Then the element
g = u−1 a = vb ∈ E maps to ((pU (a), pV (b)), which shows surjectivity.
2
In order to detect c-projective groups, we are, by (3.5.7), reduced to a special
case of embedding problems with finite abelian kernel. Next we investigate
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§5. Projective pro-c-groups
arbitrary embedding problems with finite abelian kernel A = ker(α). The group
α
G acts on A by conjugation and the group extension 1 → A → E → G → 1 is
determined by its class ε ∈ H 2 (G, A), see (1.2.4).
Let E (G, ϕ, α) be an embedding problem with finite abelian kernel A for
the profinite group G and let H = ker ϕ. Consider the diagram
@<=>?9;:
H
ψ0
G
ψ
A
1
E
α
ϕ
G
1.
We consider A as a G-module via ϕ. Then H acts trivially on A. A solution ψ
induces a Ḡ-homomorphism ψ0 : H → A. From the Hochschild-Serre spectral
sequence we get the exact sequence (cf. (1.6.7))
tg
res
inf
0 −→ H 1 (G, A)−→ H 1 (G, A) −→ H 1 (H, A)Ḡ −→ H 2 (G, A) −→ H 2 (G, A).
(3.5.9) Proposition (HOECHSMANN). Let ε ∈ H 2 (G, A) be the cohomology
class corresponding to the group extension of the embedding problem E (G) =
E (G, ϕ, α), where A = ker(α) is a finite Ḡ-module.
Then E (G) has a solution if and only if inf (ε) = 0, i.e. if there exists a
Ḡ-homomorphism ψ0 : H → A with tg(ψ0 ) = ε.
Proof: (see [80], 1.1). Consider the commutative exact diagram
1KABCHIJDEFG
A
E ×G G
pr2
A
E
1
ϕ
pr1
1
G
α
G
1
where E ×G G = {(e, σ) ∈ E × G | α(e) = ϕ(σ)} is the fibre product of
α and ϕ. The group extension above corresponds to inf (ε) ∈ H 2 (G, A). If
inf (ε) = 0, then this extension splits, i.e. there exists a homomorphism
s: G → E ×G G
such that pr2 ◦ s = id. Obviously, ψ = pr1 ◦ s is then a solution of E (G).
Conversely, a solution ψ defines a section s of pr2 by s(σ) = (ψ(σ), σ), and
therefore inf (ε) = 0.
2
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Chapter III. Duality Properties of Profinite Groups
(3.5.10) Corollary. An embedding problem E (G, ϕ, α),
LPMNO
G
ϕ
A
1
E
α
G
1,
where A is a finite abelian p-group, is solvable if and only if the corresponding
p-Sylow embedding problem
URSTQ
Gp
ϕp
A
1
Ep
αp
Gp
1
is solvable. Here the index p indicates the corresponding p-Sylow subgroups,
which are chosen in a compatible way.
Proof: The 2-cocycle εp of the Sylow problem is just the restriction of the
2-cocycle ε of the initial problem. Therefore the result follows from (3.5.9)
and the commutative diagram
[εVWXY p ]
∈
H 2 (Gp , A)
H 2 (Gp , A)
[ε]
∈
H 2 (G, A)
H 2 (G, A),
where the restriction maps are injective by (1.6.10).
2
(3.5.11) Proposition. Let E (G, ϕ, α) be an embedding problem with finite
abelian kernel A which has a solution. Then SE (G) is a principal homogeneous
space over H 1 (G, A).∗)
The homogeneous subspaces over H 1 (G, A) ⊆ H 1 (G, A) consist of all solutions ψ modulo equivalence whose restrictions to H = ker(ϕ) induce a fixed
G-homomorphism ψ0 : H ab → A.
Proof: Let ψ be a solution of E (G) and let [x] ∈ H 1 (G, A), where x: G → A
is a 1-cocycle. Then
x
ψ: G −→ E , σ 7−→ x(σ) · ψ(σ)
∗) If X is a topological space and Γ a topological group acting continuously and simply
transitively on X, then X is called a principal homogeneous space over Γ .
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§5. Projective pro-c-groups
is a solution of E (G). Indeed, x ψ is a homomorphism since
x(στ ) · ψ(στ ) = x(σ)x(τ )ψ(σ) ψ(σ)ψ(τ ) = x(σ)ψ(σ)x(τ )ψ(τ )
and α ◦ x ψ = ϕ. Another 1-cocycle x0 ∈ [x] induces an equivalent solution of
x
ψ. Thus H 1 (G, A) acts on SE (G) . This action is transitive since for any two
solutions ψ and ψ 0 we get a 1-cocycle x(σ) = ψ 0 (σ)ψ(σ)−1 , hence a class in
H 1 (G, A). Furthermore, the action is simply transitive since two solutions ψ
and ψ 0 are equivalent if and only if x is a 1-coboundary. Finally, from the exact
5-term sequence, it follows that ψ|H = ψ 0 |H if and only if [x] ∈ inf (H 1 (Ḡ, A)).
2
Now we are able to prove the theorem.
Proof of (3.5.6): (i) ⇒ (ii) is trivial and (ii) ⇒ (iii) is also trivial, noting that
the classes c and c(S(c)) contain the same abelian groups.
(iii) ⇒ (iv). Let p ∈ S(c) and let A be a finite G-module in Modp (G). By
assumption, every group extension 1 → A → E → G → 1 splits and therefore,
by (1.2.4),
H 2 (G, A) ∼
= EXT (A, G) = 0.
An arbitrary G-module A ∈ Modp (G) is the union of its finite submodules.
Taking the inductive limit, we see that H 2 (G, A) = 0, and hence cdp G ≤ 1.
(iv) ⇒ (v). Let us assume that cdp G ≤ 1 for all p ∈ S(c). Then H 2 (G, A) = 0
for all finite p-primary G-modules A, p ∈ S(c). By (3.5.9), every embedding
problem (∗) with a minimal abelian normal subgroup A ∈ c(S(c)) of E is
solvable, since such an A is necessarily of the form (ZZ/pZZ)m for some prime
number p ∈ S(c) and some integer m ≥ 1. By (3.5.7), G is S(c)-projective.
Finally, (v) ⇒ (i) is trivial, as c ⊆ c(S(c)).
2
(3.5.12) Corollary. If c ⊆ d are two full classes of finite groups and if G is a
pro-c-group which is c-projective, then it is d-projective.
Proof: We use the equivalence (3.5.6) (i)⇔(iii), thus we have to show that
every group extension
1→A→E →G→1
splits, where A is a finite abelian p-group, p ∈ S(d). By assumption, this is
true for p ∈ S(c). If p ∈ S(d) r S(c), then the orders of the groups G and A are
relatively prime, hence EXT (A, G) ∼
2
= H 2 (G, A) = 0 by (1.6.2).
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Chapter III. Duality Properties of Profinite Groups
(3.5.13) Corollary. Denoting by (solv) the full class of finite solvable groups,
the following assertions are equivalent for a profinite group G.
(i) G is projective.
(ii) G is (solv)-projective.
(iii) cd G ≤ 1.
Proof: This follows easily from (3.5.6) (i)⇔(iv).
2
Let p be a prime number. From (3.3.7) we know that the profinite groups
G with cdp G = 0 are exactly the groups of order prime to p. By (3.5.6), the
profinite groups with cdp G ≤ 1 are exactly the p-projective groups. Prototypes
of projective groups are the free pro-c-groups, which are defined as follows.
(3.5.14) Definition. A free pro-c-group over a set X is a pro-c-group F
together with a map i: X → F satisfying the following properties.
(1) Every open subgroup contains almost all elements of i(X) (i.e. all up to
a finite number).
(2) If j : X → G is any other map with the property (1) into a pro-c-group
G, then there exists a unique homomorphism f : F → G with j = f ◦ i.
The free pro-c-group F over a set X always exists and is unique up to unique
isomorphism. One obtains it by starting with the ordinary free group F0 over
X (see [73]) with the inclusion X ,→ F0 . Let U run through all normal
subgroups containing almost all elements x ∈ X and such that F0 /U ∈ c. Then
F = lim F0 /U , together with the induced map i : X → F , is a free pro-c←− U
group over X. In fact, if j : X → G is as in (2), then the universal property of
the free group F0 gives a homomorphism f0 : F0 → G such that j = f0 ◦ incl X .
If U runs through all open normal subgroups of G, then we obtain f : F → G
as the composite of the homomorphisms
F −→ lim F0 /f0−1 (U ) −→ lim G/U = G.
←−
U
←−
U
f is unique as it is continuous and F is topologically generated by i(X).
Observe that the map i: X → F in (3.5.14) is necessarily injective. The
elements of i(X) ⊆ F are called the free generators of F , the set i(X) is
called a basis of F , and rk(F ) = #X is the rank of the free pro-c-group. We
also write FX (c) for F and Fn (c), resp. Fω (c), if rk(F ) = n ∈ IN, resp. if X has
countable cardinality.
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199
§5. Projective pro-c-groups
The free pro-c-group of rank 0 is the trivial group. The free pro-c-group of
Q
rank 1 is the product ẐZ(c) = p∈S(c) ZZp .
(3.5.15) Proposition. Let c be a full class of finite groups and let F be the free
pro-c-group over a set X. Then F is projective.
Proof: By (3.5.12), it suffices to show that F is c-projective. Let
π
1→A→E →F →1
be an exact sequence with A a finite abelian p-group for some prime number p ∈
S(c). Then A ∈ c and E is a pro-c-group. We choose a continuous section
F → E of π and obtain a lift s: X → E of i: X → F , which satisfies condition
(1) of (3.5.14). By the universal property for F , s extends to a homomorphism
s: F → E such that π ◦ s is the identity on X, hence on F . This shows that
the extension 1 → A → E → F → 1 splits. By (3.5.6), F is a projective pro-cgroup.
2
(3.5.16) Corollary. Let F be the free pro-c-group over a set X. If the class c
contains the p-groups, then cdp F = 1, otherwise cdp F = 0.
Proof: By (3.3.7), cdp F = 0 if and only if the class c does not contain the
p-groups. As F is projective, (3.5.6) implies cdp F ≤ 1 for all p.
2
For pro-p-groups we have a converse:
(3.5.17) Proposition. A pro-p-group G is free if and only if cd G ≤ 1.
We shall prove this in (3.9.5). See also (4.1.5) for a more general result.
(3.5.18) Definition. The rank rk(G) of a pro-c-group G is the smallest cardinal number α such that there exists a set X of cardinality α and a surjection
FX (c) G from a free pro-c-group over X onto G.
The rank of a pro-p-group G is often denoted by d(G). In (3.9.1) we will
see that d(G) = dimIFp H 1 (G, ZZ/pZZ).
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200
Chapter III. Duality Properties of Profinite Groups
In the following, we will prove that a pro-c-group G of at most countable
infinite rank, for which every embedding problem with E a finite c-group has
a proper solution, is isomorphic to the free pro-c-group Fω (c) of countable
infinite rank. We start by showing that Fω (c) itself has this property.
(3.5.19) Lemma. Every embedding problem
^[\]Z
Fω (c)
ϕ
N
1
E
α
G
1
with E a finite c-group has a proper solution.
Proof: Let X be a basis of Fω (c). Then Y = X r ker(ϕ) is finite. For each
y ∈ Y we choose an element ey ∈ E such that α(ey ) = ϕ(y). Furthermore,
we choose a surjective map ψ00 : X ∩ ker(ϕ) → N (observe that X ∩ ker(ϕ)
is infinite) and define ψ0 : X → E by ψ0 (y) = ey for y ∈ Y and by ψ00 on
X ∩ ker(ϕ). Now ψ0 extends to a surjective homomorphism ψ: Fω (c) E
such that α ◦ ψ = ϕ.
2
The following result is due to K. IWASAWA (see [86], th. 4).
(3.5.20) Proposition. Let G be a pro-c-group of at most countable infinite
rank such that every embedding problem
`abc_
G
ϕ
1
N
E
α
G
1
with E a finite c-group has a proper solution. Then G is isomorphic to Fω (c).
Proof: Let F = Fω (c). Since G and F have at most countable rank, there
exist sequences
F = F0 ⊇ F1 ⊇ F2 ⊇ . . .
G = G0
⊇
G1 ⊇ G2
⊇
...
of open normal subgroups with trivial intersection. We inductively define two
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201
§5. Projective pro-c-groups
additional sequences
F = F00
⊇
F10
G = G00
⊇
G01
⊇
F20
⊇
G02
⊇
...
⊇
...
∼
of open normal subgroups and a sequence of isomorphisms ϕn : G/G0n →
F/Fn0 such that the diagram
G/Ggfed 0n+1
ϕn+1
∼
0
F/Fn+1
can
G/G0n
can
ϕn
∼
F/Fn0
commutes for every n. Let n be given and suppose all objects with indices up
0
to n have already been defined. If n is even, then Fn+1
:= Fn ∩ Fn0 is open in
0
F and F/Fn+1 ∈ c. The embedding problem
hikj
G
ϕ0n+1
0
F/Fn+1
can
can
F/Fn0
∼
(ϕn )−1
G/G0n
has a proper solution ϕ0n+1 by assumption. If G0n+1 = ker(ϕ0n+1 ), then ϕ0n+1
∼ F/F 0
induces an isomorphism ϕn+1 : G/G0n+1 →
n+1 which commutes with ϕn .
If n is odd, exchange the roles of G and F and use (3.5.19) in order to find an
−1
0
0
∼ G/G0 . Let ϕ
isomorphism ψn+1 : F/Fn+1
→
n+1 = ψn+1 . Since Fn+1 ⊆ Fn
n+1
0
0
0
and Gn+1 ⊆ Gn , the intersection of all groups Fn (resp. Gn ) is equal to 1. Thus
∼ F.
the isomorphisms ϕn define an isomorphism ϕ: G →
2
Exercise: Consider the diagram
pmnol
G
f
1
P
E
π
Γ
1
with P an abelian group. Two liftings f 0 , f 00 : G → E of f : G → Γ are conjugate if there
exists a σ ∈ P such that f 00 = σ ◦ f 0 ◦ σ −1 . Show that the group H 1 (G, P ) acts simply
transitively on the set of conjugacy classes of liftings, provided some liftings exist. Formulate
and prove this also for non-abelian P .
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202
Chapter III. Duality Properties of Profinite Groups
§6. Profinite Groups of scd G = 2
The profinite groups G of strict cohomological dimension scd G = 2 are
especially important for the theory of number fields, because of their close
connection with class field theory. Trivial examples are all groups of cohomological dimension 1 (see §3, ex.1).
Let G be any profinite group. We denote open subgroups of G by U, V, W
and we write V / U if V is a normal subgroup of U . In this case V ab is a
U/V -module by conjugation, and we have a group extension
1 −→ V ab −→ U/V 0 −→ U/V −→ 1,
where V 0 denotes the closure of the commutator subgroup [V, V ] of V . By
(1.2.4), the group extension defines a canonical cohomology class
uU/V
∈
H 2 (U/V, V ab ),
which plays an important role in the sequel.
(3.6.1) Lemma. Let W
⊆
V
⊆
U be open subgroups of G.
(i) If W / U , then uV /W is the image of uU/W under
res : H 2 (U/W, W ab ) −→ H 2 (V /W, W ab ).
(ii) If W / U and V / U , then (V : W )uU/W is the image of uU/V under the
map
i : H 2 (U/V, V ab ) −→ H 2 (U/W, W ab ),
induced by the pair U/W → U/V, Ver : V ab → W ab .
Proof: Assertion (i) follows from the commutative diagram
1qrstuvwxyz{
W ab
V /W 0
V /W
1
1
W ab
U/W 0
U/W
1.
For (ii) we consider the commutative diagram
1|}~€‚ƒ„…†
W ab
U/W 0
U/W
1
1
V ab
U/V 0
U/V
1.
From this, we conclude that uU/W and uU/V have the same image u0 in
Ver
H 2 (U/W, V ab ). The composite of W ab −→ V ab −→ W ab is the norm NV /W
by (1.5.9). Therefore we have a commutative diagram
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203
§6. Profinite Groups of scd G = 2
‹Šˆ‰‡ V ab )
H 2 (U/V,
inf
H 2 (U/W, V ab )
Ver
i
H 2 (U/W, W ab )
ν
H 2 (U/W, W ab ),
where ν is induced by NV /W : W ab → W ab . This yields i(uU/V ) = Ver(u0 ) =
ν(uU/W ). But if g is a finite group, h a normal subgroup, A a g-module, then
the map Ĥ n (g, A) → Ĥ n (g, A) induced by Ng/h : A → A is multiplication by
(g : h). This is trivial for n = 0 and follows for n ≥ 0 by dimension shifting.
Therefore i(uU/V ) = (V : W )uU/W .
2
(3.6.2) Lemma. For every pair V / U of open subgroups of G, we have an
exact commutative diagram
ŽŒ“”•–’‘
H2 (U/V,
ẐZ)
H0 (U/V, V ab )
i
U ab
NU/V V ab
(V ab )U/V
1
ρ
Ver
NU/V
1
(U/V )ab
Ĥ 0 (U/V, V ab )
1,
where the homomorphism ρ is given by
σ 7−→
Y
u(τ, σ),
τ ∈U/V
u(τ, σ) being a 2-cocycle representing the canonical class uU/V . If U/V is
cyclic, the homomorphism i is injective.
Proof: The Hochschild-Serre spectral sequence
H i (U/V, H j (V, Q/ZZ)) ⇒ H i+j (U, Q/ZZ)
induces the exact sequence
0 → H 1 (U/V, Q/ZZ) → H 1 (U, Q/ZZ) → H 1 (V, Q/ZZ)U/V → H 2 (U/V, Q/ZZ).
Dualizing this sequence, we obtain the upper exact sequence of the diagram.
The lower sequence is trivially exact.
Ver
The composite of V ab −→ U ab −→ V ab is the norm map NU/V by (1.5.9),
which shows that the left partial diagram of (3.6.2) is commutative. Therefore
Ver induces a homomorphism ρ: (U/V )ab → Ĥ 0 (U/V, V ab ). In order to describe it explicitly, we choose a representative σ̂ ∈ U for each coset σ ∈ U/V ,
i.e. σ = σ̂V = V σ̂. Then the function
u(τ, σ) = τ̂ σ̂ τcσ −1 mod V 0
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204
Chapter III. Duality Properties of Profinite Groups
is a cocycle representing the class uU/V . On the other hand, by definition of
the transfer Ver,
Ver(σ̂ mod U 0 ) =
Y
τ̂ σ̂ τcσ −1 mod V 0 =
τ ∈U/V
Y
u(τ, σ).
τ ∈U/V
Passing from U to U/V and then to (U/V )ab , this map induces the map
ρ : (U/V )ab → Ĥ 0 (U/V, V ab ), given by
σ 7−→
Y
u(τ, σ)
τ ∈U/V
as claimed.
If U/V is cyclic, then
H2 (U/V, ẐZ) ∼
= H 2 (U/V, Q/ZZ)∗ ∼
= Ĥ 0 (U/V, Q/ZZ)∗ = 0,
and therefore the homomorphism i in (3.6.2) is injective.
2
(3.6.3) Lemma. Consider the conditions
(1) H 1 (U/V, V ab ) = 1,
(2) H 2 (U/V, V ab ) is of order #(U/V ) and is generated by uU/V .
If they hold for all pairs V / U such that U/V is cyclic of a prime order p, then
they hold for all pairs V / U .
Proof: Assume the conditions hold whenever U/V ∼
= ZZ/pZZ. We then prove
them for the case #U/V = pn by induction on n. Since the p-group U/V has a
nontrivial center, it sits in an exact sequence
1 −→ W/V −→ U/V −→ U/W −→ 1
with W/V ∼
= ZZ/pZZ and #U/W = pn−1 . We may therefore assume the
conditions (1) and (2) for W/V and U/W . From (3.6.2), with U replaced by
W , it follows that
Ver : W ab −→(V ab )W/V
is an isomorphism, since NW/V is an isomorphism as the kernel of NW/V is
equal to Ĥ −1 (W/V, V ab ) ∼
= H 1 (W/V, V ab ) = 1 and ρ is the Nakayama map
(cf. III §1). Therefore, by (1.6.7), we obtain exact sequences
i
res
1 −→ H q (U/W, W ab ) −→ H q (U/V, V ab ) −→ H q (W/V, V ab )
for q = 1 and for q = 2 since H 1 (W/V, V ab ) = 1.
For q = 1 we obtain H 1 (U/V, V ab ) = {1}, and for q = 2
#H 2 (U/V, V ab ) ≤ pn = (U : V ).
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205
§6. Profinite Groups of scd G = 2
Furthermore, uU/V must be of order pn , since otherwise 1 = pn−1 uU/V =
i(pn−2 uU/V ) by (3.6.1). As i is injective, this implies pn−2 uU/V = 1, contradicting the induction hypothesis. This proves condition (2).
We now prove the conditions (1) and (2) for a general pair V / U . For every
prime number p, let Up /V be a p-Sylow subgroup of U/V . By (1.6.10), the
restriction map
res : H q (U/V, V ab ) −→
M
H q (Up /V, V ab )
p
for q = 1, 2, is injective. The case q M
= 1 yields H 1 (U/V, V ab ) = {1}. When
q = 2, uU/V is mapped to the element
uUp /V by (3.6.1), which generates the
p
direct sum. Therefore res is an isomorphism, uU/V generates H 2 (U/V, V ab )
and
#H 2 (U/V, V ab ) =
#H 2 (Up /V, V ab ) =
Y
Y
p
p
#Up /V = #U/V.
2
We are now able to prove the following theorem, which gives a four-fold
characterization of the profinite groups of strict cohomological dimension 2.
(3.6.4) Theorem. For a profinite group G =/ 1, the following conditions are
equivalent:
(i) scd G = 2.
(ii) For every pair V / U , the transfer
Ver : U ab −→(V ab )U/V
is an isomorphism.∗)
(iii) For every pair V / U , the U/V -module V ab is a class module with
fundamental class uU/V , i.e. the conditions (1) and (2) of (3.6.3) hold.
(iv) The G-module D2 = lim U ab is a formation module with D2U = U ab and
−→
with (uU/V ) as a system of fundamental classes.
(v) There exists a level-compact formation module C with trivial universal
norm groups NU C for all open subgroups U ⊆ G.
Proof: (i) ⇒ (ii): This follows from (3.3.11) and (1.5.9).
(ii) ⇒ (iii): Assume (ii) and let V / U be a pair such that U/V is cyclic of
prime order p. Consider the exact commutative diagram (3.6.2), in which the
homomorphism
∗) In other words, the fundamental G-modulation π ab : U 7→ U ab is a G-module (see I §5).
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Chapter III. Duality Properties of Profinite Groups
i : H0 (U/V, V ab ) → U ab
is injective and ker(NU/V ) = Ĥ −1 (U/V, V ab ). By the snake lemma, the bijectivity of Ver implies
H 1 (U/V, V ab ) ∼
= Ĥ −1 (U/V, V ab ) = 1
H 2 (U/V, V ab ) ∼
= Ĥ 0 (U/V, V ab ) ∼
= U/V ∼
= ZZ/pZZ.
In order to show that uU/V generates H 2 (U/V, V ab ), i.e. uU/V =/ 0, consider
the exact commutative diagram
1—˜™š›œžŸ ¡
V ab
U/V 0
U/V
1
1
H0 (U/V, V ab )
U ab
U/V
1.
If uU/V = 0, then the upper group extension splits, i.e. U/V 0 is a semidirect product V ab o Γ with a group Γ which is mapped isomorphically onto
U/V . Its image in U ab is nontrivial. But in the case of a semi-direct product
V ab o Γ , the transfer Ver : V ab o Γ → V ab maps Γ to 1, since for γ ∈ Γ ,
Q
Ver(γ) = β ∈Γ βγ(γβ)−1 = 1. Therefore Ver : U ab →(V ab )U/V is not injective,
a contradiction. This proves (iii) in the case U/V ∼
= ZZ/pZZ, and the general
case follows because of lemma (3.6.3).
(iii) ⇒ (i): If (iii) holds, then by §1, ex.4, the cup-product
uU/V ∪ : H p (U/V, Hom(V ab , Q/ZZ)) −→ H p+2 (U/V, Q/ZZ),
induced by the pairing V ab × Hom(V ab , Q/ZZ) → Q/ZZ is an isomorphism for
p > 0 and a surjection for p = 0. By (2.4.4), this map coincides up to sign with
the differential
d2p,1 : E2p,1 −→ E2p+2,0
of the Hochschild-Serre spectral sequence
E2p,q = H p (U/V, H q (V, Q/ZZ)) ⇒ H p+q (U, Q/ZZ).
The surjectivity implies E3p+2,0 = 0 for p
Therefore the edge morphism
≥
p,0
0 and thus E∞
= 0 for p
≥
2.
E2p,0 = H p (U/V, Q/ZZ) −→ H p (U, Q/ZZ)
p,0
is the zero map since it factors through E∞
. But this edge morphism is the
inflation map, and we obtain
H p+1 (U, ZZ) ∼
= H p (U, Q/ZZ) = lim H p (U/V, Q/ZZ) = 0
−→
V
for p ≥ 2. From this it follows that scd G = 2 (see §3, ex.6), noting that
scd G = 0 or 1 is impossible.
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207
§6. Profinite Groups of scd G = 2
(ii) ⇒ (iv): The inclusion functor
i : Mod (G) −→ M od (G)
from the category of G-modules into the category of G-modulations identifies
Mod (G) with the full subcategory of G-modulations M with “Galois descent”,
i.e. for which res : M (U ) → M (V )U/V is an isomorphism. The functor i has
as left adjoint the functor
j : M 7−→ lim M U .
−→
U
We have, in particular, D2 = j(π ab ), and the condition (ii) means that the
fundamental G-modulation π ab has Galois descent, i.e. π ab = i(D2 ). Therefore
D2U = (iD2 )(U ) = π ab (U ) = U ab .
Since (ii) is equivalent to (iii), the G-module D2 is a formation module with
respect to the isomorphisms
invU/V : H 2 (U/V, V ab ) −→ (U1:V ) ZZ/ZZ
given by uU/V 7→
classes.
1
(U :V )
mod ZZ, i.e. with (uU/V ) as a system of fundamental
(iv) ⇒ (v): Assuming (iv), D2 is a level-compact formation module, since
D2U = U ab . Therefore it suffices to show that the universal norm groups NU D2
are trivial. For this we consider, for every pair V / U , the diagram
V ©ª¨§¦¥¤¢£ ab
U ab
NU/V
i
D2V
Ver
V ab
NU/V
i
D2U
i
NU/V
D2V
The top of this diagram is commutative by (1.5.9), the bottom and the left-hand
side diagram are trivially commutative and the right-hand side diagram by the
definition of D2 . Therefore the back diagram is commutative, i.e. we have a
commutative diagram
V®«¬­
U
NU/V
D2U
D2V
T
with surjective vertical arrows. Since V ⊆U V = {1}, we obtain
NU D2 =
\
NU/V D2V = {1},
V ⊆U
what we wanted to prove.
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208
Chapter III. Duality Properties of Profinite Groups
(v) ⇒ (ii): Let C be a level-compact formation module with trivial universal
norm groups. By class field theory, we have, for every open subgroup U in
G, a canonical continuous homomorphism C U → U ab with dense image. It is
surjective, since C U is compact, and injective, since the kernel NU C is 1. If V
is an open normal subgroup of U , then we have the commutative diagram
C²¯°± U
U ab
(C V )U/V
Ver
(V ab )U/V ,
2
which shows that Ver is an isomorphism.
Remark: The equivalence (i) ⇔ (iii) in the above theorem is due to J. TATE,
as Y. KAWADA remarks. Kawada gives Tate’s proof in [101] (see ex.4). The
equivalence was independently proven by A. BRUMER (see [19]). Brumer’s
proof, however, is rather involved. Another proof is given in [72]. The
equivalence (ii) ⇔ (iii) was first proven by J.-P. SERRE (see [208], chap. VI),
and another proof of it is found in [72], 2.3. These proofs rely in an essential
way on the use of negative dimensional cohomology.
In the proof presented here, we have also shown that the (level-compact)
dualizing module D2 = lim U ab of a profinite group G of scd G = 2 has trivial
−→
universal norm groups NU D2 . Therefore we may apply POITOU’s duality
theorem (3.1.11)(i) to the G-module D2 and obtain the
(3.6.5) Corollary. Let G be a profinite group of scd G = 2. We then have a
canonical isomorphism
∼
inv : H 2 (G, D2 ) −→
1
ZZ/ZZ
#G
and the cup-product
∪
Ĥ i (G, Hom(A, D2 )) × Ĥ 2−i (G, A) −→ H 2 (G, D2 ) ∼
=
1
ZZ/ZZ
#G
induces an isomorphism
∼ Ĥ 2−i (G, A)∨
Ĥ i (G, Hom(A, D2 )) −→
for all i ∈ ZZ and every G-module A which is a finitely generated free ZZmodule. For i ≤ 0 this is true also for every G-module A, finitely generated
over ZZ.
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§6. Profinite Groups of scd G = 2
By the equivalence (i) ⇔ (v) in theorem (3.6.4), class field theory seems
to be very strongly restricted to cohomological dimension 2. This, however,
is not really true. The restriction is due to the fact that we insist on the
presence of a G-module rather than a G-modulation with trivial universal
norms and the properties of a class formation. For example, the fundamental
group G = π1 (X, x) of a smooth proper curve X over a finite field IF has
scd G = 3. Thus, by the equivalence (i) ⇔ (ii) of (3.6.4), its fundamental
G-modulation π ab is not a G-module. We have, on the other hand, the Gmodulation P ic : U 7→ Pic (X(U )), which associates to an open subgroup the
Picard group Pic (X(U )) of the unramified covering X(U ) → X determined by
U . Passing to the profinite completion Pd
ic, we obtain a canonical isomorphism
π ab ∼
ic
= Pd
of G-modulations (see [208]).
Exercise 1. A profinite group G of scd G ≤ 2 contains no closed abelian subgroups other than
procyclic ones.
Exercise 2. The group G = π1 (X, x), where X is a smooth proper curve over a finite field,
has the property of the groups of ex.1 but has scd G = 3.
Exercise 3. Not every group of scd G = 3 has the property of the groups of ex.1.
Exercise 4. Prove the implication (v) ⇒ (i) in (3.6.4) directly by means of Poitou’s duality
theorem (3.1.11)(i) for i = −1.
Hint: If C is level-compact with trivial universal norms, then by (3.1.11)(i) and (1.9.12), we
have for A = ZZ and A = ZZ/pZZ
H 3 (U, A)∗ ∼
= Ĥ −1 (U, Hom(A, C))
= Ĥ −1 (U, NU Hom(A, C))
= Ĥ −1 (U, Hom(A, NU C)) = 0.
Let Gp be a p-Sylow subgroup of G. Then
H 3 (Gp , ZZ/pZZ) = lim H 3 (U, ZZ/pZZ) = 0,
−→
U ⊇Gp
since
Gp =
\
U = lim U,
U ⊇ Gp
←−
U ⊇Gp
hence cdp G = cd Gp ≤ 2 by (3.3.2). Since H 3 (U, ZZ) = 0 for all open subgroups U , we obtain
scd G ≤ 2 by (3.3.4), and scd G = 2, since
H 2 (G, ZZ) = H 1 (G, Q/ZZ) =/ 0.
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Chapter III. Duality Properties of Profinite Groups
§7. Poincaré Groups
Poincaré groups are groups for which a cohomological duality theorem
of Poincaré type H i × H n−i → H n ,→ Q/ZZ holds with a special dualizing
module. These groups play an important role in topology as well as in number
theory. For example, the pro-p-completion of the fundamental group of a
compact Riemann surface, and the Galois group of the maximal p-extension
of a p-adic local field are typical examples of Poincaré groups. For these and
other reasons they are of special interest. We fix a prime number p for our
investigations.
In II §5 we have introduced, for every G-module A, the G-modules
Di (A) = lim H i (U, A)∗ ,
−→
U
and we have called the G-module
Dn (ZZ/pν ZZ)
D(p) = lim
−→
ν
the dualizing module of G at p if n = cdp G. Its importance lies in the
functorial isomorphism
H n (G, A)∗ ∼
= HomG (A, D(p) )
for all A ∈ Modp (G). For this section we set I = D(p) .
We have called G a duality group at p of dimension n if Di (ZZ/pZZ) = 0
for i < n (see (3.4.6)). In this case, the edge morphism of the Tate spectral
sequence
H i (G, Dn−j (A)) ⇒ H n−(i+j) (G, A)∗
for A ∈ Modp (G) is a functorial isomorphism
H i (G, Hom(A, I)) ∼
= H n−i (G, A)∗ ,
which is also obtained from the cup-product
∪
tr
H i (G, Hom(A, I)) × H n−i (G, A) −→ H n (G, I) −→ Qp /ZZp .
(3.7.1) Definition. A duality group G at p of dimension n is called a Poincaré
group at p if I ∼
= Qp /ZZp as an abelian group.
Since
Aut(I) = Aut(Qp /ZZp ) = ZZ×
p,
the group G acts on I via a continuous character
χ : G −→ ZZ×
p,
and I is determined by χ (up to isomorphism). The Serre criterion (3.4.5)
implies that scdp G = n + 1 if χ(G) is finite and scdp G = n if χ(G) is infinite.
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211
§7. Poincaré Groups
(3.7.2) Theorem. For a finitely generated profinite group G the following
assertions are equivalent.
(i) G is a Poincaré group at p of dimension 2,
(ii) cdp G = 2 and I ∼
= Qp /ZZp (as an abelian group),
(iii) cdp G = 2 and p I ∼
= ZZ/pZZ (as an abelian group).
Proof: The implications (i) ⇒ (ii) ⇒ (iii) are trivial. Now we show that
(iii) implies Di (ZZ/pZZ) = 0 for i = 0, 1. The p-Sylow subgroups Gp of G
are infinite, since 0 < cdp G = cdp Gp < ∞. This means that for every open
subgroup U , there is an open subgroup V ⊆ U such that p | (U : V ). From
this, it follows that
D0 (ZZ/pZZ) = lim H 0 (U, ZZ/pZZ)∗ = lim ZZ/pZZ = 0,
−→
U
−→
U
since the transition maps in the last direct limit are multiplication by (U : V ).
Let A ∈ Modp (G) be a finite G-module with pA = 0 and let us set
0
A = Hom (A, I). Every open subgroup U of G has the same cohomological dimension 2 as G (see (3.3.5)) and the same dualizing module I. Noting
00
that A = A, we obtain a canonical isomorphism
∼ H 2 (U, A0 ).
(∗)
ϕ∗A0 : H 0 (U, A)∗ −→
Let U ∗ = U p [U, U ], where [U, U ] is the closure of the commutator subgroup
of U . The group U/U ∗ is the largest profinite quotient of U which is abelian
and of exponent p. The restriction
res : H 1 (U, ZZ/pZZ) −→ H 1 (U ∗ , ZZ/pZZ)
is obviously the zero map. Since G is finitely generated, so is U , and thus
U/U ∗ is finite, i.e. U ∗ is open. Now let A be the finite U -module defined by
the exact sequence
∗
0 −→ ZZ/pZZ −→ IndUU (ZZ/pZZ) −→ A −→ 0.
Applying the exact functor Hom (−, I), we obtain another exact sequence
∗
0 −→ A0 −→ IndUU (ZZ/pZZ)0 −→ p I −→ 0.
By (1.6.5), the composite of the maps
∗
sh
1
∗
H 1 (U, ZZ/pZZ) −→ H 1 (U, IndUU (ZZ/pZZ)) −→
Z/pZZ)
∼ H (U , Z
is the restriction, hence the zero map. Therefore we obtain from the two exact
sequences the exact commutative diagram
´µ¶·¸¹³
H 1 (U, p I)
0
H 1 (U, ZZ/pZZ)∗
αU
∗
H 2 (U, A0 )
H 2 (U, IndUU (ZZ/pZZ)0 )
H 0 (U, A)∗
H 0 (U, IndUU (ZZ/pZZ))∗
∗
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Chapter III. Duality Properties of Profinite Groups
where the vertical arrows are the isomorphisms (∗). From this diagram, we get
on the one hand an injection
D1 (ZZ/pZZ) = lim H 1 (U, ZZ/pZZ)∗ ,→ lim im(αU ) ,
−→
U
−→
U,cor ∗
and on the other hand a surjection
im(αU ).
H 1 (U, p I) −→ lim
lim
−→
−→
U,res
U
The left-hand group is zero by (1.5.1), hence D1 (ZZ/pZZ) = 0. Hence G is a
duality group at p of dimension 2. Further, I ∼
= Qp /ZZp or ZZ/pk ZZ, k ≥ 1 (as an
abelian group) since p I ∼
= ZZ/pZZ. But I is p-divisible by (3.4.7) and therefore
G is a Poincaré group.
2
(3.7.3) Corollary. Let G be a finitely generated pro-p-group of cohomological
dimension equal to 2. Then G is a Poincaré group if and only if
dimIFp H 2 (N, ZZ/pZZ) = 1
for every open normal subgroup N of G.
Proof: Since cdp G = 2, we have for open normal subgroups N 0
group G
∼ H 2 (N, Z
H 2 (N, ZZ/pm ZZ)/p −→
Z/pZZ)
⊆
N of the
and by (3.3.11) the surjectivity of the corestriction map
H 2 (N 0 , ZZ/pZZ) −→ H 2 (N, ZZ/pZZ) .
Thus we obtain for the dualizing module I of G
pI
= p ( lim
H 2 (N, ZZ/pm ZZ)∗ ) ∼
H 2 (N, ZZ/pZZ)∗
= lim
−→
−→
m,cor ∗
cor ∗
and p I is isomorphic to ZZ/pZZ if and only if dimIFp H 2 (N, ZZ/pZZ) = 1 for every
open normal subgroup N of G. The result follows from (3.7.2).
2
Later, in §9, theorem (3.9.15), we will see that it is enough to consider only
open subgroups N of G with (G : N ) ≤ p.
The following theorem shows that the class of Poincaré groups is closed
under group extensions. It was first proved by A. PLETCH (see [167]) and later
(independently) by K. WINGBERG (see [253]).
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213
§7. Poincaré Groups
(3.7.4) Theorem. Let
1 −→ H −→ G −→ G/H −→ 1
be an exact sequence of profinite groups such that
a) H i (U, ZZ/pZZ) is finite for all open subgroups U of H and all i ≥ 0,
b) cdp G/H < ∞.
Then if two of the three groups are Poincaré groups at p, so is the third.
Moreover, in this case we have:
(i) cdp G = cdp H + cdp G/H.
(ii) There is a canonical G-isomorphism I(G)∗ ∼
= I(H)∗ ⊗ ZZp I(G/H)∗ .
Remark: Theorem (3.7.4) can be generalized in several directions, cf. [167],
[253]. The most general version known to us is the following: we call a
profinite group virtual duality group at p if some open subgroup is a duality
group at p. Then, assuming a) above, G is a virtual duality group if and only
if H and G/H are. The (virtual) cohomological dimension and the dualizing
module of G can be computed from that of H and of G/H as in (i) and (ii)
(see [SW]).
Proof of (3.7.4) (see [253]): Let d = cdp G, m = cdp H, n = cdp G/H. By
our assumption that two of the groups are Poincaré groups, d, m and n are
finite, recalling that n < ∞, m ≤ d and d ≤ m + n. Because of a) we obtain
from (3.3.8)
d = m + n.
Let g run through the open normal subgroups of G and let h = g ∩ H. Then
g/h = gH/H runs through the open normal subgroups of G/H. For a Gmodule A ∈ Modp (G), we consider the Hochschild-Serre spectral sequence
E(g, h, A) : E2ij (g, h, A) = H i (g/h, H j (h, A)) ⇒ H i+j (g, A).
If g 0 ⊆ g is another open normal subgroup of G, then the corestriction yields a
morphism
cor : E(g 0 , h0 , A) −→ E(g, h, A)
of spectral sequences, where h0 = h ∩ g 0 (see II §4, ex.3). The map
E2ij (g 0 , h0 , A) → E2ij (g, h, A)
is the composite of the maps
i
H (g
0
h
ȼ j (h0 , A)) cor h
/h , H
0
g 0 /h0
0
i
0
0
j
H (g /h , H (h, A))
cor g/h
H i (g/h, H j (h, A)),
and the map between the limit terms is the corestriction
0
cor gg : H i+j (g 0 , A) −→ H i+j (g, A).
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214
Chapter III. Duality Properties of Profinite Groups
For 2 ≤ r
≤
∞ we set
r
(G, A) = lim
Erij (g, h, A)∗ .
Dij
−→
g
0
If h runs through the open subgroups of H which are normal in G, then the
H j (h0 , A) and thus also lim 0 H j (h0 , A) are G/H-modules. As in the proof of
−→ h
(1.5.1), we see that
2
lim H i (g/h, H j (h0 , A))∗ ,
Dij
(G, A) = lim
−→ −→
(∗)
h0
g/h
where for both limits the transition maps are (induced by) cor ∗ .
By (3.4.6), G is a duality group at p of dimension d = n + m if and only if
Di+j (G, ZZ/pZZ) = lim H i+j (g, ZZ/pZZ)∗ = 0 for i + j =/ d ∗) ,
−→
g
and this is equivalent to
∞
(∗∗)
Dij
(G, ZZ/pZZ) = 0 for i + j =/ d.
Assume now that G/H is a Poincaré group at p. Then by (3.4.6)(i) the
open subgroups g/h are also Poincaré groups at p of the same dimension n
and the same dualizing module I(G/H). Furthermore, with condition a) it
follows by a standard argument that the groups H j (h0 , A) are finite for every
finite A ∈ Modp (G). Therefore we obtain by (3.4.6) (ii) (using (1.5.3)(iv) and
(1.5.1))
2
Dij
(G, A) = lim
lim
H n−i (g/h, Hom (H j (h0 , A), I(G/H)))
−→
−→
g/h,res h0 ,cor ∗
=
(∗∗∗)

 Hom (lim
←−
h0

H j (h0 , A), I(G/H))
0

 Hom (Dj (H, A)∗ , I(G/H))
=
0
for i = n,
otherwise,
for i = n,
otherwise.
Here Dj (H, A)∗ should be seen in the topological sense, i.e. as a compact
abelian group, and Hom (Dj (H, A)∗ , I(G/H)) are the continuous homomorphisms, i.e. the homomorphisms with finite image. From this, we deduce that
the following assertions are equivalent.
(1) H is a duality group at p,
(2) Dj (H, ZZ/pZZ) = 0 for j =/ m,
2
(3) Dij
(G, ZZ/pZZ) = 0 for (i, j) =/ (n, m),
∞
(4) Dij
(G, ZZ/pZZ) = 0 for (i, j) =/ (n, m),
(5) G is a duality group at p.
∗) D
Z/pZZ)
i+j (G, Z
has here a different meaning from II §1.
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215
§7. Poincaré Groups
Thus we have proved the following
(3.7.5) Lemma. Suppose that assumptions a) and b) of (3.7.4) are fulfilled
and assume that G/H is a Poincaré group at p of dimension n.
Then H is a duality group at p of dimension m if and only if G is a duality
group at p of dimension m + n. In this case the dualizing module of H is
isomorphic to the dualizing module of G regarded as an H-module.
We proceed with the proof of (3.7.4) and assume now that G and H are
duality groups at p. Since the groups H m−j (h, ZZ/pZZ) are finite, and since
lim 0 H m−j (h0 , ZZ/pZZ) = 0 for j =/ m if h0 runs through the open subgroups h0
−→ h
of h, we find for every h an h0 ⊆ h such that
res : H m−j (h, ZZ/pZZ) −→ H m−j (h0 , ZZ/pZZ)
is the zero map for all j =/ m. Since h and h0 are duality groups at p of
dimension m (as H is), we have a commutative diagram of non-degenerate
pairings
H j (h, pÀ¿¾½¼ I(H)) × H m−j (h, ZZ/pZZ)
∪
ZZ/pZZ
∪
ZZ/pZZ,
res
cor
H j (h0 , p I(H)) × H m−j (h0 , ZZ/pZZ)
which shows that the left corestriction maps are zero. Therefore we see from
(∗) that
2
Dij
(G, A) = 0,
for all i and all j =/ m,
and consequently
∞
2
(G, p I(H)),
Dim
(G, p I(H)) = Dim
for all i.
If we assume, in addition, that H is a Poincaré group, it follows that
p I(H)
∼
= ZZ/pZZ
(as an abelian group), and there exists an open subgroup g of G which acts
trivially on p I(H). Therefore (∗∗) implies
2
2
(G, p I(H)) = Dim
(G, ZZ/pZZ) = 0,
Dim
for all i =/ n.
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Chapter III. Duality Properties of Profinite Groups
Together with (∗), this implies
Di (G/H, ZZ/pZZ) = lim H i (g/h, ZZ/pZZ)∗
−→
g/h
H i (g/h, H 0 (h0 , ZZ/pZZ)∗ ))∗
= lim
( lim
−→
←−
g/h,cor ∗ h0 ,res ∗
= lim ( lim H i (g/h, H m (h0 , p I(H))))∗
−→
←−
g/h,cor ∗ h0 ,cor
= lim
lim H i (g/h, H m (h0 , p I(H)))∗
−→
−→
g/h,cor ∗ h0 ,cor ∗
2
(G, p I(H)) = 0.
= Dim
We have thus shown that G/H is a duality group at p of dimension n.
The proof of (ii) goes as follows. As before let h0 run through all open
subgroups of H which are normal in G. Since m = cdp H, we have by (2.1.4),
H m+n (g, A) = H n (g/h, H m (h, A)),
and we obtain similarly as in (∗∗∗)
I(G) = lim lim H m+n (g, ZZ/pν ZZ)∗
−→ −→
ν
g
lim lim H n (g/h, H m (h0 , ZZ/pν ZZ))∗
= lim
−→ −→ −→
ν
h0
g/h
= lim lim lim H 0 (g/h, Hom (H m (h0 , ZZ/pν ZZ), I(G/H)))
−→ −→ −→
ν
h0 g/h,res
Hom (lim
H m (h0 , ZZ/pν ZZ), I(G/H))
= lim
−→
←−
ν
h0
= lim Hom (Dm (H, ZZ/pν ZZ)∗ , I(G/H))
−→
ν
= Hom (I(H)∗ , I(G/H)) = (I(H)∗ ⊗ I(G/H)∗ )∗ .
ZZp
In particular, it follows that
rank ZZp I(G)∗ = rank ZZp I(H)∗ · rank ZZp I(G/H)∗ .
This completes the proof of the theorem.
Let us finally consider the case of a pro-p-group G. We set
H i (G) = H i (G, ZZ/pZZ)
and consider these groups as IFp -vector spaces.
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217
§7. Poincaré Groups
(3.7.6) Proposition (SERRE). For an infinite pro-p-group G the following
statements are equivalent:
(i) G is a Poincaré group of dimension n.
(ii) dimIFp H i (G) < ∞ for all i ≤ n, H n (G) ∼
= IFp , and the cup-product yields
∗
)
a non-degenerate pairing
H i (G) × H n−i (G) −→ H n (G) ∼
= IFp
for all 0 ≤ i ≤ n.
Proof: (i) ⇒ (ii). Let G be a Poincaré group of dimension n. Since
I ∼
= ZZ/pZZ as abelian groups, and consequently also
= Qp /ZZp , we have p I ∼
as G-modules, since G is a pro-p-group and #Aut(p I) = p − 1 is prime to p.
The G-module Hom(ZZ/pZZ, p I) is also isomorphic to ZZ/pZZ, hence the duality
isomorphism (3.4.6) (ii) yields an isomorphism
H i (G) ∼
= Hom (H n−i (G), IFp ) = H n−i (G)∗ ,
given by the cup-product
∪
tr
H i (G) × H n−i (G) −→ H n (G) −→ IFp .
For i = n, we obtain H n (G) ∼
= IFp and for i ≥ 0,
H i (G)∗∗ ∼
= H n−i (G)∗ ∼
= H i (G),
from which follows that H i (G) is finite. (The canonical homomorphism from
a vector space to its bidual is an isomorphism if and only if the dimension is
finite!)
(ii) ⇒ (i). Let A be any finite G-module such that pA = 0 and let 0
We claim that the pairing
∪
H i (G, A) × H n−i (G, A∗ ) −→ H n (G) ∼
= IFp
≤
i
≤
n.
∼ H n−i (G, A)∗ . This is true for
induces an isomorphism αiA : H i (G, A∗ ) →
A = ZZ/pZZ by the assumption (ii). We first show that for an arbitrary A (with
pA = 0) αiA is surjective for i = 0, bijective for i = 1, . . . n − 1 and injective
for i = n. We proceed by induction on dimIFp A.
Since ZZ/pZZ is the only simple p-primary G-module, there is an exact
sequence of G-modules
0 −→ A0 −→ A −→ ZZ/pZZ −→ 0.
This, together with the dual sequence
0 −→(ZZ/pZZ)∗ −→ A∗ −→ A∗0 −→ 0 ,
yields an exact diagram for 0 ≤ i ≤ n
∗) A pairing A×B → Q/ZZ is non-degenerate if it induces injections A ,→ B ∗ and B ,→ A∗ .
Clearly, these are isomorphisms if A and B are finite.
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218
Chapter III. Duality Properties of Profinite Groups
H i−1ÍÌËÊÁÂÃÄÅÆÇÈÉ (A∗0 )
H i ((ZZ/pZZ)∗ )
A
ZZ /p ZZ
0
αi−1
αi
H j+1 (A0 )∗
H j (ZZ/pZZ)∗
H i (A∗ )
H i (A∗0 )
H i+1 ((ZZ/pZZ)∗ )
A0
αA
i
ZZ /p ZZ
αi+1
αi
H j (A)∗
H j (A0 )∗
H j−1 (ZZ/pZZ)∗ ,
where i + j = n and H k (A) means H k (G, A), H k = 0 for negative k and the
map αk is the zero map for k < 0 and k > n. Now the induction step follows
by diagram chasing.
It remains to show duality in dimension i = 0 and i = n and by symmetry it
suffices to deal with the case i = 0. Let A be a finite G-module with pA = 0
and let U be an open subgroup with AU = A. Choose an open subgroup
V $ U (this is where we use that G has infinite order) strictly contained in U .
Consider the exact sequence
φ
0 −→ A−1 −→ Ā −→ A −→ 0,
where Ā := Map(G/U, A), φ is the map f 7→ g∈G/U f (g) and A−1 := ker φ
(compare with the remark following (1.3.8)). We claim that the map
P
H 0 (φ) : H 0 (G, Ā) → H 0 (G, A)
is zero. Indeed, via the identification Ā ∼
= IndUG (A) (see I §6), it corresponds
0
to the corestriction map cor : H (V, A) → H 0 (G, A), which factors through
cor : H 0 (V, A) → H 0 (U, A). Since U acts trivially on A, the last map is
multiplication by (U : V ), hence trivial as pA = 0.
We know that the functor H 0 (G, −) is coeffaceable (see II §2) on G-modules
annihilated by p. Now choose an injection A ,→ B into a G-module B with
pB = 0 such that the map H 0 (G, B ∗ ) → H 0 (G, A∗ ) is the zero map. Then we
get a commutative exact diagram
×ÖÕÏÐÑÒÓÔÎ ∗ )
H 0 (B
αB
0
H n (B)∗
0
H 0 (A∗ )
αA
0
H n (A)∗
H 1 ((B/A)∗ )
B/A
α1
H n−1 (B/A)∗
H 1 (B ∗ )
αB
1
H n−1 (B)∗ .
The α1 ’s are bijective, hence α0A is injective. So we have proved duality for
modules annihilated by p and for i = 0, . . . , n.
We apply this result to A = ZZ/pZZ[G/U ], where U runs through the open
normal subgroups of G. By (3.4.1), Shapiro’s lemma and (1.6.5), we obtain
for 0 ≤ i < n
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219
§7. Poincaré Groups
Di (ZZ/pZZ) = lim H i (G, ZZ/pZZ[G/U ])∗
−→
U
H n−i (G, Hom (ZZ/pZZ[G/U ], ZZ/pZZ))
= lim
−→
U
= lim H n−i (G, IndUG (ZZ/pZZ))
−→
U
H n−i (U, ZZ/pZZ) = 0,
= lim
−→
U,res
by (1.5.1).
Next we show cdp G ≤ n. Let x ∈ H n+1 (G, A), A ∈ M odp (G), pA = 0. By
(1.5.1), we have lim
H n+1 (U, A) = 0, i.e. there exists an open subgroup U
−→
U
of G such that x becomes zero in H n+1 (U, A) = H n+1 (G, IndUG (A)). From the
exact sequence
0 −→ A −→ IndUG (A) −→ B −→ 1,
it follows that there is an exact sequence
H n (G, IndUG (A)) −→ H n (G, B) −→ H n+1 (G, A) −→ H n+1 (G, IndUG (A)).
The functor H n (G, −) is right exact on G-modules annihilated by p since it is
dual to the functor H 0 (G, −) which is left exact. Therefore the last arrow is
injective, and x = 0. This proves cdp G ≤ n.
Thus we have shown that G is a duality group at p of dimension n. It remains
to determine the dualizing module I. By (3.4.7), I is a divisible p-torsion group
and therefore it suffices to show p I ∼
= ZZ/pZZ. By (3.4.7), p I ∼
= Dn (ZZ/pZZ) and
the same calculation as above for Di , i < n, shows
Dn (ZZ/pZZ) = lim H 0 (U, ZZ/pZZ) = ZZ/pZZ.
−→
U,res
This proves the proposition.
2
The pro-p-groups G which are Poincaré groups of dimension 2 are called
Demuškin groups. We give an explicit description and classification of them
in §9.
Exercise 1. Let U be an open subgroup of the profinite group G. If cdp G < ∞ then G is a
duality group at p of dimension n if and only if U is.
Exercise 2. If a p-Sylow subgroup Gp of a profinite group G is a duality group at p of
dimension n, then so is G. Is the converse true?
Exercise 3. Let G be a Poincaré group of dimension n > 0.
(i) If H is a proper closed subgroup of G, then the restriction map H n (G, ZZ/pZZ) −→
H n (H, ZZ/pZZ) is 0.
(ii) If H is a closed, but not open subgroup of G, then cd H ≤ n − 1.
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Chapter III. Duality Properties of Profinite Groups
§8. Filtrations
In this section G is always a pro-p-group. In the following we introduce the
notion of the q-central series of G.
(3.8.1) Definition. Let G be a pro-p-group and let q be a power of p. Then the
descending q-central series of G is the filtration {Gi }i≥1 recursively defined
by
G1 = G, Gi+1 = (Gi )q [Gi , G],
where [Gi , G] and (Gi )q are the closed subgroups topologically generated by
the commutators
(x, y) = x−1 y −1 xy,
x ∈ Gi , y ∈ G, and by the q-th powers of elements of Gi , respectively. ∗)
If q = 0, then we denote this series by {Gi }i≥1 , i.e.
G1 = G,
Gi+1 = [Gi , G] .
It is called descending central series of G.
(3.8.2) Proposition. Let G be a pro-p-group. Then the subgroups Gi of the
q-central series are normal in G and Gi /Gi+1 is contained in the center of
G/Gi+1 . Furthermore,
\
Gi = 1 .
i
If G is finitely generated and q =/ 0, then the subgroups Gi form a fundamental
system of open neighbourhoods of 1.
Proof: The first two statements are obvious. For the third, let U be an open
normal subgroup of G. The projection G → G/U maps the q-central series
(Gi ) of G into a subseries of the q-central series of G/U , which terminates
with {1}. Therefore Gi ⊆ U for i sufficiently large.
Now assume that G is finitely generated and q =/ 0. We will show that
the subgroups Gi are of finite index, hence open, and proceed by induction
on i. For i = 1, this is trivial. Assume that (G : Gi ) is finite. Then Gi is
a finitely generated pro-p-group and Gi /(Gi )q [Gi , Gi ] is a finitely generated
abelian group of exponent q, thus finite, and has Gi /(Gi )q [Gi , G] = Gi /Gi+1
as quotient. This shows that (G : Gi+1 ) is finite, so Gi+1 is open in G.
2
We collect some formulae for commutators and p-powers of the pro-p-group G.
∗) The reader should take care about the different meaning of the superscripts i and q.
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§8. Filtrations
(3.8.3) Proposition. If x ∈ Gi , y
∈
Gj , a ∈ q r ZZp , then
a
(i) (xy)a ≡ xa y a (y, x)( 2 ) mod Gi+j+max(1,r) ,
a
(ii) (xa , y) ≡ (x, y)a ((x, y), x)( 2 ) mod Gi+j+1+max(1,r) ,
a
(iii) (x, y a ) ≡ (x, y)a ((x, y), y)( 2 ) mod Gi+j+1+max(1,r) .
Proof: Recall the identities
(xy, z) = (x, z)((x, z), y)(y, z),
(x, yz) = (x, z)(x, y)((x, y), z)
(∗)
and the Hall-Witt identity (where xy = y −1 xy):
((x, y −1 ), z)y ((y, z −1 ), x)z ((z, x−1 ), z)x = 1.
Using these identities, we first observe that [Gi , Gj ] ⊆ Gi+j . For x ∈ Gi , y
and a ∈ IN, we obtain
a
(xy)a ≡ xa y a (y, x)( 2 ) mod Gi+j+1 .
∈
Gj
By passing to the limit, we obtain (i) for a ∈ q r ZZp , r = 0, 1.
We proceed by induction on r. Let r ≥ 2 and a = bq, b ∈ q r−1 ZZp . Then
a
bq
b b
(xy) = (xy) = (x y (y, x)
b
2
fr−1 )q with fr−1
∈
Gi+j+r−1 .
By induction and using (∗), we obtain
(xy)a ≡ xa y a (y, x)
Since
b
2
q≡
a
2
b
2
q
mod Gi+j+r .
mod q r+1 , we obtain (i). Using this result, we get
a
y −1 xa y = (y −1 xy)a = (x(x, y))a ≡ xa (x, y)a ((x, y), x)( 2 ) mod Gi+j+1+max(1,r) ,
hence
a
(xa , y) = x−a y −1 xa y ≡ (x, y)a ((x, y), x)( 2 ) mod Gi+j+1+max(1,r) .
2
This shows (ii) and (iii) follows similarly.
(3.8.4) Corollary. For i, j
r
s
≥
1 and r, s ≥ 0, where r + s =/ 0, we have
[(Gi )q , (Gj )q ] ⊆ (Gi+j )q
r+s
· (Gi+j+1 )q
r+s−1
· Gi+j+r+s+1 .
The quotient Gi /Gi+1 is an abelian group, which we now write additively.
We denote this additive group by gri (G). The direct sum
gr(G) =
∞
M
gri (G)
i=1
has the structure of a Lie algebra over k = ZZp /q ZZp . The Lie bracket [ , ] is
induced by the commutator, that is, if ξ = x̄ ∈ gri (G) and η = ȳ ∈ grj (G), then
[ξ, η] is the image of (x, y) = x−1 y −1 xy in gri+j (G). It is convenient to carry
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Chapter III. Duality Properties of Profinite Groups
out commutator calculations additively with respect to the filtration {Gi } in
the Lie algebra gr(G).
Proposition (3.8.3) shows that the map x 7→ xq of Gi into Gi+1 induces a
mapping πi : gri (G) → gri+1 (G). The family (πi ) then induces a map π∗ :
gr(G) → gr(G). Let π be an indeterminate over the ring k = ZZ/q ZZ if q =/ 0
and the zero element of k = ZZp if q = 0. Then there exists a unique mapping
Φ : k[π] × gr(G) −→ gr(G),
which is k-linear in the first variable, such that Φ(π i , ξ) = π∗i (ξ). If we set
α · ξ = Φ(α, ξ), we have π i · (π j · ξ) = π i+j · ξ. Proposition (3.8.3) with a = q
now yields the
(3.8.5) Proposition. Let ξ
∈
gri (G) and η
(i) π · (ξ + η) = π · ξ + π · η
(ii) π · (ξ + η) = π · ξ + π · η +
∈
grj (G). Then
if i = j > 1,
q
2
[ξ, η]
(iii) π · [ξ, η] = [π · ξ, η]
if i = j = 1,
if i =/ 1,
π · [ξ, η] = [ξ, π · η]
if j =/ 1,
(iv) [π · ξ, η] = π · [ξ, η] +
q
2
[[ξ, η], ξ]
if i = j = 1,
(v) [ξ, π · η] = π · [ξ, η] +
[[ξ, η], η]
if i = j = 1.
q
2
Remark:
If G = F is a free pro-p-group and if q is not a power of 2, then
q
≡ 0 mod q and gr(F ) is a free Lie algebra over k[π] (see [125]).
2
For a free pro-p-group F , we obtain the following formula for the intersection
of the p-central series and the central series of F :
(3.8.6) Proposition. Let F be a free pro-p-group and let {F i } be the p-central
series and {Fi } the central series of F . Then for all i ≥ j ≥ 1, there is the
equality
i−j
i−j−1
F i ∩ Fj = (Fj )p · (Fj+1 )p
· . . . · Fi .
Proof: We proceed by induction on j. For j = 1 we have to prove that
i−1
F i = F p · . . . · Fi .
First, we show that
i−1
F i = F p · . . . · Fi · F i+1 .
This is obviously true for i = 1. So let i > 1 and assume that the assertion is
true for i − 1. Then, using (3.8.4),
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223
§8. Filtrations
F i = (F i−1 )p [F i−1 , F ]
i−2
i−2
= (F p · . . . · Fi−1 · F i )p [F p · . . . · Fi−1 · F i , F ]
i−1
p
· Fi · F i+1 .
= F p · . . . · Fi−1
i−1
Assume we have proved that F i = F p · . . . · Fi · F k for k ≥ i + 1. Then
i−1
F i = F p · . . . · Fi · (F k−1 )p [F k−1 , F ]
i−1
k−2
k−2
= F p · . . . · Fi · (F p · . . . · Fk−1 · F k )p [F p · . . . · Fk−1 · F k , F ]
i−1
= F p · . . . · Fi · F k+1 .
T
Since k F k = 1, we have proved the case j = 1.
Now let i ≥ j > 1. Then
i−j+1
F i ∩ Fj = F i ∩ Fj−1 ∩ Fj = (Fj−1 )p
· . . . · Fi ∩ Fj
by the induction assumption. Let
i−j
pi−j+1
x = zj−1
y ∈ Fj with zj−1 ∈ Fj−1 , y ∈ (Fj )p · . . . · Fi .
Claim: Fj−1 /Fj is ZZp -torsion-free.
Proof: Using a limit argument we may assume that F is finitely generated.
Since the free pro-p-group F is the completion of a free discrete group, Fj /Fj+1
is the completion of a free (finitely generated) abelian group by [260], Satz 4.
Hence it is a free ZZp -module.
i−j
Using this claim, we see zj−1 ∈ Fj , so that x ∈ (Fj )p · . . . · Fi . This proves
the proposition.
2
We now define a refinement {G(i,j) } of the descending p-central series of G.
(3.8.7) Definition. Let {Gi } and {Gj } be the p-central series and the central
series of the pro-p-group G, respectively. We set, for i, j ≥ 1,
G(i,j) := (Gi ∩ Gj ) Gi+1 .
Obviously we have
G(i,1) = Gi and G(i,j) = Gi+1 for j > i ≥ 1.
We introduce the following notational convention:
The letter ν always stands for a pair (i, j), i ≥ j ≥ 1, and we order these
pairs lexicographically. We say that
ν + 1 = (i, j + 1) if i > j ,
ν + 1 = (i + 1, 1) if ν = (i, i) .
The descending chain {G(ν) } of normal characteristic subgroups is a refinement
of the descending p-central series. In particular, G(ν) /G(ν+1) is an IFp -vector
space for all ν.
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Chapter III. Duality Properties of Profinite Groups
(3.8.8) Proposition. For every ν = (i, j), the IFp -vector space homomorphism
ψν :
(G/G2 )⊗j
−→ G(ν) /G(ν+1)
i−j
x̄1 ⊗ · · · ⊗ x̄j 7−→ ([x1 , [x2 , [· · · , xj ]] · · ·])p mod G(ν+1)
is well-defined and surjective.
Proof: Let S = {xα }α∈I be a minimal system of generators of G. Then their
image S̄ := S mod Gp [G, G] in G/G2 = G/Gp [G, G] is an IFp -basis of G/G2 .
(In fact, if R is any proper subset of S, then R generates a closed subgroup
=/ G, which sits in a maximal, i.e. normal, subgroup M of index p, so that
R̄ ⊆ M/G2 =/ G/G2 cannot generate G/G2 .) Thus the tensors x̄α1 ⊗ · · · ⊗ x̄αj ,
α1 , . . . , αj ∈ I define a basis of (G/G2 )⊗j . Recalling the definition of the (ν)filtration, it follows from (3.8.6) (for free groups and then for all pro-p-groups)
that the elements
i−j
([xα1 , [xα2 , [· · · , xαj ]] · · ·])p ,
α1 , . . . , αj ∈ I
generate G(ν) modulo G(ν+1) . Finally, an inductive application of (3.8.3) shows
that ψν is well-defined.
2
Remark: The same proof shows that we can define ψν as a homomorphism
(G/G2 )⊗j −→ G(ν) /Gi+1 ⊆ Gi /Gi+1
if either j > 1 or if p is odd (hence p |
p
2
).
§9. Generators and Relations
For this entire section G is a pro-p-group. We set
H n (G) = H n (G, ZZ/pZZ)
and regard these groups as IFp -vector spaces. By the dimension, dim H n (G),
we mean the cardinality of a basis. H 1 (G) is the Pontryagin dual of the group
G/G∗ , where G∗ = Gp [G, G], i.e. the closure of the subgroup generated by
commutators and p-th powers.∗) Indeed, the group G/G∗ is the largest profinite
abelian quotient of exponent p of G, so that
H 1 (G) = Hom(G, ZZ/pZZ) = Hom(G/G∗ , Q/ZZ).
A generator system of G is a convergent family S = (gi )i∈I , gi ∈ G, which
generates G as a topological group. By convergent we mean convergent to 1,
∗) G∗ is the Frattini subgroup of G, i.e. the intersection of all maximal closed subgroups.
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225
§9. Generators and Relations
i.e. every open subgroup of G contains gi for almost all i. The rank of a pro-pgroup G is the infimum over the cardinalities card(S ) = card(I) of generator
systems S of G, and is denoted by d(G). A generator system is minimal if
no proper subfamily is a generator system.
(3.9.1) Proposition. A convergent family S is a generator system of G if and
only if the family S = (g i )i∈I of residue classes g i = gi mod G∗ generates
G/G∗ . S is minimal if and only if S is. Every minimal system of generators
has cardinality d(G) and we have the equality
d(G) = dim H 1 (G).
In particular, G is finitely generated if and only if H 1 (G) is finite.
The first assertion of the proposition is often called Frattini argument.
Proof: Let H be the closed subgroup of G generated by S . Then we have
the equivalences
the inclusion H → G is surjective
⇔ H 1 (G) → H 1 (H) is injective
(by (1.6.14)(ii))
⇔ H/H ∗ → G/G∗ is surjective
(by Pontryagin duality).
2
This shows the statement.
Let N be a normal subgroup of G. A generator system of N as a normal
subgroup is a convergent family S such that N is the smallest closed normal
subgroup of G containing all elements of S . In the case N = G, (3.9.1) shows
that S generates G as a normal subgroup (of itself) if and only if S generates
G as a pro-p-group.
(3.9.2) Proposition. Let N be a normal closed subgroup of G, S a convergent
family of elements of N and H the closed subgroup generated by S . Then
the restriction map
res : H 1 (N )G −→ H 1 (H)
is injective if and only if S is a generator system of N as a normal subgroup.
Proof: Let N 0 be the normal closed subgroup of G generated by S = (ni )i∈I
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Chapter III. Duality Properties of Profinite Groups
(as a normal subgroup), and consider the commutative diagram
ÙØÚ )G
H 1 (N
res1
res2
H 1 (H)
res3
1
0 G
H (N ) .
As a pro-p-group, N 0 is generated by the family S˜ = (σni σ −1 )i∈I,σ∈G . For
χ ∈ H 1 (N 0 )G , we have
res3 (χ) = 0 ⇒ χ(σni σ −1 ) = χσ (ni ) = χ(ni ) = 0 for all σ and all i
⇒ χ(S˜) = 0 ⇒ χ = 0,
i.e. res3 is injective. Therefore res1 is injective if and only if res2 is injective. But this is equivalent with the injectivity of the homomorphism
1
1
0
res N
N 0 : H (N ) → H (N ) since
G
N
ker(res2 ) = ker(res N
N 0 ) = 0 ⇐⇒ ker(res N 0 ) = 0
0
by (1.6.12). Finally, the injectivity of res N
N 0 is equivalent with N = N by
(1.6.14).
2
(3.9.3) Corollary. Let N be a normal closed subgroup of G. Then a convergent
family S of elements of N generates N as a normal subgroup if and only if
the family S of residue classes modulo N p [G, N ] generates N/N p [G, N ]. In
particular, S is minimal if and only if S is, and in this case
card(S ) = dim H 1 (N )G .
Proof: Let H be the closed subgroup in N generated by S . Taking into
account that N/N p [G, N ] = (H 1 (N )G )∨ , we the have following equivalences
S generates N as a normal subgroup
⇔ H 1 (N )G → H 1 (H) is injective
∗
p
⇔ H/H → N/N [G, N ] is surjective
(by (3.9.2))
(by Pontryagin duality).
This shows the statement.
2
Let S = (gi )i∈I be a system of generators of the pro-p-group G and let F
be the free pro-p-group with basis (xi )i∈I , see (3.5.14). Sending xi to gi , we
obtain an exact sequence
1 −→ R −→ F −→ G −→ 1.
A relation system, also called a system of defining relations with respect
to S , is a generator system R of R as a normal subgroup of F .
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§9. Generators and Relations
(3.9.4) Proposition. Suppose that the pro-p-group G is finitely generated and
let S be a finite system of generators of G. Then a finite system R of defining
relations with respect to S exists if and only if H 2 (G) is finite. If R is finite
and minimal, we have the equality
card(S ) − card(R) = dim H 1 (G) − dim H 2 (G).
Proof: Applying (3.9.3) to the closed subgroup R of F , we obtain
card(R) = dim H 1 (R)F = dim H 1 (R)G .
Consider the five term exact sequence
0 −→ H 1 (G) −→ H 1 (F ) −→ H 1 (R)G −→ H 2 (G) −→ H 2 (F ).
Since F is free, H 2 (F ) = 0, and we obtain the result by counting IFp -dimensions.
2
In particular, given G, the cardinality of a minimal system of defining
relations only depends on the cardinality of the system S of generators. If S
is a minimal system of generators, we call the cardinality of a minimal relation
system the relation rank of G. We denote it by r(G).
(3.9.5) Proposition. Let G be a pro-p-group. Then the relation rank of G
satisfies the formula
r(G) = dim H 2 (G).
In particular, G is a free pro-p-group if and only if cd G ≤ 1.
Proof: Let S be a minimal system of generators of G. Then the exact sequence
∼ H 1 (F ) −→ H 1 (R)G −→ H 2 (G) −→ 0
0 −→ H 1 (G) −→
∼ H 2 (G), and so r(G) = dim H 1 (R)G =
yields the isomorphism H 1 (R)G →
dim H 2 (G).
2
From (3.3.16) we obtain the
(3.9.6) Corollary. Let G be a finitely generated pro-p-group of rank d(G) and
let U be a cofinal set of open subgroups of G. Then G is a free pro-p-group if
and only if
d(U ) − 1 = (G : U )(d(G) − 1) for all U
∈
U.
Remark: If G is a finitely generated pro-p-group, i.e. d(G) < ∞, then the
relation rank r(G) need not be finite. We have the following example occurring
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Chapter III. Duality Properties of Profinite Groups
in number theory. Let G = G(k(p)|k) be the Galois group of the maximal pextension of a p-adic local field k (cf. VII §5) and let Gt , t ∈ IR, t ≥ 0, be
the higher ramification groups of G (see [130] for the definition). Since G is
finitely generated (cf. (7.5.11)), the quotients G/Gt are also finitely generated.
But N. L. GORDEEV proved in [58] that r(G/Gt ) is infinite if t > 1. This
extended the result of E. MAUS who showed this for t ∈/ ZZ[ p1 ], t > 1, see [131].
The rank and the relation rank of a pro-p-group G played an important role
in the solution of the famous and long standing class field tower problem in
number theory. The question is whether the maximal unramified p-extension
K∅ (p)|K of a number field K need always be of finite degree. It was a
great surprise when, in 1964, the Russian mathematicians E. S. GOLOD and
I. R. ŠAFAREVIČ proved that this is not always the case. The Galois group G
of K∅ (p)|K is a pro-p-group, and the crucial point in the proof (presented in
X §10) was the discovery that for a finite p-group G the relation rank has to be
very large in comparison with the rank. More precisely, we have the
(3.9.7) Theorem. If G is a finite p-group, then
r(G) >
1
4
d(G)2 .
For the proof we need the following
(3.9.8) Lemma. Let G be a finite p-group and Λ = IFp [G]. For every finite
G-module A such that pA = 0, there is a resolution
∂
∂
∂
0 −→ A −→ Λb0 −→ Λb1 −→ Λb2 −→ · · · ,
where bn = dim H n (G, A), and ∂((Λbn )G ) = 0.
Proof: We have canonically ΛG ∼
= IFp . Let a1 , . . . , ab0 be a basis of the
G
IFp -vector space A . Then the isomorphism AG →(Λb0 )G = IFbp0 , ai 7→ ei ,
extends to an injective G-homomorphism
j : A −→ Λb0 .
Indeed, the map HomG (A, Λb0 ) → HomG (AG , Λb0 ) is surjective, since in the
exact sequence
0 −→ Hom(A/AG , Λb0 ) −→ Hom(A, Λb0 ) −→ Hom(AG , Λb0 ) −→ 0
of induced G-modules, H 1 of the first term is zero. The extension j is automatically injective, since from ker(j|AG ) = ker(j)G = 0, it follows that ker(j) = 0
by (1.6.12). Since Λb0 is an induced G-module, from the exact sequence
0 −→ A −→ Λb0 −→ B −→ 0 ,
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§9. Generators and Relations
we obtain isomorphisms
H i (G, B) ∼
= H i+1 (G, A)
for i ≥ 1. The same holds for i = 0, since in the exact sequence
AG −→(Λb0 )G −→ B G −→ H 1 (G, A) −→ 0
the first arrow is bijective, i.e. (Λb0 )G is mapped to zero.
Proceeding in the same way with the G-module B in place of A and noting
that dim H 0 (G, B) = dim H 1 (G, A) = b1 , we obtain an exact sequence
0 −→ B −→ Λb1 −→ C −→ 0
such that B G →(Λb1 )G is an isomorphism, i.e. (Λb1 )G is mapped to zero. If we
define ∂ : Λb0 → Λb1 to be the composite map Λb0 → B → Λb1 , then ∂((Λb0 )G ) =
0. Continuing this process, the lemma follows by induction.
2
Proof of theorem (3.9.7): For every finite G-module A such that pA = 0, we
define the “ascending central series”
0 = A0
⊆
A1
⊆
A2
⊆
· · · ⊆ Am = A
by A0 = 0, A1 = AG and An+1 /An = (A/An )G for n ≥ 1. By (1.6.12),
An =/ An+1 , unless An = A. If h : A → B is an injective G-homomorphism,
then we see inductively
An = h−1 (Bn ).
We set cn (A) = dim(An+1 /An ) and form the Poincaré polynomial
PA (t) =
X
cn (A)tn .
n≥0
If 0 < t < 1 is a real variable, then
PA (t)
where sn (A) =
X
1
=
sn (A)tn ,
1 − t n≥0
n
X
ci (A) = dim(An+1 ).
i=0
We apply the lemma to the G-module A = IFp . Recalling that IFp = ΛG = Λ1 ,
we obtain an exact sequence
∂
∂
(∗)
0 −→ E −→ D −→ R,
where E = Λ/Λ1 , D = Λd , R = Λr ,
d = dim H 1 (G) = d(G), r = dim H 2 (G) = r(G),
such that ∂(D1 ) = R0 = 0. From this it follows inductively that ∂(Dn ) ⊆ Rn−1 ,
and we obtain the sequences
(∗∗)
0 −→ En −→ Dn −→ Rn−1 ,
n ≥ 1.
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Chapter III. Duality Properties of Profinite Groups
These sequences are again exact, since (∗) is exact and En = ∂ −1 (Dn ). Now
consider the Poincaré polynomials associated to E, D, R. Setting P (t) = PΛ (t),
we obtain
P (t) − 1
PE (t) =
, PD (t) = dP (t), PR (t) = rP (t) .
t
From (∗∗) we get the inequalities
sn (D) ≤ sn (E) + sn−1 (R),
(where s−1 (R) = 0), so that
1
t
1
≤ PE (t)
+ PR (t)
.
PD (t)
1−t
1−t
1−t
It follows that
P (t) − 1
dP (t) ≤
+ rtP (t) for 0 < t < 1,
t
and so
1 ≤ P (t)(rt2 − dt + 1) if 0 < t < 1.
Since P (t) has positive coefficients, we obtain
0 < rt2 − dt + 1 if 0 < t < 1.
Substituting t 7→ 2rd , we get
r > 41 d2 ,
as asserted. This substitution is valid, since the exact sequence
p
0 → H 1 (G) → H 2 (G, ZZ) → H 2 (G, ZZ) → H 2 (G)
shows that d ≤ r < 2r, i.e. 0 < 2rd < 1.
2
Remarks: 1. GOLOD and ŠAFAREVIČ had originally proved only that r(G) >
1
(d(G) − 1)2 (see [57]). The sharper inequality r(G) > 14 d(G)2 was obtained
4
independently by W. GASCHÜTZ and E. B. VINBERG [241]. The proof given
here is dual to the proof presented by P. ROQUETTE in [183], who uses homology
instead of cohomology. There is another more general proof (based, however,
on the same idea as in [183]) with more far reaching results given by H. KOCH
in [110] and [72], and yet another proof was given by J.-P. SERRE in [210].
2. We want to present the result of KOCH without proof. For that we need
the notion of the Zassenhaus filtration of a pro-p-group. Let G be a finitely
generated pro-p-group and for n ≥ 1 let the ideal I n (G) of IFp [[G]] be the n-th
power of the augmentation ideal I(G). The filtration
G(n) = {g | g − 1 ∈ I n (G)},
n ≥ 1,
is called the Zassenhaus filtration of G. The normal subgroups G(n) form a
full system of neighbourhoods of the identity of G. For the basic properties of
{G(n) }n≥1 , see [110], §7.4. We mention only that
(G(n) )p ⊆ G(np) and [G(n) , G] ⊆ G(n+1) .
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231
§9. Generators and Relations
Now let
1 −→ R −→ F −→ G −→ 1
be a minimal presentation of the finite p-group G by a free pro-p-group F and
assume that R ⊂ F(m) for some m, where {F(n) }n≥1 is the Zassenhaus filtration
of F . Then
d(G)m
r(G) >
(m − 1)m−1 .
m
m
This formula shows that for a finite p-group we get a better bound for r(G) if
we have information on the complexity of the relations.
We are now aiming at the classification of the Demuškin groups, which are
defined as follows.
(3.9.9) Definition. A pro-p-group G is called a Demuškin group if its cohomology H i (G) has the following properties:
(i) dimIFp H 1 (G) < ∞,
(ii) dimIFp H 2 (G) = 1,
(iii) the cup-product H 1 (G) × H 1 (G) → H 2 (G) is non-degenerate.
By (3.7.6), an infinite Demuškin group has necessarily cohomological dimension 2 and is precisely a Poincaré group of dimension 2. The finite
Demuškin groups are classified by the following
(3.9.10) Proposition. The group G = ZZ/2ZZ is the only finite Demuškin
group.
Proof: The relation rank r = r(G) of a Demuškin group G is equal to 1
by definition. Therefore if G has generator rank d = d(G) ≥ 2, then its
abelianization Gab has a ZZp -rank of at least d − 1 ≥ 1. We conclude that
d = 1 for a finite Demuškin group, i.e. G is cyclic. Hence H 1 (G) is a one
dimensional IFp -vector space and for every x ∈ H 1 (G) we have 2 · (x ∪ x) =
x ∪ x + x ∪ x = 0 by the anti-symmetry of the cup-product. This gives a
contradiction for odd p ∗) and therefore G ∼
= ZZ/2k ZZ for some k ≥ 1. A
∗) In other words: the cup-product is always anti-symmetric, which implies in odd characteristics that it is alternating. A vector space with a non-degenerate alternating bilinear form
is necessarily of even dimension. In characteristic 2 the cup-product may induce a symmetric,
non alternating form.
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Chapter III. Duality Properties of Profinite Groups
straightforward computation, which we leave to the reader, shows that the cup∪
product H 1 (G) × H 1 (G) −→ H 2 (G) ∼
= IF2 is trivial for k > 1, and nontrivial,
hence non-degenerate, for k = 1.
2
The appearance of G = ZZ/2ZZ as the only finite Demuškin group is interesting from the arithmetic point of view, since G is the absolute Galois group of
the field IR.
Let us consider more generally finitely generated pro-p-groups G with only
one defining relation, i.e.
n = dimIFp H 1 (G) < ∞
and
dimIFp H 2 (G) = 1.
Such a group is called a one-relator pro-p-group. We have an exact sequence
involving G,
1 −→ R −→ F −→ G −→ 1,
where F is the free pro-p-group of rank n and R is generated by one element
ρ as normal subgroup of F . We also write R = (ρ). Passing to the abelianized
groups, we obtain Gab as a quotient of F ab ∼
= ZZnp by a subgroup which is
either isomorphic to ZZp or zero. Noting that H 1 (G) = H 1 (Gab ) = (ZZ/pZZ)n ,
we conclude that
Gab ∼
or
Gab ∼
(f ≥ 1).
= ZZn
= ZZ/pf ZZ × ZZn−1
p
p
f
We set q = p in the second case and q = 0 in the first case. The numbers n
and q are invariants of the group G. Since F/F q [F, F ] → G/Gq [G, G] is an
isomorphism by definition of q, we have
R ⊆ F q [F, F ] = F 2 ,
where F i is the q-central series of F .
The central result which we want to prove is the following
(3.9.11) Theorem (DEMUŠKIN). Let G be a one-relator pro-p-group. Suppose
that the invariant q of G is =/ 2. Then G is a Demuškin group if and only if it
is isomorphic to the pro-p-group defined by n generators x1 , . . . , xn subject to
the one relation
xq1 (x1 , x2 )(x3 , x4 ) · · · (xn−1 , xn ) = 1,
where (x, y) is the commutator x−1 y −1 xy. In particular, G is then determined
by the two invariants n and q.
The proof of the theorem is not easy. We follow in essence the presentation
in [117] by J. LABUTE, where the case q = 2 is also treated.
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233
§9. Generators and Relations
Before considering one-relator pro-p-groups and proving the result stated
above, we consider the case where G is an arbitrary finitely generated prop-group of rank n. If Gab has a nontrivial torsion subgroup, then we set
k = ZZp /q ZZp , where q is the smallest elementary divisor of Gab , i.e. q = pf
is the maximal p-power such that Gab /q is a free ZZp /q ZZp -module. When
Gab ∼
= ZZnp we set k = ZZp and change our usual notation and denote by
i
H (G, ZZp ) the continuous cochain cohomology (see II §7). Since G is finitely
generated, (2.7.6) implies H i (G, ZZp ) = lim H i (G, ZZ/pν ZZ) for i = 0, 1, 2.
←− ν
Let
1 −→ R −→ F −→ G −→ 1
be a minimal presentation of the pro-p-group G. The inflation
inf : H 1 (G, k) −→ H 1 (F, k)
is an isomorphism with which we identify the two groups.
Since F is free, we have H 2 (F, k) = 0, and the five term exact sequence
(1.6.7) shows that also the transgression tg : H 1 (R, k)G → H 2 (G, k) is an
isomorphism. More generally, we obtain isomorphisms
ab
∼ H 2 (G, Z
tg : Homcts (RG
, ZZ/pn ZZ) = H 1 (R, ZZ/pn ZZ)G −→
Z/pn ZZ)
for all n ≤ f . Therefore every element ρ ∈ R gives rise to trace maps
tr = trρ : H 2 (G, ZZ/pn ZZ) −→ ZZ/pn ZZ
which are defined by ϕ 7→ (tg−1 ϕ)(ρ). If q = 0, then these maps are defined
for all n ≥ 1 and also with ZZp -coefficients.
(3.9.12) Proposition. Let
1 −→ R −→ F −→ G −→ 1
be a minimal presentation of the finitely generated pro-p-group G.
(i) Every basis x1 , . . . , xn of F defines a k-basis χ1 , . . . , χn of the k-module
H 1 (F, k) = H 1 (G, k) such that χi (xj ) = δij .
(ii) Given an element ρ ∈ R, the bilinear form induced by the cup-product
∪
trρ
H 1 (G, k) × H 1 (G, k) −→ H 2 (G, k) −→ k
is non-degenerate if and only if it is non-degenerate with k replaced by
ZZ/pZZ, and in this case trρ is surjective.
(iii) If G is a one-relator group and if ρ generates R as a normal subgroup,
then trρ is injective.
Proof: (i) By definition of q, a minimal generator system x1 , . . . , xn of F
defines an isomorphism F/F q [F, F ] ∼
= k n , where [F, F ] is the closure of the
commutator subgroup of F . (i) follows from this.
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Chapter III. Duality Properties of Profinite Groups
(ii) Consider the commutative diagram
âÛáÜÝÞßà k)
H 1 (G,
H 1 (G, k)
×
H 1 (G, ZZ/pZZ) × H 1 (G, ZZ/pZZ)
∪
H 2 (G, k)
trρ
∪
H 2 (G, ZZ/pZZ)
trρ
k
ZZ/pZZ ,
where the vertical arrows are induced by the reduction map k → ZZ/pZZ. If B
is a matrix for the upper bilinear form, then B = B mod p is a matrix for the
lower one. The upper one is non-degenerate if and only if B is invertible over
k, i.e. det (B) ∈ k × , and this is the case if and only if det(B) =/ 0, i.e. if the
lower one is non-degenerate. Clearly trρ must be surjective in this case.
ab
. Therefore
(iii) If ρ generates R as a normal subgroup in G, then it generates RG
−1
if tg (ϕ)(ρ) = 0, then ϕ = 0, so that trρ is injective.
2
Let R = {ρi | i ∈ I} be a minimal system of defining relations of the finitely
generated pro-p-group G = F/R.
(3.9.13) Proposition. Let
1 −→ R −→ F −→ G −→ 1
be a minimal presentation of the finitely generated pro-p-group G. Let
x1 , . . . , xn be a basis of F and let χ1 , . . . , χn be the corresponding k-basis
of H 1 (F, k) = Hom(F, k) = Hom(G, k), i.e. χi (xj ) = δij .
(i) Every element ρ ∈ F 2 has a representation
ρ=
n
Y
qaj
xj
j=1
·
Y
(xk , xl )akl · ρ0 ,
ρ0
∈
F 3 , aj , akl
∈
k.
1≤k<l≤n
The akl are uniquely determined by ρ, and if q =/ 0 so are the aj .
(ii) If R = {ρi | i ∈ I} is a minimal system of defining relations of G and
ρi =
n
Y
qaij
xj
j=1
·
Y
i
(xk , xl )akl · ρ0i ,
ρ0i
∈
F 3 , aij , aikl
∈
k, i ∈ I ,
1≤k<l≤n
then the bilinear form
trρ
∪
i
H 1 (G, k) × H 1 (G, k) −→ H 2 (G, k) −→
k
is given by the matrix Bi = (bikl ) with respect to the basis χ1 , . . . , χn ,
where

−aikl
if
k < l,



ailk


bikl = trρi (χk ∪ χl ) = 
−
q
2
aik
if
k > l,
if
k = l.
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235
§9. Generators and Relations
Proof: The existence of a representation (i) for a finite word in the letters
x1 , . . . , xn is obtained by a simple collecting process (see M. HALL [73], chap.
11.1). For an arbitrary ρ ∈ F 2 it follows by a limit process, noting that the
discrete free group generated by the x1 , . . . , xn is a dense subgroup of F . If
q =/ 0, then k = ZZ/q ZZ and
2
F/F q [F, F ] = (ZZ/q 2 ZZ)n .
The image of ρ in this last group is (qa1 , . . . , qan ), so that the aj are uniquely
determined in k by ρ. The uniqueness of the akl follows from the proof of
(ii) The cohomology class χk ∪ χl ∈ H 2 (G, k) is represented by the inhomogeneous 2-cocycle
c0 (σ, τ ) = χk (σ)χl (τ ).
Let c be the inflation of c0 to F . Since H 2 (F, k) = 0, there exists an inhomogeneous cochain
u = ukl : F −→ k
such that c = ∂u and, moreover, by subtracting the homomorphism h : F → k,
h(xj ) = u(xj ), we can suppose u(xj ) = 0, j = 1, . . . , n. Then
u(xy) = u(x) + u(y) − χk (x)χl (y),
(1)
x, y
∈
F.
In particular, u(xy) = u(x) + u(y) whenever x or y is contained in F 2 =
F q [F, F ]. Therefore u is a homomorphism on F 2 which vanishes on F 3 , since
F 3 is topologically generated by the products xq (x, y), x ∈ F 2 , y ∈ F , on
which u behaves multiplicatively.
The restriction v of u to R is an element of H 1 (R, k)G since (y, x) ∈ F 3 for
y ∈ R and x ∈ F , and so
v(x−1 yx) = u(y(y, x)) = u(y) + u((y, x)) = v(y).
By the definition (1.6.6) of the transgression, we now obtain
tg(v) = [∂u] = χk ∪ χl ,
and by definition of the matrix Bi = (bikl ) and of trρi , we get
bikl = trρi (χk ∪ χl ) = v(ρi ) = ukl (ρi ).
Since Bi is anti-symmetric, it therefore remains to show that
(
−aikl
if
k < l,
(2)
u(ρi ) =
− q2 aik
if
k = l.
We compute the values u(xm
j ) and u((xν , xµ )) by means of (1).
−1
m+1
m
If k =/ l, we have u(xj ) = u(xm
j ) and u(xj ) = 0, which implies u(xj ) = 0
for any m ∈ ZZ. If k = l, we have
m
m
u(xm+1
) = u(xm
j
j ) − χk (xj )χl (xj ) = u(xj ) − mδkj .
This implies
u(xm
j ) = −
m
2
δkj
for
m = 1, 2, 3, . . .
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Chapter III. Duality Properties of Profinite Groups
Furthermore, noting that u(x−1 ) + u(x) + χk (x)χl (x) = 0, we have for ν < µ
−1
u((xν , xµ )) = u(x−1
ν ) + u(xµ xν xµ ) + χk (xν )χl (xν )
= −δkν + u(x−1
µ xν xµ ) + δkν δlν
= u(x−1
µ ) + u(xν xµ ) + χk (xµ )χl (xν ) + χk (xµ )χl (xµ )
= −δkν δlµ + u(xν ) + u(xµ ) − χk (xν )χl (xµ ) + δkµ δlν
= δkµ δlν − δkν δlµ ,
i.e. for ν < µ and k < l
(
u((xν , xµ )) =
−1
if k = ν, l = µ,
0
otherwise.
2
Now applying the homomorphism u : F /F 3 → k to
ρi ≡
n
Y
qaij
xj
j=1
Y
i
(xν , xµ )aνµ mod F 3
1≤ν<µ≤n
yields the desired result (2) and also the uniqueness of the aikl .
2
If q =/ 0 and k = ZZ/q ZZ, we define the Bockstein homomorphism
B : H 1 (G, k) −→ H 2 (G, k)
as the connecting homomorphism in the long exact cohomology sequence
associated to
q
0 −→ ZZ/q ZZ −→ ZZ/q 2 ZZ −→ ZZ/q ZZ −→ 0 .
We get a second interpretation of the exponents aij in the expression for the
relations of the group G in (3.9.13) (ii):
(3.9.14) Proposition. With the notation of (3.9.13) and assuming that q =/ 0,
we have
trρi (B(χj )) = −aij .
We leave the proof to the reader, since it follows the same lines as the proof
of (3.9.13).
Before we determine explicitly the defining relation of a Demuškin group, we
first sharpen the result (3.7.3) concerning the characterization of a Demuškin
group by properties of its subgroups.
(3.9.15) Theorem. Let G be a finitely generated one-relator pro-p-group of
rank d(G) > 1. Then the following assertions are equivalent:
(i) G is a Demuškin group.
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237
§9. Generators and Relations
(ii) cdp G = 2 and the dualizing module of G is isomorphic to Qp /ZZp as an
abelian group.
(iii) cdp G = 2 and dimIFp H 2 (N, ZZ/pZZ) = 1 for every open subgroup N
of G.
(iv) cdp G = 2 and d(N ) − 2 = (G : N )(d(G) − 2) for every open subgroup
N of G.
(v) cdp G = 2 and dimIFp H 2 (N, ZZ/pZZ) = 1 for every open subgroup N of G
with (G : N ) = p.
(vi) d(N ) − 2 = (G : N )(d(G) − 2) for every open subgroup N of G with
(G : N ) = p.
Remark: Observe that in assertion (vi) we do not assume that cdp G = 2. The
equivalences between (i), (v) and (vi) were first proved by I. V. ANDOŽSKII
and later independently by J. DUMMIT and J. LABUTE (without the assertion
cdp G = 2 in (v)) - see [3] and [44].
Proof: The equivalences between (i), (ii) and (iii) follow from (3.7.2) and
(3.7.3). Using (3.3.13), we get the implications (iii) ⇒ (iv) and (v) ⇒ (vi). The
assertions (v) and (vi) are trivial consequences of (iii) and (iv), respectively.
So let us assume that (vi) holds, and try to prove (i).
Let
,
Gab ∼
= ZZp /q ZZp × ZZd(G)−1
p
where q is equal 0 if Gab is torsion-free. Furthermore, let
1 −→ R −→ F −→ G −→ 1
be a minimal presentation of G by a free pro-p-group F of rank d = d(G).
Suppose that the cup-product pairing
H 1 (G, ZZ/pZZ) × H 1 (G, ZZ/pZZ) → H 2 (G, ZZ/pZZ) ∼
= ZZ/pZZ
is degenerate, and let χ1 be an element of the radical of this pairing. Extend
this element to a basis χ1 , . . . , χd of H 1 (G, ZZ/pZZ) ∼
= H 1 (F, ZZ/pZZ) and let
x̄1 , . . . , x̄d be the corresponding dual basis of F/F 2 (F i denotes the p-central
series of F ). We define
E = (xp1 , x2 , . . . , xd )F / F
to be the subgroup of F generated by all elements between the brackets and
their conjugates, where xi is an arbitrary lifting of x̄i to F . Then R ⊆ E and
if we put N = E/R ⊆ G and G = G/N = F/E = hx̄1 i ∼
= ZZ/pZZ, we get the
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Chapter III. Duality Properties of Profinite Groups
commutative exact ãäåæçèéêëìíîïð diagram
G
G
1
R
F
G
1
1
R
E
N
1.
We consider the G-module R̄ := RE 2 /E 2 which fits into the exact sequence
0 −→ R̄ −→ E/E 2 −→ N/N 2 −→ 0 .
Since R/Rp [R, F ] ∼
= ZZ/pZZ surjects onto R̄G , the G-module R̄ is generated
by one element. Observe that
(∗)
F2
⊆
E and F 3
⊆
[[E, F ], F ] · E 2 .
It follows from (3.9.13)(ii) and the fact that χ1 ∪ χi = 0 for all i ≥ 1, that
R̄ ⊆
and so
R̄ ⊆



hxp1 i[[E, hx1 i], hx1 i]E 2 /E 2 if q = p =/ 2 ,


[[E, hx1 i], hx1 i]E 2 /E 2



hxp1 iE 2 /E 2 · (E/E 2 )(x̄1 −1)


(E/E 2 )(x̄1 −1)
Thus in both cases
(since (E/E 2 )(x̄1
−1)p
otherwise ,
2
2
if q = p =/ 2 ,
otherwise .
p−2
R̄(x̄1 −1)
=0
= 0) and therefore
dimIFp R̄ ≤ p − 2 .
From the exact sequence (∗), we obtain using (3.9.6)
dimIFp N/N 2 = dimIFp E/E 2 − dimIFp R̄
≥ p(d − 1) + 1 − (p − 2)
> p(d − 2) + 2 .
This contradicts the assumption (vi) and therefore the cup-product pairing has
to be non-degenerate, i.e. G is a Demuškin group.
2
Now we are going to determine a defining relation of a Demuškin group
explicitly. Let G = F/R be a one-relator pro-p-group of rank n where R = (ρ).
First we want to put the relation ρ in the shape
ρ ≡ xq1 (x1 , x2 )(x3 , x4 ) · · · (xn−1 , xn ) mod F 3
and for this, we have to find an appropriate basis of H 1 (F, k).
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§9. Generators and Relations
If q =/ 0, then there is a uniquely determined class σ mod [F, F ] ∈ F ab =
F/[F, F ] such that
ρ ≡ σ q mod [F, F ].
We deduce this from the fact that F ab ∼
= ZZnp and that the image ρ̄ of ρ
ab
in F topologically generates a closed subgroup (ρ̄) such that F ab /(ρ̄) ∼
=
n−1
ZZ/q ZZ × ZZp .
We need a result concerning symplectic bilinear forms in which we write
χ ∪ χ0 for tr(χ ∪ χ0 ).
(3.9.16) Proposition. Assume that q =/ 2 and that the bilinear pairing induced
tr
by the cup-product H 1 (G, k) × H 1 (G, k) → H 2 (G, k) → k is non-degenerate.
Then n is even and there exists a k-basis χ1 , . . . , χn of H 1 (G, k) such that
χ1 ∪ χ2 = χ3 ∪ χ4 = . . . = χn−1 ∪ χn = 1
and χi ∪ χj = 0 for all other i < j, and χi (σ) = δ1i when q =/ 0.
∪
Proof: By (3.9.9), the cup-product H 1 (G) × H 1 (G) → H 2 (G) ∼
= IFp is a
/
non-degenerate bilinear form over the field IFp . If p = 2, it is alternating,
hence n = dimIFp H 1 (G) is even. When q = 2f , f > 1, n is also even, since
the cup-product on H 1 (G, ZZ/2f −1 ZZ) is still alternating and non-degenerate,
which implies that H 1 (G, ZZ/2ZZ) decomposes into a direct sum of hyperbolic
planes. For this, we refer to [106], chap.I, §4.
We start with any k-basis χ1 , . . . , χn of H 1 (G, k) such that χi (σ) = δ1i when
q =/ 0. To find such a basis when q =/ 0, one only has to extend the image
of σ in F/F q [F, F ] to a basis of F/F q [F, F ] and then take the dual basis.
The nondegeneracy of the cup-product means that the matrix B = (χk ∪ χl ) is
invertible over k. Therefore one of the elements χ1 ∪ χi with i > 1 must be a
unit of k. After a permutation, we may assume that χ1 ∪ χ2 is a unit, and after
multiplying χ2 by a unit, we may even assume χ1 ∪ χ2 = 1. If χ1 ∪ χi = ai =/ 0
for some i > 2, replace χi by χi − ai χ2 . Since the condition χi (σ) = δ1i when
q =/ 0 is not altered, we may assume χ1 ∪ χi = 0 for i > 2.
Now if V is the subspace spanned by χ3 , . . . , χn , our cup-product restricted
to V × V is non-degenerate and alternating. Hence we may inductively choose
χ3 , . . . , χn such that
χ3 ∪ χ4 = χ5 ∪ χ6 = . . . = χn−1 ∪ χn = 1
and χi ∪ χj = 0 for all other 2 < i < j. When q =/ 0 the condition χi (σ) = δ1i
is still satisfied, χ1 ∪ χ2 = 1 and χ1 ∪ χi = 0 for i > 2. If we replace χ2 by
χ 2 + a3 χ 3 + · · · + an χ n
with a2i = χ2 ∪ χ2i−1 and a2i−1 = −χ2 ∪ χ2i , we have, in addition, χ2 ∪ χi = 0
for i > 2. This proves the proposition.
2
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Chapter III. Duality Properties of Profinite Groups
The following corollary is a first approximation to our main theorem (3.9.11)
on Demuškin groups.
(3.9.17) Corollary. Let G = F/(ρ) be a finitely generated one-relator pro-pgroup with the invariants (n, q). Let q =/ 2.
Then G is a Demuškin group if and only if there exists a basis x1 , . . . , xn of
F such that
ρ ≡ xq1 (x1 , x2 )(x3 , x4 ) · · · (xn−1 , xn ) mod F 3 .
Proof: For any minimal generator system x1 , . . . , xn of F with corresponding
k-basis χ1 , . . . , χn of H 1 (G, k), we have by (3.9.13)(i)
n
Y
qai
ρ≡
xi
i=1
Y
·
(xk , xl )akl mod F 3
1≤k<l≤n
with akl ∈ k uniquely determined (and ai ∈ k also, when q =/ 0). Furthermore,
by (3.9.13) (ii), the matrix B = (bkl ) of the cup-product pairing χ ∪ χ0 is given
by

−akl
if
k < l,


bkl = χk ∪ χl =

alk
if
k > l,



if
k = l.
−
q
2
ak
Assume that
ρ ≡ xq1 (x1 , x2 )(x3 , x4 ) · · · (xn−1 , xn ) mod F 3 .
Then b12 = b34 = · · · = bn−1,n = −1 and bkl = 0 for all other k < l, and
bkk = − q2 , i.e. B is the matrix

B=









−
q
2
−1
1
−

q
2
0
...
0
−
q
2
−1
−







1 

q
2
.
2
It has determinant det(B) = (1 + q2 )n/2 mod q ZZp , which is 1 if p =/ 2, or if
q = 2f =/ 2. Hence the cup-product is non-degenerate and G is thus a Demuškin
group.
Conversely, if G is a Demuškin group, then the cup-product χ ∪ χ0 is
non-degenerate and we may choose a k-basis χ1 , . . . , χn of H 1 (G, k) as in
proposition (3.9.16). Let ξ1 , . . . , ξn be the dual basis of F/F q [F, F ] and
x1 , . . . , xn a lift to F . Then x1 , . . . , xn is a minimal generator system of F
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241
§9. Generators and Relations
with corresponding basis χ1 , . . . , χn of H 1 (G, k). When q =/ 0, let
σ≡
n
Y
âi
xi mod [F, F ],
âi
∈
ZZp .
i=1
In the representation
ρ≡
n
Y
qai
xi
i=1
·
Y
(xk , xl )akl mod F 3 , ai , akl
∈
k,
1≤k<l≤n
we have
ai ≡ âi mod q ZZp = χi (σ) = δ1i
when q =/ 0, because of the uniqueness of the ai , and
a12 = a34 = · · · = an−1,n = 1
and akl = 0 for all other k < l. We therefore get
2
ρ ≡ xq1 (x1 , x2 )(x3 , x4 ) · · · (xn−1 , xn ) mod F 3 .
Our last task is to make an equation from the congruence
ρ ≡ xq1 (x1 , x2 )(x3 , x4 ) · · · (xn−1 , xn ) mod F 3 .
This is achieved by a successive approximation process recalling that by (3.8.2),
the q-central series (F i ) of the finitely generated free pro-p-group F , i.e.
F 1 = F, F i+1 = (F i )q [F i , F ], where q is any nontrivial p-power, form a
fundamental system of open neighbourhoods of 1.
From now on we assume that the rank of F , i.e. the dimension n =
dimIFp H 1 (F ), is even. For every n-tuple y = (y1 , . . . , yn ) ∈ F n , we set
r(y) = y1q (y1 , y2 )(y3 , y4 ) · · · (yn−1 , yn ).
The following lemma is the most subtle part of the proof of theorem (3.9.11).
(3.9.18) Lemma. Let q =/ 2 and let ρ ∈ F 2 . Then for every j
basis x = (x1 , . . . , xn ) of F such that
≥
3 there exists a
ρ ≡ r(x) mod F j ,
provided this is true for j = 3.
Proof: Let x = (x1 , . . . , xn ) be any minimal generator system of F . Let
t1 , . . . , tn ∈ F j−1 and set yi = xi t−1
i . Then y = (y1 , . . . , yn ) is again a minimal
generator system of F since yi ≡ xi mod F q [F, F ]. We may write
r(x) = r(y)dj−1 (t1 , . . . , tn ),
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Chapter III. Duality Properties of Profinite Groups
where dj−1 (t1 , . . . , tn ) is a uniquely determined element of F j . A simple
calculation using proposition (3.8.5) shows that if τi is the image of ti in
grj−1 (F ), then the image of dj−1 (t1 , . . . , tn ) in grj (F ) is
(∗)
π · τ1 +
q
2
[τ1 , ξ1 ] + [τ1 , ξ2 ] + [ξ1 , τ2 ] + · · · + [τn−1 , ξn ] + [ξn−1 , τn ],
where ξi is the image of xi in gr1 (F ). Hence dj−1 induces a k-linear homomorphism
δj−1 : grj−1 (F )n −→ grj (F ), j ≥ 3.
Claim: The map δj−1 is surjective.
Let Hj = im(δj−1 ). In order to prove that Hj = grj (F ), it suffices to
show π · τ ∈ Hj for every τ ∈ grj−1 (F ). In fact, grj (F ) is generated by
the elements π · τ and [τ, ξi ] with τ ∈ grj−1 (F ); the explicit expression (∗)
for δj−1 (τ1 , . . . , τn ) tells us that we have [τ, ξi ] ∈ im(δj−1 ) for i ≥ 3 and
π · τ + [τ, ξ2 ], [τ, ξ1 ] ∈ im(δj−1 ).
We now proceed by induction. Assume that we have shown that Hj =
im(δj−1 ) for some j ≥ 3. If τ ∈ grj−1 (F ), then
τ = δj−1 (τ1 , . . . , τn )
with τ1 , . . . , τn
∈
grj−1 (F ). But then, using (3.8.5),
π · τ = δj−1 (π · τ1 , . . . , π · τn ),
which implies π · τ ∈ Hj+1 for τ ∈ Hj .
Thus we are reduced to proving the lemma for j = 3, that is, π · τ ∈ H3
for τ ∈ H2 . Moreover, it suffices to take τ in the form π · ξi , [ξi , ξj ], since
these elements generate gr2 (F ) by (3.9.13) (i). The ring k is a local ring with
maximal ideal m = pk. Hence by Nakayama’s lemma, it suffices to prove
π · gr2 (F ) ⊆ H3 + m gr3 (F ), since then we would have gr3 (F ) = H3 + m gr3 (F ).
Set M = m gr3 (F ). Then by proposition (3.8.5), we have
π · [ξi , ξj ] = [π · ξi , ξj ] + m = [ξi , π · ξj ] + m0 ,
where m, m0 ∈ M . Therefore, since [τ, ξi ] ∈ im(δ2 ) if i =/ 2, we have π·[ξi , ξj ] ∈
H3 + M for any i, j. Moreover, as π · τ + [τ, ξ2 ] ∈ H3 for any τ ∈ gr2 (F ), we
have π 2 · ξi ∈ H3 + M for any i. This proves the surjectivity of δj−1 .
Now let j ≥ 3 and assume that there exists a minimal generator system
x = (x1 , . . . , xn ) of F such that ρ ≡ r(x) mod F j , i.e. ρ = r(x)ej , ej ∈ F j .
Let εj be the image of ej in grj (F ). Since δj−1 is surjective, there exists
t1 , . . . , tn ∈ F j−1 such that δj−1 (τ1 , . . . , τn ) = −εj , i.e. dj−1 (t1 , . . . , tj ) = e−1
j
mod F j+1 . Put yi = xi t−1
.
Then
y
=
(y
,
.
.
.
,
y
)
is
a
minimal
generator
1
n
i
system of F such that
ρ = r(x)ej = r(y)dj−1 (t1 , . . . , tn )ej ≡ r(y) mod F j+1 .
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243
§9. Generators and Relations
Proof of theorem (3.9.11): The proof is an immediate consequence of the
corollary (3.9.17) and the above lemma. Let G = F/(ρ) be a one-relator pro-pgroup with invariants (n, q). If there exists a basis x = (x1 , . . . , xn ) of F such
that ρ = r(x), then a fortiori ρ ≡ r(x) mod F 3 , and from (3.9.17) it follows
that G is a Demuškin group.
Conversely, let G be a Demuškin group. Then by (3.9.17) and (3.9.18), there
(j)
exists for every j ≥ 3 a minimal generator system x(j) = (x(j)
1 , . . . , xn ) such
that
ρ ≡ r(x(j) ) mod F j .
When q = 0 this remains valid if we replace the central series F j by the
p-central series, i.e. it is always true that
(∗)
ρ ≡ r(x(j) ) mod U (j),
where (U (j)) is a series of open normal subgroups which form a basis of
neighbourhoods of 1 by (3.8.2). If X (j) denotes the finite nonempty set of
(j)
minimal generator systems x(j) = (x(j)
1 , . . . , xn ) of F/U (j) satisfying (∗), then
(j)
lim X is nonempty and any x = (x1 , . . . , xn ) in this projective limit is a basis
←−
of F such that ρ = r(x). This concludes the proof of theorem (3.9.11).
2
Remark: We have proved a sharper result than the assertion of theorem
(3.9.11). Namely, we have shown that for any defining relation ρ ∈ F of G,
i.e. G = F/(ρ), there exists a basis x1 , . . . , xn of F , such that
ρ = xq1 (x1 , x2 )(x3 , x4 ) · · · (xn−1 , xn ).
Along the same lines, but with more complicated modifications in the individual steps, one can also settle the case q = 2. This was originally proved
by J.-P. SERRE in the case when n is odd, and later J. LABUTE obtained the
general case. For this we refer the reader to LABUTE’s article [117]. The result
is the following
(3.9.19) Theorem. Let G = F/(ρ) be a Demuškin group of rank n such that
q = 2.
(i) If n is odd, there exists a basis x1 , . . . , xn of F such that
f
ρ = x21 x22 (x2 , x3 )(x4 , x5 ) · · · (xn−1 , xn )
for some f = 2, 3, . . . , ∞ (f = ∞ means 2f = 0).
(ii) If n is even, there exists a basis x1 , . . . , xn of F such that
f
2
ρ = x2+α
1 (x1 , x2 )x3 (x3 , x4 )(x5 , x6 ) · · · (xn−1 , xn )
for some f = 2, 3, . . . , ∞ and α ∈ 4ZZ2 .
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244
Chapter III. Duality Properties of Profinite Groups
For further results on one-relator pro-p-groups, see [110], [118], [255].
In arithmetic applications one sometimes meets the situation that one is given
a canonical system of generators of a pro-p-group G. Even if the group G is
free, we are interested in a presentation of G in terms of generators and relations,
which is not necessarily minimal, but uses the given system of generators (or
at least generators which are closely related to the given ones). The following
theorem provides a typical example of such a situation. Its proof, for which we
refer the reader to [251], Lemma 2.4, uses an approximation process similar to
that in the proof of theorem (3.9.11).
(3.9.20) Theorem. Let
1 −→ H −→ G −→ D −→ 1
be an exact sequence of pro-p-groups. Assume that G is free and that D is a
Demuškin group of rank n with torsion-free abelianization. Let D be generated
by x̄1 , . . . , x̄n with the one defining relation
(x̄1 , x̄2 ) · · · (x̄n−1 , x̄n ) = 1.
Suppose that the free ZZp -module H/[H, G] has the basis
{ỹj mod [H, G] | j = 1, . . . , s}
and that
s
Y
ỹj
∈
[G, G].
j=1
Then there exist generators x1 , . . . , xn , y1 , . . . , ys of G with
(i) xi mod H = x̄i , i = 1, . . . , n ,
(ii) yj = (ỹj )ατj with α ∈ ZZ×
p , τj ∈ G, j = 1, . . . , s ,
subject to the one relation
(x1 , x2 ) · · · (xn−1 , xn )
s
Y
yj = 1.
j=1
Exercise: Let G be a pro-p-group.
Consider the filtration G = G1 ⊇ G2 ⊇ . . . defined by
\
∗
Gn = (Gn−1 ) . Show that
Gn = {1} and that the Gn are open if and only if d(G) < ∞.
n≥1
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Chapter IV
Free Products of Profinite Groups
§1. Free Products
As in III §5, we let c be a full class of finite groups and we consider the
category of pro-c-groups. A family
κi : Gi −→ G, i ∈ I,
of homomorphisms of pro-c-groups is called convergent (to 1) if every open
subgroup U of G contains the images κi (Gi ) for almost all i, i.e. all but a
finite number. The free products of pro-c-groups are defined by the following
universal property.
(4.1.1) Definition. The free pro-c-product of a family Gi , i ∈ I, of pro-c-groups
is a pro-c-group G together with a convergent family of homomorphisms
κi : Gi −→ G, i ∈ I,
such that for every other convergent family of homomorphisms κ0i : Gi → G0
there is a unique homomorphism of pro-c-groups κ0 : G → G0 such that the
diagrams
κi
Góñò i
G
κ0
κ0i
0
G
are commutative. The group G is then denoted by
G=
∗G.
i∈I
i
The free pro-c-product is clearly unique up to isomorphism, if it exists. Its
discr
existence is based on the usual free product G = ∗ Gi of the Gi in the
i∈ I
category of groups (see [73]), which contains the Gi as subgroups. Let BG
denote the family of all normal subgroups N of G such that
(1) G/N ∈ c,
(2) N ⊇ Gi for almost all i ∈ I,
(3) N ∩ Gi is an open subgroup of the pro-c-group Gi .
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Chapter IV. Free Products of Profinite Groups
Taking BG as a basis of open neighbourhoods of the identity of G defines
a group topology TG on G ∗) , which induces the profinite topology on Gi .
Indeed, if N ∈ BG , then N ∩ Gi is open in Gi , and, conversely, if U is any
open normal subgroup of Gi , we obtain a homomorphism G → Gi /U whose
kernel N is in BG and has the property that N ∩ Gi = U . This homomorphism
is obtained when we map Gj onto 1 ∈ Gi /U for j =/ i and Gi onto Gi /U by
the canonical projection. Since Gi is compact, it is a closed subgroup of G in
the topology TG . We now take the completion
G = lim G/N,
←−
N∈BG
which comes equipped with a canonical continuous homomorphism λ : G → G,
and with the family of composite maps
λ
κi : Gi ,→ G −→ G.
These satisfy the universal condition of a free pro-c-product. In fact, if we have
any convergent family of homomorphisms κ0i : Gi → G0 into a pro-c-group
G0 , then the diagram
Gøù÷õöô i
G
κi
λ
G
κ0
κ0i
κ0
G0
of solid arrows is commutatively completed by the dotted arrows; κ0 exists
because of the universal property of the free product G in the category of
groups, and κ0 exists by the universal property of the completion: because of
the conditions (1), (2), (3) for the normal subgroups N ∈ BG , κ0 is continuous
and thus defines a homomorphism κ0 : G → G0 of pro-c-groups.
Example: Let ẐZ(c) be the free pro-c-group of rank 1 (see III §5). The free
pro-c-group F over a set X is the free pro-c-product
F =
∗ ẐZ(c).
x∈X
This follows at once from the universal properties, as do the following remarks
and propositions.
∗) The topology T is Hausdorff, by remark 1 at the end of §3. But we don’t need this here.
G
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247
§1. Free Products
If J is a subset of I, then we have an exact sequence which splits canonically
ûú
1 −→ N −→
(∗)
∗G
i∈I
∗ G −→ 1.
π
s
i
i∈ J
i
The homomorphism π is obtained by choosing the canonical maps κi : Gi →
Gi for i ∈ J and the trivial homomorphism Gi → {1} ⊆ Gi for i ∈ I r J.
∗
∗
i∈ J
i∈ J
The homomorphism s is obtained by the canonical maps κj : Gj →
j
∈
i
for
∗ G as embedded in ∗ G ,
∗G ∗G.
as subgroups of G = ∗ G . They generate G
J. We have π ◦ s = id and consider
i
i∈J
i∈J
In particular, we view the Gi
∗G
i∈ I
i ⊆
i∈I
i
i
i∈ I
i∈I
i
topologically, and the kernel N in (∗) is the smallest closed normal subgroup
of G containing the Gi , i ∈ I r J.
For an arbitrary index set I, we consider the family of finite subsets of I
partially ordered by inclusion.
(4.1.2) Proposition. The free pro-c-product G =
∗G
i∈ I
G = lim
i
is the projective limit
∗G ,
i
←− i∈S
S
where S runs through the finite subsets of I.
Proof: It is sufficient to show that the projective limit on the right-hand side
satisfies the universal property of the free product with respect to homomorphisms to finite groups in c. But for a finite group H we have
Hom(lim
∗ G , H) = lim Hom( ∗ G , H) = lim
←− i∈S
S ⊆I
i
−→
S ⊆I
i∈S
i
M
−→
S ⊆I i∈S
Hom(Gi , H).
This concludes the proof, because for every convergent family of homomorphisms (κi : Gi → H)i∈I , there exists a finite subset S ⊆ I such that κi = 0 for
all i ∈/ S.
2
If c0 ⊆ c is a full subclass of c, then to each pro-c-group G we may associate
the maximal pro-c0 -factor group G(c0 ). This is the projective limit
G(c0 ) = lim G/U
←−
U
over all open normal subgroups U of G such that G/U ∈ c0 . It is characterized
by the universal property that every homomorphism of G into a pro-c0 -group
factors through G(c0 ). From this, we obtain the
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Chapter IV. Free Products of Profinite Groups
∗ the free pro-c -product, we have
( ∗ G )(c ) = ∗ G (c ).
0
(4.1.3) Proposition. Denoting by
0
i
i∈I
0
0
i∈I
i
0
We now turn to the cohomology of free pro-c-products. It behaves in a
particularly nice way. Let G =
Gi and let A be a G-module. Then the
∗
i∈ I
n
restriction maps res i : H (G, A) → H n (Gi , A) define a homomorphism into
the direct sum
M
H n (G, A) −→
H n (Gi , A).
i∈I
In fact, if S runs through the finite subsets of I, then by (4.1.2) and (1.5.1) we
obtain
H n (G, A) = H n (lim Gi , A) = lim H n ( Gi , A)
∗
→ lim
−→
S
∗
←− i∈S
S
−→
S
M
M
H n (Gi , A) =
i∈ S
H n (Gi , A).
i∈I
i∈S
We will now prove the following remarkable isomorphism theorem.
(4.1.4) Theorem. Let G =
∗ G . Then for every torsion G-module A we have
i∈ I
i
a canonical exact sequence
0 → AG (c) → A(c) →
M
δ
res
(A/AGi )(c) → H 1 (G, A) →
and for all n
H 1 (Gi , A) → 0,
i∈I
i∈ I
≥
M
2
H n (G, A) ∼
=
M
H n (Gi , A).
i∈I
Proof: By a direct limit argument we may assume that A is finite. Moreover,
we may assume A ∈ c. Namely, if we decompose A into its p-primary components A(p), then the cohomology groups decompose into direct summands.
These are zero if p does not divide the order of G (i.e. the order of a finite
quotient G/U ). Otherwise one sees at once that A ∈ c. Moreover we have
A/AGi = 0 for almost all i ∈ I, since there is an open subgroup U of G which
acts trivially on A.
The exact sequence of the theorem will follow by the snake lemma from the
exact commutative diagram
∂
0ýþÿü
A/AG
Z 1 (G, A)
H 1 (G, A)
0
res
0
M
i∈I
A/AGi
∂
M
i∈ I
Z 1 (Gi , A)
res
M
H 1 (Gi , A)
i∈I
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249
§1. Free Products
once we have shown that the restriction map
res : Z 1 (G, A) −→
(∗)
M
Z 1 (Gi , A)
i∈I
on inhomogeneous 1-cocycles is bijective (its image is contained in the direct
sum by the same argument as for H n (G, A)). To this end, consider the exact
commutative diagram
1
A
Ĝi
1
A
Ĝ
πi
si
Gi
1
π
s
G
1,
(∗∗)
where Ĝ is the semi-direct product of A by G with the canonical homomorphic
section s, Ĝi = π −1 (Gi ) and si = s|Gi . The map χ 7→ s0 = χs gives a 1-1
correspondence between the 1-cocycles χ ∈ Z 1 (G, A) and the homomorphic
sections s0 : G → Ĝ of π. If res i χ = 1 for all i ∈ I, then s0 |Gi coincides with
si for all i ∈ I, hence s0 = s by the universal property of the pro-c-product, and
so χ = 1. This proves the injectivity of (∗).
Conversely let (χi )i∈I be a family of cocycles χi ∈ Z 1 (Gi , A) such that
χi = 1 for almost all i ∈ I. Then we obtain new sections s0i = χi si : Gi → Ĝi ,
hence a family of homomorphisms s0i : Gi → Ĝ which is convergent since
s0i = si for almost all i ∈ I. This family defines a homomorphism s0 : G → Ĝ
such that s0 |Gi = s0i by the universal property of the free pro-c-product. From
πi ◦ s0i = id it follows that π ◦ s0 = id, i.e. s0 is a new section of π and defines a
1-cocycle χ(σ) = s0 (σ)s(σ)−1 such that χ|Gi = χi for all i ∈ I. This proves the
surjectivity of (∗).
It remains to prove the second assertion of the theorem. By dimension
shifting, we obtain the surjectivity of the map
res : H n (G, A) −→
M
H n (Gi , A)
i∈I
for n ≥ 1. It suffices to prove the injectivity for n = 2, again by dimension
shifting. And again we may assume that A is finite and in c, and moreover
that I is finite, by (4.1.2). Let x ∈ H 2 (G, A) be such that xi = res i x = 0
for all i ∈ I. By (1.2.4), x and the xi define group extensions, and we have
a commutative diagram (∗∗) with homomorphic sections si : Gi → Ĝi ⊆ Ĝ,
since the upper group extensions split. By the universal property of the free
pro-c-product G = Gi , the si define a homomorphic section s : G → Ĝ such
∗
i∈I
that s|Gi = si . Hence the lower group extension splits, i.e. x = 0. This shows
the injectivity.
2
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250
Chapter IV. Free Products of Profinite Groups
For pro-p-groups G we have a converse result to the above theorem. We set
H n (G) = H n (G, ZZ/pZZ).
(4.1.5) Theorem. Let G be a pro-p-group and let Gi , i ∈ I, be a convergent
subgroup family of G. Then the following conditions are equivalent.
(i) G =
∗G
i∈I
i
(free pro-p-product).
(ii) The homomorphism
res : H n (G) −→
M
H n (Gi )
i∈I
is bijective for n = 1 and injective for n = 2.
Proof: The implication (i) ⇒ (ii) follows from (4.1.4) since G acts trivially
on ZZ/pZZ.
Conversely assume that (ii) holds. The inclusions Gi → G are by assumption
a convergent family of homomorphisms, and thus define a homomorphism of
the free pro-p-product
Ĝ =
∗ G −→ G.
π
i∈ I
i
By (4.1.4) we have
H j (Ĝ) =
M
H j (Gi )
i∈I
for j = 1, 2, and so, by assumption,
H j (G) −→ H j (Ĝ)
is an isomorphism for j = 1 and injective for j = 2. Now the result follows
from (1.6.15).
2
The concept of free pro-c-product has generalizations in many directions (see
[154]). One can define restricted free pro-c-products (Gi , Hi ) with respect
∗
i∈I
to subgroups Hi ⊆ Gi and one obtains an idèle version. Further, one can
consider free pro-c-groups with amalgamated subgroup G1 ∗ G2 . We mention
H
some facts about the last concept in the exercises.
However, one important generalization will be introduced in §3. While the
groups occurring as factors in the free product have been independent of each
other so far, we will introduce there the free pro-c-product over a family of
pro-c-groups which vary continuously over a topological base space.
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251
§1. Free Products
Exercise 1. Let i1 : H ,→ G1 , i2 : H ,→ G2 be two injective homomorphisms of pro-c-groups.
The push-out of i1 and i2 is a commutative diagram of pro-c-groups
H
i2
G2
κ2
i1
G1
G
κ1
with the universal property that any two homomorphisms κ01 : G1 → G0 , κ02 : G2 → G0 into
a pro-c-group G0 such that κ01 ◦ i1 = κ02 ◦ i2 determine a unique homomorphism f : G → G0
such that κ1 ◦ f = κ01 and κ2 ◦ f = κ02 . One speaks of an amalgamated free pro-c-product
along H if κ1 and κ2 are injections. One writes in this case
G = G1 ∗ G2 .
H
Show that the push-out always exists. The amalgamated product does not always exist (see
[179] for an example).
Exercise 2. If c is the class of all finite groups, the amalgamated free profinite product exists
in the following cases:
1) H is central in either G1 or G2 ;
2) H is normal in both G1 and G2 and is topologically finitely generated;
3) H is finite.
The amalgamated free pro-c-product G1 ∗ G2 of two pro-p-groups G1 , G2 along a common
H
procyclic subgroup H exists (see [179]).
For the next exercises (see [178]), assume that the amalgamated free pro-c-product
G = G1 ∗ G2 exists and consider the cohomology groups of pairs as defined in I §6, ex.4
H
for a torsion G-module A.
Exercise 3. (Excision) The canonical homomorphism
H n (G, G1 , A) −→ H n (G2 , H, A)
induced by the inclusion (G2 , H) ,→ (G, G1 ) is an isomorphism for all n ≥ 0.
Exercise 4. The inclusions (G1 , H) ,→ (G, H) and (G2 , H) ,→ (G, H) induce an isomorphism
H n (G, H, A) ∼
= H n (G1 , H, A) ⊕ H n (G2 , H, A)
(n ≥ 0).
Exercise 5. (Mayer-Vietoris sequence) We have an exact sequence
∆
· · · −→ H n (H, A) −→ H n+1 (G, A) −→ H n+1 (G1 , H, A) ⊕ H n+1 (G2 , H, A)
Ψ
−→ H n+1 (H, A) −→ H n+2 (G, A) −→ · · · ,
where ∆ is the composite of the canonical maps
δ
ι
∼
H n (H, A) −→ H n+1 (G2 , H, A) −→
H n+1 (G, G1 , A) −→ H n+1 (G, A),
and Ψ (x1 ⊕ x2 ) = h1 (x1 ) − h2 (x2 ), where h1 and h2 are induced by the inclusions H ,→ G1
and H ,→ G2 .
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252
Chapter IV. Free Products of Profinite Groups
§2. Subgroups of Free Products
A very useful result is the following analogue of the well-known Kurosh
subgroup theorem in (discrete) group theory. In the form presented here it is
due to E. BINZ, J. NEUKIRCH and G. WENZEL (see [12]). There are different
proofs and generalizations by several other authors (see [56], [74], [135]).
∗G
(4.2.1) Theorem. Let G =
i∈I
i
be the free pro-c-product of the Gi and let
H be an open subgroup of G. Then there exist systems Si of representatives
S
si of the double coset decomposition G = . si ∈Si Hsi Gi for all i and a free
pro-c-group F ⊆ G of the finite rank
rk(F ) =
X
[(G : H) − #Si ] − (G : H) + 1,
i∈I
such that the natural inclusions induce a free product decomposition
H=
∗ (G
i,si
si
i
∩ H) ∗ F,
where Gsi i (= si Gi s−1
i ) denotes the conjugate subgroup.
Remarks: 1. If N ⊆ H is an open subgroup which is normal in G, then
Gi ⊆ N for almost all i ∈ I, and for such i we have #Si = (G : H). Hence
rk(F ) is finite.
2. It is not true in general that we may choose any systems Si of representatives
of the double coset decompositions. Only if c is the class of p-groups the
theorem is true for any choice of the Si by (4.1.5).
discr
Proof of theorem (4.2.1):
Let G =
∗
i∈I
Gi be the (discrete) free product
of the Gi in the category of groups. We give G the topology TG which is
defined by the family BG of open normal subgroups N ⊆ G satisfying the
three conditions
(1) G/N ∈ c,
(2) N ⊇ Gi for almost all i and
(3) N ∩ Gi is an open subgroup of the pro-c-group Gi for all i.
Then G is the completion of G with respect to TG , and the restriction of TG to
Gi is the given profinite topology on Gi (compare the existence proof for free
products in §1). Let κ : G → G be the canonical completion homomorphism.
We denote the isomorphic images of the groups Gi in G and G also by Gi .
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253
§2. Subgroups of Free Products
Now let H be an open subgroup in G. We denote the open subgroup
κ−1 (H) ⊆ G by H. In particular, H is of finite index in G and by the Kurosh
subgroup theorem for discrete groups (see [215], chap.I §5.5 th. 14), there
exist systems Si of representatives σi of the double coset decomposition G =
S
. Hσ G and a (discrete) free group F ⊆ G of rank
i i
rk(F) =
X
[(G : H) − #Si ] − (G : H) + 1,
i∈ I
such that the natural inclusions induce a free product decomposition
discr
H=
∗ (G
i,σi
σi
i
discr
∩ H) ∗ F.
By the argument of remark 1 above, we conclude that the sum on the right side
of the rank equation is finite.
Since TG induces on Gi , and therefore also on Gσi i , the pro-c-topology, and
since H is open, Gσi i ∩ H is an open subgroup of the pro-c-group Gσi i . Now let
TH be the topology on H which is induced by the family BH of open normal
subgroups I ⊆ H satisfying the three conditions
(1) H/I ∈ c,
(2) I ⊇ Gσi i ∩ H for almost all i, σi and
(3) I ∩ Gσi i ∩ H is an open subgroup of the pro-c-group Gσi i ∩ H
for all i, σi .
We first note that TH induces on Gσi i ∩ H the pro-c-topology of Gσi i ∩ H,
and induces on F the topology given by all normal subgroups IF ⊆ F with
F/IF ∈ c. Therefore the completion of H with respect to the topology TH has a
free product decomposition as in the statement of the theorem. Thus we have
to show:
Claim: The restriction of the topology TG to H is equal to TH .
For an N ∈ BG , one easily observes N ∩ H ∈ BH and therefore the topology
TH is finer than TG ∩ H. To show that also TG ∩ H is finer than TH , let I ∈ BH .
We have to find a group N ∈ BG such that N ∩ H ⊆ I. We claim that we can
take for N the intersection Ĩ of all conjugates of I in G. Since I is of finite
index in H and H is of finite index in G, I is of finite index in G. Therefore I
has only finitely many conjugates Iτj in G, i.e. Ĩ is open and of finite index in
G. Denoting by H̃ ∈ BG the intersection of the finitely many conjugates of H
in G, we obtain an exact sequence
1 → H̃/Ĩ → G/Ĩ → G/H̃ → 1.
The group H̃/Ĩ is canonically embedded into the group τj Hτj /Iτj ∈ c, and
the class c is full. Therefore G/Ĩ ∈ c. Next we have to show that Gi ∩ Ĩ is open
in the pro-c-topology of Gi . It is sufficient to show that Gi ∩ Iτj is open in Gi
Q
τ −1
τ −1
or, equivalently, that Gi j ∩ I is open in Gi j . If we write τj−1 = h · σi · gi ,
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254
Chapter IV. Free Products of Profinite Groups
τ −1
Gi , we get Gi j ∩ I = (Gσi i ∩ I)h and this group
τj−1 ∗
i gi
) . At the same time,
is open in the pro-c-group (Gσi i )h = Ghσ
=
G
i
i
h
∈
H, σi
∈
Si and gi
∈
τ −1
τ −1
we see that Gi j ∩ I = (Gσi i ∩ I)h = (Gσi i )h = Gihσi gi = Gi j for almost all
i ∈ I. This proves the claim. The proof of the theorem is then completed by
the observation that the images si ∈ H of the σi under κ form systems Si of
S
representatives of the double coset decomposition . Hsi Gi in G.
2
The next corollary is the profinite analogue of the Nielson-Schreier formula
for discrete groups. If G is a pro-p-group, we have already proved it in III §5
using partial Euler-Poincaré characteristics (see (3.9.6)).
(4.2.2) Corollary. An open subgroup H of a free pro-c-group G is again a free
pro-c-group. The ranks satisfy the equation
rk(H) − 1 = (rk(G) − 1) (G : H).
∗
Proof: A free pro-c-group G is the free pro-c-product G = i∈I Gi , where
(c) denotes the free pro-c-groups of rank 1 (see III §5). Since (loc.cit.)
Gi ∼
= ẐZQ
ẐZ(c) = p∈S ZZp , where S is the set of prime numbers which divide the order
of a group in c, we see that every open subgroup of the free pro-c-group of
rank 1 is isomorphic to the free pro-c-group of rank 1 again. By (4.2.1), there
exists a system Si of representatives si of the double coset decomposition
S
G = . si ∈Si Hsi Gi and a free pro-c group F ⊆ G of the finite rank rk(F ) =
P
i∈I [(G : H) − #Si ] − (G : H) + 1, such that the natural inclusions induce
a free product decomposition H = i,si (Gsi i ∩ H) ∗ F. Therefore H is a free
pro-c-group (as a free pro-c-product of free pro-c-groups) and
X
rk(Gsi i ∩ H) + rk(F )
rk(H) =
∗
i,si
=
X
#Si + rk(F )
i∈ I
=
X
(G : H) − (G : H) + 1
i∈ I
= (G : H) (rk(G) − 1) + 1.
2
Having proved several results which are quite similar to those in discrete
group theory, the reader should be warned of some curious pathologies in the
profinite case, which are (more or less) due to the existence of infinite words
in the profinite products.
∗) Observe the rule (Ga )b = Gba .
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255
§2. Subgroups of Free Products
For example, if a group P is the free product of its subgroups G and H, then
one should expect that for every pair σ, τ ∈ P , the conjugates Gσ and H τ also
form a free product. However, look at the following example due to D. HARAN
(see [74]).
(4.2.3) Proposition. Assume that ZZ/2ZZ, ZZ/3ZZ ∈ c and let G (resp. H) be a
finitely generated pro-c-group having a normal subgroup of index 2 (resp. 3).
Let
P = G ∗ H.
Then there exist elements σ, τ ∈ P such that the subgroups Gσ and H τ do not
form a free pro-c-product in P .
Proof: Let Q be the set of closed subgroups Q ⊆ P such that there exist
σ, τ ∈ P with Gσ , H τ ⊆ Q. For Q ∈ Q the sets {σ ∈ P | Gσ ⊆ Q} and
{τ ∈ P | H τ ⊆ Q} are closed in P . We deduce that Q is closed under
descending chains. By Zorn’s lemma, it has a minimal element, say Q.
Assume that Q = Gσ ∗ H τ for suitable σ, τ ∈ P . Then, by our assumptions on
G and H and by the universal property of the free product, there would exist
a homomorphism φ : Q → S4 (the symmetric group on four elements), such
that φ(Gσ ) = h(12)i and φ(H τ ) = h(134)i. Since S4 is generated by the cycles
(12) and (134), φ is surjective. Choose ρ ∈ Q with φ(ρ) = (1234). Then
φ(hGσ , H ρτ i) = h(12), (134)(1234) i = h(12), (124)i $ S4 .
This shows that hGσ , H ρτ i $ Q is contained in Q. This contradicts the
minimality of Q.
2
Remarks: 1. It follows from the construction of the free pro-c-product that
P = Gσ ∗ H τ whenever σ and τ are finite words in elements of G and H (i.e.
σ and τ are the products of finitely many elements which are in G or H).
2. This kind of pathology does not occur if c is the class of pro-p-groups. Then
P = G ∗ H is the free pro-p-product of Gσ and H τ for every pair σ, τ ∈ P .
This follows from (4.1.5). However, by the result of exercise 3 below, this is
the only good case.
Exercise 1. Assume that we are given pro-c-groups G, H and an open normal subgroup
N ⊆ G. Show that the kernel of the homomorphism
G ∗ H −→ G/N ,
which, by the universal property of the free pro-c-product, is given by the projection G → G/N
and the trivial homomorphism H → {1} ⊆ G/N , is of the form
N ∗ ( ∗ H σ ),
σ ∈R
where R is a system of representatives of the cosets of N in G.
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Chapter IV. Free Products of Profinite Groups
Exercise 2. Let p be a prime number and assume ZZ/pZZ ∈ c. Determine the kernels of the
homomorphisms
ZZ/pZZ ∗ ZZ/pZZ → ZZ/pZZ ,
which, by the universal property of the free pro-c-product, are associated to the pairs (id, id)
and (0, id).
Exercise 3. Assume that the full class c is strictly larger than the class (p-groups) and assume
that G, H are pro-p-groups. Let P = G ∗ H be the free pro-c-product of G and H.
(i) Show that P is not a pro-p-group if G and H are nontrivial.
(ii) Show that there are σ, τ ∈ P such that the closed subgroup U = hGσ , H τ i, generated
by Gσ and H τ in P , is a pro-p-group.
(iii) Show that the closed subgroup U constructed in (ii) is isomorphic to the free pro-pproduct of G and H.
Hint: Use (4.2.1) for (i), the Sylow theorems (1.6.9) for (ii) and (4.1.5) for (iii).
Exercise 4. Assume that the finite group G is the semi-direct product of its subgroup H and
its normal subgroup N . Consider the canonical surjection
ε : N ∗ H −→ G,
which is induced by the inclusions of N and H into G. Show that the kernel of ε is a free
profinite group of rank (#N − 1)(#H − 1) with basis
{hnh−1 n−1
h
∈
N ∗ H | h ∈ H\{1}, n ∈ N \{1}},
where nh := (hnh−1 ) ∈ N , i.e. ker(ε) is the free profinite group on the symbols which
represent the difference between formal and real conjugation of nontrivial elements of N by
nontrivial elements of H.
§3. Generalized Free Products
In this section we define a vast generalization of the free products of §1.
Instead of (discrete) families of profinite groups we now consider families
of profinite groups which vary continuously over a topological base space.
Such generalizations were introduced by D. GILDENHUYS and L. RIBES ([56])
for locally constant families and by D. HARAN ([74]) and (independently) by
O. MELNIKOV ([135]) in its most general form.
As before, let c be a full class of finite groups. Given continuous maps
f : X → Z, g : Y → Z between profinite spaces, we will write
X ×Z Y = {(x, y) ∈ X × Y |f (x) = g(y)}
for the fibre product of X and Y over Z, which is a profinite space.
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§3. Generalized Free Products
(4.3.1) Definition. Let T be a profinite space. A bundle of pro-c-groups
over T is a group object in the category of profinite spaces over T , such that
the fibre over every point of T is a pro-c-group. In other words, a bundle of
pro-c-groups over T is a tuple (G, p, T, m, e, ι), where G is a profinite space
together with continuous maps
p : G → T (the structure map),
m : G ×T G → G (the multiplication),
e : T → G a section to p (the unit),
ι : G → G (the inversion),
such that the fibre Gt := p−1 (t), together with the induced maps mt : Gt ×Gt →
Gt , ιt : Gt → Gt and the unit element et := e(t), is a pro-c-group for every
point t ∈ T .
Because of the existence of the unit e, the structure map p is necessarily
surjective. Usually we will denote such a bundle by (G, p, T ), or by G if no
confusion is possible.
Example 1. If G is a pro-c-group and T is a profinite space, then we always have
the constant bundle (G × T, prT , T ), where prT is the projection G × T → T
and the maps m, ι, e are those induced by the group operations in G.
Example 2. Assume that (Gi )i∈I is a (discrete) family of pro-c-groups. We
define the associated bundle of pro-c-groups as the bundle over the one point
S
compactification I¯ := I . {∗} of I which is given as a set by
[
S
G := . Gi . {∗} ,
i∈I
and which has the following topology: Gi ⊆ G (together with its profinite
topology) is open in G for all i, and for every open neighbourhood U ⊆ I¯ of
¯ let
∗ ∈ I,
[
. G S. {∗}
i
i∈U
be an open neighbourhood of ∗ ∈ G. The space G is profinite and one checks
that the map
¯ Gi 3 gi 7→ i, ∗ 7→ ∗
p : G → I;
is continuous. Viewing {∗} as the group with one element, we see that the
group operations on the Gi ’s induce the structure of a bundle of pro-c-groups
¯
on the triple (G, p, I).
Assume that we are given a pro-c-group G and a profinite space T .
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Chapter IV. Free Products of Profinite Groups
(4.3.2) Definition. A family (Gt )t∈T of subgroups of the profinite group G,
indexed over the profinite space T , is a continuous family of subgroups if for
every open subgroup U ⊆ G the set T (U ) := {t ∈ T | Gt ⊆ U } is open in T .
Given such a continuous family of subgroups, we define a bundle of pro-cgroups (G, p, T ) over T by putting
G = {(g, t) ∈ G × T | g
∈
Gt }
and defining p to be the restriction to G of the projection of G × T onto T . We
define the maps m, ι, e by restricting the corresponding maps from the constant
bundle.
(4.3.3) Lemma. (G, p, T ) is a bundle of pro-c-groups.
Proof: Once we have proved that G is a profinite space, all properties follow
from the corresponding properties of the constant bundle G × T → T . Hence
it remains to show that G is closed in G × T . Let (h, s) ∈/ G, i.e. h ∈/ Gs .
Then there exist open subgroups V, W ⊆ G with Gs ⊆ W and hV ∩ W = ∅.
Since the family (Gt ) is continuous, T (W ) is open in T . But then (h, s) ∈
hV × T (W ) ⊆ G × T \ G, which finishes the proof.
2
(4.3.4) Definition. A morphism of bundles
φ : (G, pG , T ) → (H, pH , S)
is a pair φG : G → H, φT : T → S of continuous maps such that
(i) the diagram
G
φG
pH
pG
T
H
φT
S
commutes and
(ii) for every t ∈ T the associated map φt : Gt → HφT (t) is a group homomorphism.
We say that φ is surjective if φG (and hence also φT ) is surjective.
We will not distinguish between the pro-c-group G and the bundle (G, p, {∗})
over the one point space {∗}. In particular, a morphism from a bundle (G, p, T )
to a group G is a continuous map φ : G → G such that the induced maps
φt : Gt → G are group homomorphisms for every t ∈ T . One easily verifies
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§3. Generalized Free Products
that the family ( φ(Gt ) )t∈T is a continuous family of subgroups of G indexed
by T .
(4.3.5) Definition. The free pro-c-product of a bundle (G, p, T ) of pro-c-groups
is a pro-c-group
G= G
∗
T
together with a morphism ω : G → G, which has the following universal
property: for every morphism f : G → H from G to a pro-c-group H there
exists a unique homomorphism of pro-c-groups φ : G → H with f = φ ◦ ω.
In categorical language this means that the functor “free product over T ” is a
left adjoint to the functor “constant bundle” from (pro-c-groups) to (bundles of
pro-c-groups over T ).
The map ω induces a canonical map ωt : Gt → G for every t ∈ T . We will
see in (4.3.12) that ωt is an injective group homomorphism for every t ∈ T and
that the images of the Gt in G are in some sense independent from each other.
(4.3.6) Proposition. The free pro-c-product
∗ G exists and is unique up to
T
unique isomorphism.
Proof: The uniqueness assertion is clear by the universal property. The
existence follows from the following construction:
Let L be the abstract free product of the family of groups (Gt )t∈T and let
λ : G → L be the map which is given on every Gt as the natural inclusion
of Gt into L. Then define the free product G = T G as the completion of L
with respect to the topology which is given by the family of normal subgroups
N ⊆ L of finite index for which (a) L/N ∈ c and (b) the composition pN ◦ λ of
λ with the natural projection pN : L → L/N is continuous. It is easily verified
that G has the required universal property. (Compare with the construction of
the free product in §1.)
2
∗
(4.3.7) Definition. Assume that (Gt )t∈T is a continuous family of subgroups
of the pro-c-group G indexed over the profinite space T and let G be the
pro-c-group bundle which is associated to the family (Gt )t∈T by (4.3.3). We
say that G is the free pro-c-product over the family (Gt )t∈T if the canonical
homomorphism G → G is an isomorphism.
∗
T
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Chapter IV. Free Products of Profinite Groups
Example 3. Assume that G is a pro-c-group and that T is a profinite space.
Then we denote the free pro-c-product over the constant bundle G × T → T
by T G.
Assume, in addition, that a profinite group Γ acts continuously on T . Then,
by the universal property of the free product, Γ also acts continuously on T G,
i.e. T G becomes a pro-c-Γ operator group, which is defined as follows.∗)
∗
∗
∗
(4.3.8) Definition. Let Γ be a profinite group. A pro-c-Γ operator group
is a pro-c-group G together with a homomorphism
φ : Γ −→ Aut(G),
such that the action: Γ × G → G, (γ, g) 7→ φ(γ)(g), is continuous.
If Γ = T acts on itself by left multiplication and if G = Fr is the free pro-cgroup of rank r, then Fr is the free pro-c-Γ operator group of rank r.
∗
Γ
Free pro-c-Γ operator groups satisfy the obvious universal property in the
category of pro-c-Γ operator groups.
Example 4. Assume that we are given a (discrete) family (Gi )i∈I of pro-cgroups and let G be the associated bundle over the one point compactification
I¯ of I (see example 2). Then to give a morphism φ : G → G is the same as
to give a convergent (see §1) family of group homomorphisms φi : Gi → G.
Therefore we obtain a canonical isomorphism
∗ G ∼= ∗ G .
I¯
i∈I
i
¯
If I is finite, it is compact and we can omit the point ∗ ∈ I.
More generally, from the universal property follows the
(4.3.9) Proposition. Let (G, p, T ) be a bundle of pro-c-groups and assume that
S
S
T = T1 . · · · . Tk is a finite disjoint decomposition of T into closed and open
subsets. Then there is a canonical isomorphism
∗G = ∗G
T
T1
1
∗···∗
∗G ,
Tk
k
where Gi denotes the restricted bundle p−1 (Ti ) for i = 1, . . . , k.
∗) The notion of a profinite operator group was introduced by H.KOCH
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261
§3. Generalized Free Products
One of the basic facts of the theory of profinite groups is the assertion that
a topological group which is profinite as a topological space is already the
inverse limit of finite groups (see (1.1.3)). The following theorem extends this
fact to continuous families of profinite groups.
We call a bundle of pro-c-groups finite if it is finite as a space, i.e. it is a
finite disjoint union of groups in c.
(4.3.10) Theorem. Every pro-c-group bundle is the inverse limit of a system
of finite bundles with surjective transition morphisms.
Proof: Denote the given bundle by (G, p, T ). As a topological spaces, G and
T are inverse limits of finite discrete spaces. We will make use of the following
claim:
S
S
For every pair of disjoint decompositions G = U1 . · · · . Un of G and
S
S
T = V . · · · . V of T into open and closed subsets there exists a
1
m
morphism
φ : (G, p, T ) → (H, p0 , S)
to a finite bundle (H, p0 , S), such that the decompositions of G (resp. T )
into the pre-images of the (finitely many) points of H (resp. S) are finer
than the given decomposition.
The proof of this claim will be given below.
Observe that the set of finite quotients of (G, p, T ) is filtered. Indeed, assume
that we are given two finite quotients (Hi , pi , Si ) (i = 1, 2) of (G, p, T ). Then
we first take their pullbacks to a suitable chosen common finite basis S and
then we take the fibre product over S. Now it follows from the claim that the
canonical morphism from (G, p, T ) to the inverse limit of its finite quotients is
an isomorphism.
It remains to show the claim. We construct φ in several steps.
Step 1. Fix a point t ∈ T . Then, by (1.1.3), there exists a finite group
Ht and a homomorphism φt : Gt → Ht , such that the decomposition of Gt
into the pre-images of the elements of Ht is finer than the decomposition
S
Gt = ni=1 (Ui ∩ Gt ). (In order to be accurate, we should remove the indices i
with Ui ∩ Gt = ∅.)
Step 2. We can extend φt : Gt → Ht to a bundle morphism φt : GWt → Ht ,
where Wt is a sufficiently small open and closed neighbourhood of t ∈ T and
GWt denotes the restriction of G to Wt (see exercise 1 below).
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Chapter IV. Free Products of Profinite Groups
Step 3. Making Wt smaller, if necessary, we may assume that the decomposition of GWt into the pre-images of the elements of Ht is finer than the
S
decomposition GWt = ni=1 (GWt ∩ Ui ).
Step 4. The open and closed sets Wt (t ∈ T ) cover T . Replacing Wt by
Wt ∩ Vj (j = 1, . . . , m), we may assume that every Wt is contained in one of
the closed and open subsets V1 , . . . , Vm . Since T is compact, there exists a
finite subcovering T = Wt1 ∪ · · · ∪ Wtk , which we may assume to be disjoint,
Sj−1
by replacing Wtj by Wtj \ ν=1
Wtν .
S
S
.
Step 5. The bundle H
· · · . H (together with the obvious structure map
t1
tk
to the set of k elements) is a finite bundle to which we have a bundle morphism
from (G, p, T ) of the required type.
2
The next proposition is the natural generalization of (4.1.2).
(4.3.11) Proposition. Let
(G, p, T ) = lim (Gi , pi , Ti )
←−
i∈I
be the inverse limit of the pro-c-group bundles Gi . Then
∗ G = lim ∗ G .
←− Ti
i∈I
T
i
Proof: It suffices to show that the inverse limit on the right-hand side satisfies
the universal property of the free product with respect to homomorphisms into
finite groups in c. To begin with, we show the statement in the special case
that all bundles Gi are finite and all transition maps are surjective. In this case,
we have equalities for every finite group H ∈ c :
Hom(
∗ G, H)
T
= Mor((G, p, T ), H) = Mor(lim (Gi , pi , Ti ), H)
←−
i
= lim Mor((Gi , pi , Ti ), H) = lim Hom(
−→
i
= Hom(lim
∗ G , H).
←− Ti
i
−→
i
∗ G , H)
Ti
i
i
Returning to the general case, represent every Gi as the inverse limit of its finite
quotients: Gi = lim Gij . Every morphism
←− j
φi1 ,i2 : Gi1 → Gi2
is realized as a limit of morphisms on finite levels. Therefore we can replace Gi
by the system Gij without changing either side in the stated equality. In other
words (changing the index set), we may assume that all Gi are finite. Again,
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263
§3. Generalized Free Products
without changing the projective limit, we can add the images of all transition
maps to the inverse system and since all Gi are finite, the intersections
\
im(Gj ) ⊆ Gi
j ≥i
stabilize and set up a cofinal subsystem. Finally, we arrive at the point that all
bundles in the system are finite and all transition maps are surjective, and this
special case was already solved at the beginning of the proof.
2
Using (4.3.10) and (4.3.11), many results for finite free products carry over
to the case of generalized free products. For example, we have the
(4.3.12) Proposition. Let ω : G → G, G =
∗ G be a free pro-c-product.
T
Then
(i) ω|Gt : Gt → G is an injective group homomorphism for every t ∈ T ,
(ii) if s, t ∈ T and g
∈
G satisfy ω(Gs ) ∩ ω(Gt )g =/ 1, then s = t.
Proof: Since all statements are compatible with inverse limits, we may
assume by (4.3.10), (4.3.11) that G is finite. Then (i) is trivial by the results of
§1. In order to prove (ii), assume s =/ t. It suffices to construct a group H and
a homomorphism f : G → H whose restriction to ω(Gs ) is injective and such
that the images of ω(Gs ) and gω(Gt )g −1 have trivial intersection in H. For
example, we can choose H := Gs × Gt . Let f be the homomorphism which
corresponds via the universal property to the bundle homomorphism which
is given by Gs → H, gs 7→ (gs , 1), Gt → H, gt 7→ (1, gt ), and the trivial
homomorphism on Gu for every other u. Then we have f (ω(Gs )) = Gs × 1
and f (gω(Gt )g −1 ) = f (g)f (ω(Gt ))f (g)−1 = 1 × Gt .
2
Remarks: 1. In fact a much stronger statement than (4.3.12) is true: the
canonical homomorphism from the discrete free product of the Gt to G = G
∗
T
is injective. Using (4.3.10) and (4.3.11), this can be deduced from the results
in [56].
2. There are also variants of the Kurosh subgroup theorem for the generalized
free products (see [56], [74], [135], [261]).
Now we calculate the cohomology of free products. We start with the
definition of restricted direct sums.
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Chapter IV. Free Products of Profinite Groups
(4.3.13) Definition. Let (G, T ) be a bundle of pro-c-groups, G =
a torsion G-module. For each n ≥ 0 we denote by
M0
∗
TG
and A
H n (Gt , A)
Q t∈T n
H (G
the subgroup of elements in t∈T
t , A) which are in the image of some
map
M
Y
H n (G0t0 , A0 ) −→
H n (Gt , A),
t0 ∈T 0
0
t∈T
0
where (G , T ) is a finite quotient bundle of (G, T ) and A0 is a finite submodule
of A which is a T 0 G 0 -module.
∗
Remark: If (Gi )i∈I is a discrete family of pro-c-groups and G is the associated
bundle over the one point compactification I¯ = I ∪ {∗} of the discrete set I,
see example 2 page 257, then
M0
H n (Gi , A) =
i∈I¯
M
H n (Gi , A).
i∈I
In many arithmetical applications the considered bundle will be the projective
limit of bundles which are associated to discrete families (Gλ,i )i∈Iλ , λ ∈ Λ,
of pro-c-groups, i.e. (G, T ) is the projective limit of the associated bundles
(Gλ , I¯λ ) over the one point compactifications I¯λ = Iλ ∪ {∗} of the discrete sets
Iλ , λ ∈ Λ.
Assume in this situation that the torsion T G-module A is a I¯λ Gλ -module
L
L0
n
for all λ ∈ Λ. Since 0t∈T H n (Gt , A) = lim
i∈I¯λ H (Gλ,i , A), we have in
−→ λ∈Λ
this case
M
M0
H n (Gt , A) = lim
H n (Gλ,i , A).
∗
∗
−→
λ∈Λ i∈Iλ
t∈T
(4.3.14) Theorem. Let (G, T ) be a bundle of pro-c-groups, G =
a torsion G-module. Then we have a canonical exact sequence
0 → AG (c) → A(c) →
M0
M0
t∈T
t∈T
(A/AGt )(c) → H 1 (G, A) →
∗
TG
and A
H 1 (Gt , A) → 0,
and for all n ≥ 2
H n (G, A) ∼
=
M0
H n (Gt , A).
t∈T
Proof: By a direct limit argument, we may assume that A is finite. The
composite map G → G → Aut(A) is continuous, hence locally constant.
Using the claim in the proof of (4.3.10), we find a presentation
(G, p, T ) = lim
(Gi , pi , Ti )
←−
i∈ I
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§3. Generalized Free Products
S
of the bundle G as an inverse limit of finite bundles Gi = . s∈Ti Gi,s , i ∈ I,
such that A is a Ti Gi -module for all i. We set Gi = Ti Gi . By (4.3.11),
G = lim Gi , and by (4.1.4), we obtain an exact sequence
∗
∗
←− i∈I
0 → A/AG → lim
−→
M
A/AGi,s → H 1 (G, A) → lim
M
−→
i∈I s∈Ti
i∈I s∈Ti
H 1 (Gi,s , A) → 0
and isomorphisms for all n ≥ 2
M
∼ lim
H n (G, A) −→
H n (Gi,s , A).
−→
i∈I s∈Ti
For n ≥ 1 we consider the natural map
φ: lim
M
−→
i∈I s∈Ti
H n (Gi,s , A) −→
M0
H n (Gt , A).
t∈T
Each finite quotient bundle (G 0 , p0 , T 0 ) of (G, p, T ) is a quotient of (Gi , pi , Ti ) for
L
sufficiently large i ∈ I. Therefore φ is surjective. Let x ∈ s∈Ti0 H n (Gi0 ,s , A)
L
n
represent a nonzero element in lim
s∈Ti H (Gi,s , A). Then, for each i ≥ i0 ,
−→ i∈I
the set Si ⊆ Ti of elements s ∈ Ti such that the s-component of the image of
L
x in s∈Ti H n (Gi,s , A) is nonzero is nonempty. As the inverse limit of finite
nonempty sets is nonempty, φ is injective, hence an isomorphism. Similarly,
the natural map
M0
M
A/AGt
lim
A/AGi,s −→
−→
i∈I s∈Ti
t∈T
is an isomorphism. This completes the proof of the theorem.
2
(4.3.15) Corollary. Let (G, p, T ) be a bundle of pro-c-groups and let p be a
prime number. Then
cdp G = sup {cdp Gt } and scdp G = sup {scdp Gt }.
∗
T
∗
T
t∈T
t∈T
Proof: By (4.3.14), we obtain the inequality cdp G ≤ sup t∈T {cdp Gt }. By
(4.3.12), each Gt is a closed subgroup of G, hence, by (3.3.5), cdp G ≥
sup t∈T {cdp Gt }. It remains to show the statement on the strict cohomological
dimension. By the same argument as for cdp , we obtain the inequality scdp G ≥
sup t∈T {scdp Gt } =: n. Assume that n is finite. If all Gt are trivial, so is the
free product. As profinite groups of strict cohomological p-dimension 1 do not
exist, we may assume n ≥ 2. We claim that H m (G, A)(p) = 0 for all discrete
G-modules A and all m ≥ n + 1. By the first part of the proof,
cdp G = sup {cdp Gt } ≤ sup {scdp Gt } = n.
t∈T
t∈T
Therefore we may assume that A is torsion-free, hence ZZ-flat. Tensoring the
exact sequence 0 → ZZp → Qp → Qp /ZZp → 0 by A and using the fact that
uniquely divisible modules are cohomologically trivial, we obtain
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Chapter IV. Free Products of Profinite Groups
H m (G, A)(p) ∼
= H m (G, A) ⊗ ZZp ∼
= H m (G, A ⊗ ZZp )
∼
= H m−1 (G, A ⊗ Qp /ZZp )
∼
=
M0
H m−1 (Gt , A ⊗ Qp /ZZp ).
t∈T
For each t ∈ T , we have H m−1 (Gt , A ⊗ Qp /ZZp ) ∼
= H m (Gt , A)(p) = 0. Hence
H m (G, A)(p) = 0, and scdp G = n.
2
Exercise 1. Let (G, p, T ) be a bundle of pro-c-groups and assume that for some t ∈ T we
are given a homomorphism f : Gt → H into a finite group H ∈ c. Show that there exists a
closed and open neighbourhood U of t in T and a morphism F : GU → H extending f , i.e.
F |Gt = f .
Hint: By I §1, ex.2 there exists a continuous map F : G → H extending f . The two
maps from G ×T G → H which are given by (g1 , t) × (g2 , t) 7→ F (g1 , t) · F (g2 , t) and by
(g1 , t) × (g2 , t) 7→ F ((g1 , t) · (g2 , t)) coincide on an open neighbourhood of Gt ⊆ G ×T G.
Exercise 2. Assume that the profinite space T is the topological quotient of the profinite space
T 0 by the action of the profinite group P . Let G be a bundle of profinite groups over T and let
G 0 := G ×T T 0
0
be the pull-back of G to T . We set G := ∗ G and G0 := ∗ G 0 . Then for every abelian group A,
T
T0
the cohomology groups H i (G0 , A) are discrete P -modules in a natural way. Show that there
exist isomorphisms for all i
H i (G0 , A) = IndP H i (G, A).
The next exercises deal with profinite operator groups. For more information about operator
groups, we refer the reader to [256] and [136].
Exercise 3. Let Γ be a profinite group, G a pro-c-group and let
φ : Γ → Aut(G)
be a homomorphism. Show that the action: Γ × G → G, (γ, g) 7→ φ(γ)(g), is continuous
if and only if G possesses a system of neighbourhoods of the identity consisting of open
Γ -invariant normal subgroups.
Exercise 4. Prove the assertion that the free pro-c-Γ operator group ∗Γ Fr is equal to the
maximal pro-c-factor group of the kernel of the surjection Fr ∗ Γ Γ , where ∗ denotes the
free product in the category of profinite groups.
Hint: See [94] Satz 3.4.
Exercise 5. Let p be a prime number and let Γ = ZZp be the additive group of p-adic integers.
Let G be a pro-p-Γ operator group. Show that the following conditions are equivalent:
(i) G is a free pro-p-Γ operator group of rank r,
(ii) G is a free pro-p-group, the fixed module (Gab )Γ is trivial and the cofixed module Gab
Γ
is a free ZZp -module of rank r.
Hint: In order to show the difficult implication (ii) ⇒ (i), choose a homomorphism
φ : ∗Γ Fr → G
∼
which induces an isomorphism ZZrp → Gab
Γ . Use (1.6.14) and the topological Nakayama
lemma (see 5.2.18) in order to show that φ is surjective. Then apply Γ -homology to the exact
sequence 0 → (ker φ)ab
Zp [[Γ ]]r → Gab → 0 and observe that H1 (Γ, Gab ) = (Gab )Γ = 0.
G →Z
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Chapter V
Iwasawa Modules
The Iwasawa algebra, usually denoted by the Greek letter Λ, is the complete
group algebra ZZp [[Γ ]] of a group Γ , which is noncanonically isomorphic to
∼ Z
ZZp . This means that we will not specify a particular isomorphism φ : Γ →
Zp
or, equivalently, we will not fix a topological generator γ of the procyclic
group Γ .
We can view Λ from many different algebraic aspects. By definition, it is
a complete group algebra and we will see that, choosing a generator γ ∈ Γ ,
it can be identified with a power series ring over ZZp . However, Λ is also a
commutative, two-dimensional regular local ring, it is a complete local ring
and it is a compact ZZp -algebra. Using all these properties of Λ, we will develop
the structure theory for Λ-modules, so-called Iwasawa modules. With an eye
on applications to generalized Iwasawa algebras, we will treat each aspect of
Λ separately and in greatest possible generality as long as this does not require
additional work. We have, however, omitted the interpretation of the elements
of Λ as ZZp -valued measures on Γ . For this point of view, which is of particular
importance for the analytic part of Iwasawa theory, we refer the reader to [122]
or [246].
In the first and third sections (after preparing some basic properties of
complete group rings in section 2) we will treat the classification of modules up to pseudo-isomorphism which was developed by K. IWASAWA and
J.-P. SERRE. To obtain a finer structure theory, we will apply the homotopy
theory of modules, developed in section 4, to the case of Λ-modules. This will
be done in section 5 closely following work of U. JANNSEN. Finally, in section 6
we will investigate some further properties of complete group algebras which
will be needed in the arithmetic applications.
Throughout this chapter, we will assume basic knowledge of commutative
algebra, as can be found in [16] or in many other text books about this subject.
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268
Chapter V. Iwasawa Modules
§1. Modules up to Pseudo-Isomorphism
I. In this subsection, let A be a commutative, noetherian and integrally closed
domain with quotient field K. For every prime ideal p in A, one has a canonical
embedding Ap ,→ K and Ap is integrally closed.
Let P (A) be the set of prime ideals of height ht(p) = dim Ap = 1. Since A
is integrally closed, the localization Ap with respect to p ∈ P (A) is a discrete
valuation ring and
\
A=
Ap
p∈P (A)
(see [16], chap. VII, §1, no.6, th. 4).
(5.1.1) Definition. An A-module M is called reflexive if the canonical map
ϕM : M −→ M ++ = HomA (HomA (M, A), A) ,
m 7−→ ϕM (m) : α 7−→ α(m) ,
of M to its bidual is an isomorphism.
Remark: A reflexive module is torsion-free because the dual M + =
HomA (M, A) of an A-module M is always torsion-free.
If M is a finitely generated and torsion-free A-module, then the localization
Mp of M with respect to a prime ideal p of A is a torsion-free Ap -module and
we have injections
M ,→ Mp ,→ Mp ⊗Ap K = M ⊗A K =: V
M + ,→ (M + )p ,→ (M + )p ⊗Ap K = M + ⊗A K = HomK (V, K) =: V ∧ .
We see that
M+ ∼
= {λ ∈ V ∧ | λ(m) ∈ A for all m ∈ M }
(M + )p ∼
= {λ ∈ V ∧ | λ(m) ∈ Ap for all m ∈ Mp } ∼
= (Mp )+ .
Identifying (M + )p and (Mp )+ , we write Mp+ for this module and consider it
as a submodule of V ∧ .
(5.1.2) Lemma. Let M be a finitely generated torsion-free A-module. Then
\
(i) M + =
Mp+ .
p∈P (A)
(ii) M
++
=
\
Mp ,
p∈P (A)
(iii) M
=
\
Mp
if and only if M is reflexive.
p∈P (A)
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§1. Modules up to Pseudo-Isomorphism
\
Proof: Let λ ∈ Mp+ . Then for every m ∈ M we get λ(m) ∈ Ap for all
p ∈ P (A), so that λ(m) ∈ A and therefore λ ∈ M + . This proves (i) because the
other inclusion is obvious.
Since Mp is a finitely generated torsion-free module over the discrete valuation ring Ap , p ∈ P (A), Mp is free and Mp → Mp++ is an isomorphism. Via the
identification of V with V ∧∧ , we prove (ii), from which (iii) follows immediately.
2
(5.1.3) Corollary. If M is finitely generated, then M + is reflexive.
(5.1.4) Definition. A finitely generated A-module M is called pseudo-null if
the following equivalent conditions are fulfilled:
(i) Mp = 0 for all prime ideals p in A of height ht(p) ≤ 1.
(ii) If p is a prime ideal with a = annA (M ) ⊆ p, then ht(p) ≥ 2.
Remarks: 1. We have Mp = 0 if and only if there is an s ∈ A r p such that
sM = 0, hence annA (M ) 6⊂ p. This shows the above equivalence.
2. A pseudo-null module is torsion because M(0) = M ⊗A K = 0.
3. If A is a Dedekind domain, then M is pseudo-null if and only if M = 0.
4. If A is a 2-dimensional, noetherian, integrally closed local domain with
finite residue field, then M is pseudo-null if and only if M is finite. Indeed,
if M is finite, then there exists an r ∈ IN such that mr M = 0, hence supp(M )
⊆ {m},where m denotes the maximal ideal of A. Conversely, if supp(M ) =
{p ∈ Spec A | a ⊆ p} is contained in {m}, then mr ⊆ a for some r ∈ IN and M
is a finitely generated A/mr -module. But A/mr is finite, thus M is finite.
(5.1.5) Definition. A homomorphism f : M → N of finitely generated Amodules is called a pseudo-isomorphism if ker(f ) and coker(f ) are pseudonull or, equivalently, if
∼ N
fp : Mp −→
p
is an isomorphism for all p of height ≤ 1. We write
≈
f : M −→ N.
(5.1.6) Lemma. Let M be a finitely generated A-torsion module and let α ∈ A
be a non-zero element such that supp(A/αA) is disjoint to supp(M ) ∩ P (A).
Then the multiplication on M by α is a pseudo-isomorphism.
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270
Chapter V. Iwasawa Modules
Proof: This is clear, since α is a unit in Ap for every p
and Mp = 0 for p = (0) or p ∈ P (A) r supp(M ).
∈
supp(M ) ∩ P (A)
2
For a finitely generated A-module M let
TA (M ) be the torsion submodule and
FA (M ) = M/TA (M ) be the maximal torsion-free quotient of M .
(5.1.7) Proposition. If the A-module M is finitely generated, then
(i) there exists a pseudo-isomorphism
≈
f : M −→ TA (M ) ⊕ FA (M ),
(ii) there exists a finite family (pi )i∈I of prime ideals of height 1 in A, a finite
family (ni )i∈I of natural numbers and a pseudo-isomorphism
≈
g : TA (M ) −→
M
A/pni i .
i∈I
The families (pi ) and (ni ) are uniquely determined by TA (M ) up to
renumbering.
Proof: Let {p1 , . . . , ph } = supp(TA (M )) ∩ P (A). If h = 0, then TA (M ) is
pseudo-null and the maps f : M can FA (M ) and g : TA (M ) −→ 0 are of the
T
S
required form. Now let h > 0 and S = hi=1 A r pi = A r hi=1 pi . Then S −1 A
is a semi-local Dedekind domain and therefore a principal ideal domain.
The S −1 A-module S −1 TA (M ) is the torsion module of S −1 M . Using
the structure theorem for modules over principal ideal domains, we see that
S −1 TA (M ) is a direct summand of S −1 M . Since M is finitely generated, we
have
HomS −1 A (S −1 M, S −1 TA (M )) = S −1 HomA (M, TA (M )).
Hence there exists a morphism f0 : M → TA (M ) and s0 ∈ S such that
f0
: S −1 M −→ S −1 TA (M )
s0
is the projector of S −1 M onto its direct summand S −1 TA (M ). Therefore
f0
|
= idS −1 TA (M ) and thus there exists an s1 ∈ S such that for f1 = s1 f0
s0 S −1 TA (M )
f1 |TA (M ) = s1 s0 idTA (M ) .
Now let
f = (f1 , can) : M −→ TA (M ) ⊕ FA (M ) .
The commutative exact diagram
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271
§1. Modules up to Pseudo-Isomorphism
0!#$%&'"
TA (M )
M
f1 |TA (M )
0
TA (M )
FA (M )
0
FA (M )
0
f
TA (M ) ⊕ FA (M )
shows ker(f ) = ker(f1 |TA (M ) ) and coker(f ) = coker(f1 |TA (M ) ). But f1 |TA (M ) is
a pseudo-isomorphism by (5.1.6). This proves (i).
In order to prove (ii), let
E :=
ri
h M
M
n
A/pi ij
i=1 j=1
for natural numbers nij such that there exists an isomorphism
∼ S −1 E.
g0 : S −1 TA (M ) −→
We use again the structure theorem for modules over principal ideal domains,
and the fact that S −1 A is a semi-local ring with maximal ideals S −1 pi , i =
1, . . . , n. Using
HomS −1 A (S −1 TA (M ), S −1 E) = S −1 HomA (TA (M ), E),
we obtain a morphism g : TA (M ) → E and an s ∈ S such that g = sg0 . Again
by (5.1.6) we see that g is a pseudo-isomorphism.
2
Remarks: 1. The same argument as in the proof of (ii) shows that for a
pseudo-isomorphism
≈
f : M −→ N
of finitely generated torsion modules there exists a pseudo-isomorphism
≈
g : N −→ M.
Therefore we will use in this case the notion M ≈ N .
For general finitely generated A-modules, the existence of a pseudo-isomor≈
phism M −→ N does not imply the existence of a pseudo-isomorphism in the
other direction; see §3, ex.1 for an example.
2. Let
0 −→ M 0 −→ M −→ M 00 −→ 0
be an exact sequence of finitely generated A-torsion modules such that the
associated sets of prime ideals of height 1 of M 0 and M 00 are disjoint. Then
there exists a pseudo-isomorphism
≈
M −→ M 0 ⊕ M 00 .
The proof is similar to the proof of (5.1.7) (ii).
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Chapter V. Iwasawa Modules
(5.1.8) Proposition. Let M be a finitely generated torsion-free A-module.
Then there exists an injective pseudo-isomorphism of M into a reflexive Amodule M 0 .
Proof: The canonical morphism
ϕM : M −→ M ++
is a pseudo-isomorphism because Mp is a free finitely generated Ap -module,
∼ M ++ for all p of height ≤ 1. Furthermore, by (5.1.3), M ++ is
hence Mp →
p
reflexive and ker(ϕM ) ⊗A K = 0, thus ker(ϕM ) is torsion and therefore zero.
2
II. Now let A be a 2-dimensional regular local ring ∗) . The following
proposition is essential for the structure theory of Iwasawa modules.
(5.1.9) Proposition. Let A be an n-dimensional regular local ring, 2 ≤ n < ∞,
let (p1 , . . . , pn ) be a regular system of parameters generating the maximal ideal
of A and let p0 := 0. For a finitely generated A-module M , the following
assertions are equivalent.
(i) For every i = 0, . . . , n − 2, the A/(p0 , . . . , pi )-module M/(p0 , . . . , pi )M
is reflexive.
(ii) M is a free A-module.
In particular, a reflexive A-module M over a 2-dimensional regular local ring
A is free.
Proof (DIEKERT [40]): In order to prove the nontrivial implication we assume
(i). In particular, M is reflexive, hence torsion-free. Therefore multiplication
M is a minimal free presentation of M ,
by p1 is injective on M . If ϕ : ( Ar
then we obtain the following commutative exact diagram
0.+,-)*/2301
Ar
p1
ϕ
0
M
Ar
ϕ
p1
M
(A/p1 )r
0
ϕ̄
M/p1
0.
∗) For the definition and properties of regular local rings and regular systems of parameters
see [129].
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273
§2. Complete Group Rings
Assume that M/p1 is a free A/p1 -module. Then Nakayama’s lemma implies that ϕ̄ is an isomorphism, hence multiplication by p1 on ker(ϕ) is an
isomorphism. Again by Nakayama’s lemma, we obtain ker(ϕ) = 0.
It remains to show that M/p1 is a free A/p1 -module. We are reduced to the case
n = 2 since for n > 2 we argue by induction applied to the (n − 1)-dimensional
regular local ring A/p1 , the regular system of parameters (p̄2 , . . . , p̄n ), where
p̄i = pi + p1 A, and the module M/p1 .∗)
So let n = 2. Thus A/p1 is regular of dimension 1, i.e. a discrete valuation ring, in particular, an integral domain. Therefore the A/p1 -module
HomA (M + , A/p1 ) is torsion-free. Since A is an integral domain, the map
M ++ /p1 = HomA (M + , A) ⊗ A/p1 ,→ HomA (M + , A/p1 )
is injective and, because M is reflexive, we see that M/p1 = M ++ /p1 is a
torsion-free module over the discrete valuation ring A/p1 , so that M/p1 is free.
This finishes the proof of the proposition.
2
From (5.1.7), (5.1.8) and (5.1.9) we obtain the
(5.1.10) Structure Theorem. Let A be a 2-dimensional regular local ring and
let M be a finitely generated A-module. Then there exist finitely many prime
ideals pi , i ∈ I, of height 1, a nonnegative integer r, natural numbers ni ∈ IN
and a pseudo-isomorphism
≈
f : M −→ Ar ⊕
M
A/pni i .
i∈I
The prime ideals pi and the numbers r, ni are uniquely determined by M :
r = dimK M ⊗A K,
{pi | i ∈ I} = supp(M ) ∩ P (A).
§2. Complete Group Rings
In this section, let O be a commutative local ring which is complete in its
m-adic topology, where m is the maximal ideal. We further assume that O/mn
is finite for all n; in particular, O is compact. Let p be the characteristic of the
finite residue field k = O/m.∗∗) . Furthermore, let G be a profinite group.
∗) see [129], th. 14.2.
∗∗) In the applications O will always be the ring of integers in a finite extension field K|Q .
p
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Chapter V. Iwasawa Modules
(5.2.1) Definition. The complete group algebra of G over O is the topological inverse limit
O[[G]] := lim O[G/U ],
←−
U
where U runs through the open normal subgroups of G.
Since O[[G]] is a compact O-algebra, the assignment G → O[[G]] defines
a covariant functor from the category of profinite groups to that of compact
O-algebras. In particular, if N ⊆ G is a closed normal subgroup, then we
4
have an epimorphism O[[G]]
O[[G/N ]], whose kernel I(N ) is a two-sided
closed ideal in O[[G]]. It is the closed left (right) ideal generated by the
elements x − 1, x ∈ N . For the particular case N = G, we set
(5.2.2) Definition. The kernel IG := I(G) of the canonical epimorphism, the
augmentation map
5
O[[G]]
O,
is called the augmentation ideal of G.∗)
By a (left) O[[G]]-module M , we always understand a separated topological
module, i.e. M carries the structure of a Hausdorff abelian topological group
and the structure of an O[[G]]-module such that the action O[[G]] × M → M
is continuous. In other words, M is a Hausdorff topological O-module with a
continuous G-action. By
MG = M/IG M,
we denote the maximal quotient module of M on which G acts trivially. We
call MG the module of coinvariants of M (cf. II §6). If M is an O[[G]]-module,
then a finitely generated submodule N ⊆ M is the continuous homomorphic
image of the compact module O[[G]]n for some n, and hence closed. Thus the
notions ‘topologically finitely generated’ and ‘finitely generated’ coincide for
O[[G]]-modules, and finitely generated modules are compact.
If we are given a left O[[G]]-module M , we can define a right O[[G]]module M 0 keeping the O-module structure and letting g ∈ G act as g −1 .
This establishes an equivalence between the categories of left and right O[[G]]modules, and we will sometimes ignore the difference between left and right
modules using this natural equivalence.
The category C of compact O[[G]]-modules and the category D of discrete
O[[G]]-modules will be of particular importance. Both are abelian categories,
∗) The reader should not confuse I
N
subgroup N ⊆ G.
⊆
O[[N ]] with I(N )
⊆
O[[G]] for a closed normal
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275
§2. Complete Group Rings
and Pontryagin duality defines a contravariant equivalence of categories between C and D.
(5.2.3) Definition. For a set X we define the free compact O[[G]]-module
with basis X as
Y
F (X) = O[[G]]X =
O[[G]]
x∈X
6
with the product topology. If F (X)
M is a continuous surjection onto
a compact O[[G]]-module M , then we call the image of X in M a set of
topological generators.
Remark: The compact O[[G]]-module F (X) has the following universal property: for every compact module M ∈ C and every convergent family (mx )x∈X
of elements of M (i.e. for every open neighbourhood U of 0 ∈ M , one has
mx ∈ U for all but finitely many x ∈ X) there exists a unique continuous O[[G]]module homomorphism f : F (X) → M such that f (1x ) = mx (compare with
the non-abelian situation in IV §1).
(5.2.4) Proposition. (i) Every compact O[[G]]-module is the projective limit
of finite modules, in particular, it is an abelian pro-p-group. The category C
has sufficiently many projectives and exact inverse limits.
(ii) Every discrete O[[G]]-module is the direct limit of finite modules, in particular, it is an abelian p-torsion group. The category D has sufficiently many
injectives and exact direct limits.
Proof: Assume that N ∈ D and n ∈ N . Then annO[[G]] (n) is an open ideal
in O[[G]]. Therefore O[[G]] · n ⊆ N is an O/mk [G/U ]-module for some k
and some open normal subgroup U ⊆ G. This shows the first statement of (ii)
and therefore also the first statement of (i) by duality. Hence every M ∈ C
has a convergent set of topological generators. This implies that free compact
modules are projective and that every compact module is the quotient of a
free module. By duality we find that D has sufficiently many injectives. The
statement about the exactness of limits only depends on the underlying abelian
topological groups and is well-known for discrete modules. It follows for
compact modules by duality.
2
(5.2.5) Corollary. A compact O[[G]]-module has a fundamental system of
neighbourhoods of zero consisting of open submodules.
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Chapter V. Iwasawa Modules
A tensor product for compact O[[G]]-modules is defined by its universal
property. Explicitly, let M be a compact right and N be a compact left
O[[G]]-module. Then the complete tensor product is a compact O-module
ˆ O[[G]] N coming along with an O[[G]]-bihomomorphism ∗)
M⊗
ˆ O[[G]] N
α : M × N −→ M ⊗
with the following property: given any O[[G]]-bihomomorphism f of M × N
into a compact O-module R, there is a unique O-module homomorphism
ˆ → R such that f = g ◦ α.
g : M ⊗N
The complete tensor product is constructed as follows:
ˆ O[[G]] N = lim M/U ⊗O[[G]] N/V,
M⊗
←−
U,V
where U (resp. V ) run through the open O[[G]]-submodules of M (resp. N ).
ˆ is a compact O-module.
Observe that M/U and N/V are finite, so that M ⊗N
The natural bihomomorphisms M × N → M/U ⊗ N/V induce the desired
ˆ by passing to the limit. The exact
bihomomorphism α : M × N → M ⊗N
sequence
0 −→ im(M ⊗ V + U ⊗ N ) −→ M ⊗ N −→ M/U ⊗ N/V −→ 0
ˆ is the completion of M ⊗ N in the topology induced by
shows that M ⊗N
taking im(M ⊗ V + U ⊗ N ) as a fundamental system of open neighbourhoods
of 0.
ˆ O[[G]] − by
We denote the left derived functors of the (right exact) functor −⊗
O[[G]]
∼
ˆ O[[G]] M induces the
Tor
(−, −). The canonical isomorphism MG = O⊗
.
(5.2.6) Proposition. There are canonical isomorphisms for all i
M ∈ C:
Hi (G, M ) ∼
(O, M ).
= TorO[[G]]
i
≥
0 and all
Proof: Since both functors agree for i = 0, it suffices to show that a free
O[[G]]-module F has trivial G-homology and one easily reduces to the case
F = O[[G]]. We have
Hi (G, O[[G]]) = lim
Hi (G/U, O[G/U ]),
←−
U ⊆G
where U runs through the open normal subgroups in G. But the G/U -module
O[G/U ] ∼
2
= O ⊗ ZZ ZZ[G/U ] is induced, hence homologically trivial.
∗) i.e. α is a continuous O-homomorphism such that α(mλ, n) = α(m, λn) for m ∈ M, n ∈
N and λ ∈ O[[G]].
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277
§2. Complete Group Rings
Since C (resp. D) is an abelian category with sufficiently many projectives
(resp. injectives), we have Ext-functors
.
Ext.
ExtO[[G]] (−, −) : C × C −→ Ab
O[[G]] (−, −)
: D × D −→ Ab
in the usual way. In the following, we will make use of intermediate homomorphism groups. Let M ∈ C , N ∈ D and let f : M → N be an O[[G]]homomorphism. Then f has finite image which is therefore invariant under an
open ideal in O[[G]]. Thus we have a functor
HomO[[G]] (−, −) : C × D −→ (discrete O-modules)
and we can use either projective resolutions in C or injective resolutions in D
to define the functors
.
ExtO[[G]] (−, −) : C × D −→ (discrete O-modules).
Noting the canonical isomorphism
NG ∼
= HomO[[G]] (O, N )
for N
∈
D, we have the
(5.2.7) Proposition. There are canonical isomorphisms
H i (G, N ) ∼
(O, N )
= Exti
O[[G]]
for all i ≥ 0 and all N
∈
D.
Proof: For N ∈ D the induced module IndG (N ) is also in D, i.e. carries the
structure of an O[[G]]-module in a natural way. By the arguments of II §6, we
therefore see that the functor H (G, −) is universal as a δ-functor on D. The
2
same is true for ExtiO[[G]] (O, −) and both functors agree in degree 0.
.
(5.2.8) Proposition. If M = lim Mi
←−
i∈I
∈
C and N = lim Nj
−→
j ∈J
∈
D, then
ExtnO[[G]] (M, N ) = lim ExtnO[[G]] (Mi , Nj )
−→
i,j
for every n ≥ 0.
Proof: It suffices to show that the functor commutes with limits in the first
and in the second variable separately. Let N ∈ D be fixed and represent
every Mi as an inverse limit over its finite quotients: Mi = lim Mi,k . Every
←− k
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Chapter V. Iwasawa Modules
homomorphism from Mi into N has finite image and therefore factors through
some Mi,k , i.e.
HomO[[G]] (Mi , N ) = lim HomO[[G]] (Mi,k , N ).
−→
k
Thus the statement of the proposition is true for n = 0 in the first variable
if we have a surjective system of finite modules. If i1 ≥ i2 , then for every
7 Mi
Mi2 ,k2 factors through
finite quotient Mi2 ,k2 of Mi2 , the map Mi1 −→
2
some finite quotient Mi1 ,k1 of Mi1 . Therefore we can write M in the form
M = lim
Mi,k and we have
←−
i,k
lim
HomO[[G]] (Mi , N ) = lim
HomO[[G]] (Mi,k , N )
−→
−→
i
i,k
from what we have already shown. By this consideration, it suffices to consider
the case that all Mi are finite. But then we may change the projective system
T
Mi to i0 ≥i im(Mi0 → Mi ) again without changing the limits. Thus we have
reduced to the case of a system of finite modules with surjective transition maps.
As shown above, in this case Hom(−, −) commutes with inverse limits in the
first argument and using a fixed injective resolution of the second argument
shows the same for Ext(−, −). The proof for limits in the second variable is
formally dual to that for the first variable, and will be omitted.
2
(5.2.9) Corollary. For M ∈ C and N ∈ D, there are canonical isomorphisms
for all i ≥ 0
TorO[[G]]
(M, N ∨ ) ∼
= ExtiO[[G]] (M, N )∨ ,
i
where ∨ denotes the Pontryagin dual.
Proof: Let U (resp. V ) run through the open submodules of M (resp. N ∨ ).
Then we obtain by (5.2.8)
ˆ O[[G]] N ∨
M⊗
∨
= lim M/U ⊗O[[G]] N ∨ /V
∨
←−
U,V
= lim
(M/U ⊗O[[G]] N ∨ /V )∨
−→
U,V
= lim HomO[[G]] (M/U, (N ∨ /V )∨ )
−→
U,V
= HomO[[G]] (M, N ).
This proves the corollary using the methods of homological algebra.
From now on we will use the following notational convention:
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§2. Complete Group Rings
If a given ring (with unit) A has a natural topology, we tacitly assume that all Amodules are Hausdorff topological modules. If A is compact, the unspecified
term module will always mean compact A-module.
(5.2.10) Definition. Let A be a ring and let M be an A-module. The projective
dimension pdA M of M is the minimal number n such that there exists a
projective resolution
0 −→ Pn −→ Pn−1 −→ . . . −→ P0 −→ M −→ 0
for M of length n. We set pdA M = ∞ if no such resolution exists. For the
trivial A-module 0, we set pdA 0 = −1.
The projective dimension of the ring A, denoted by pd(A), is defined as
sup{pdA M | M an A-module}.
Observe that simple discrete or compact O[[G]]-modules are finite, i.e. compact and discrete.
(5.2.11) Proposition. For M
(i) pdO[[G]] M ≤ n,
∈
C the following assertions are equivalent:
(ii) Extn+1
O[[G]] (M, N ) = 0 for all simple N ,
(iii) Extn+1
O[[G]] (M, N ) = 0 for all simple N ,
(iv) TorO[[G]]
n+1 (M, N ) = 0 for all simple N .
Proof: Using a projective resolution of M , we have for N simple
n+1
Extn+1
O[[G]] (M, N ) = ExtO[[G]] (M, N ).
Thus (ii) ⇔ (iii). The equivalence (ii) ⇔ (iv) follows from (5.2.9). Using dimension shifting, the remaining equivalence (i) ⇔ (iii) reduces to the
statement
P
∈
C is projective ⇐⇒ Ext1O[[G]] (P, N ) = 0 for all simple N.
Therefore we have to show that P
∈
C is projective if every exact sequence
0 −→ N −→ N 0 −→ P −→ 0
in C with N simple splits. Assume that 0 → A → B → P → 0 is any exact
sequence in C and consider the collection S of pairs (C, s) consisting of a
closed submodule C ⊆ A and a splitting morphism s : P → B/C such that
8
π ◦ s = idP , where π : B/C
P is the canonical projection. Obviously, S
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Chapter V. Iwasawa Modules
T
is not empty and, since B/ Ci = lim B/Ci , it is an inductively ordered set.
←−
By Zorn’s lemma, there exists a minimal element (C, s) in S. If C =/ 0, we
can find an open submodule C 0 ⊆ C such that C/C 0 is simple. By assumption
there exists a morphism t : P → B/C 0 which makes the diagram
9:;<=>
P
t
C/C 0
0
B/C 0
s
B/C
0
commutative. The element (C 0 , t) ∈ S is strictly smaller than (C, s). This
contradiction proves that C = 0, thus P is projective.
2
Using (5.2.11) and (5.2.8), we obtain the
(5.2.12) Corollary. For M = lim
Mi
←−
∈
i
pdO[[G]] M
≤
C , one has
sup{pdO[[G]] Mi }.
i
In particular, the inverse limit of projective modules is projective.
pdO[[G]] O = cdp G.
(5.2.13) Corollary.
Proof: From (5.2.7) and (3.3.2) we obtain the inequality pdO[[G]] O ≤ cdp G.
The other inequality does not follow directly. The difficulty (which does not
occur for O ∼
= ZZp ) lies in the fact that a simple G-module need not be an
O[[G]]-module. We overcome the problem as follows. Let N be a simple
G-module with pN = 0. Then N is a finite dimensional IFp -vector space. By
our assumptions, k = O/m is a finite extension of IFp . Then N ⊗IFp k is an
O[[G]]-module and we obtain for all i
H i (G, N ) ⊗IF k ∼
(O, N ⊗IF k).
= H i (G, N ⊗IF k) ∼
= Exti
p
p
O[[G]]
p
i
Hence H (G, N ) = 0 for i > pdO[[G]] O. This finishes the proof.
2
Now we are in the position to identify continuous cochain cohomology of a
compact O[[G]]-module with Ext-groups.
(5.2.14) Proposition. For M
∈
C , we have isomorphisms
i
(G, M ) ∼
Hcts
= ExtiO[[G]] (O, M ).
for every i ≥ 0.
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§2. Complete Group Rings
Proof: Consider, for every open normal subgroup U
of free O[G/U ]-modules given by
⊆
.
G, the complex P U
PnU = O[(G/U )n+1 ]
U
with differentials dn : PnU → Pn−1
defined for n > 0 by
dn (g0 , . . . , gn ) =
n
X
(−1)i (g0 , . . . , gi−1 , gi+1 , . . . , gn ).
i=0
.
The complex P U is a free resolution of the trivial O[G/U ]-module O. For a
finite O[G/U ]-module A, we have a natural isomorphism of complexes
Hom
(P U , A) ∼
= C (G/U, A),
.
O[G/U ]
.
.
.
where C (G/U, A) is the homogeneous cochain complex of G/U with coefficients in A as defined in I §2. Passing to the inverse limit over U , we obtain
from the complexes P U a complex P which is a resolution of the trivial
O[[G]]-module O by compact O[[G]]-modules. Since every compact O[[G]]module is the inverse limit of finite ones, we obtain natural isomorphisms of
complexes for every M ∈ C
Hom
(P , M ) ∼
= C (G, M ),
.
.
.
O[[G]]
.
.
.
cts
where Ccts (G, M ) is the continuous homogeneous cochain complex of G with
coefficients in M as defined in II §7. By definition, the cohomology of the
complex on the right is continuous cochain cohomology. Thus it remains to
show that Pn is a projective O[[G]]-module for every n ≥ 0, because then the
cohomology of the left complex is the required Ext-group.
In order to prove that Pn is projective, it suffices to show that Ext1O[[G]] (Pn , N )
vanishes for every finite simple N (see (5.2.11)). Let U ⊆ G be open and
normal. A class x ∈ Ext1O[[G]] (PnU , N ) corresponds to an extension of O[[G]]modules
0 −→ N −→ N 0 −→ PnU −→ 0.
Since N is finite, N 0 is an O[G/V ]-module for some open subgroup V ⊆ U
which is normal in G. Consider the pull-back of the above sequence via the
natural surjection
PnV → PnU .
We obtain an exact sequence of O[G/V ]-modules which splits because PnV is
a free O[G/V ]-module. In other words, the image of x in Ext1O[[G]] (PnV , N )
vanishes. Using a projective resolution of the first argument, we see that
for finite N and compact A the groups ExtO[[G]] (A, N ) and ExtO[[G]] (A, N )
coincide. By (5.2.8), we therefore obtain
2
Ext1 (Pn , N ) ∼
= lim Ext1 (P U , N ) = 0.
O[[G]]
−→
U
O[[G]]
n
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Chapter V. Iwasawa Modules
The next result is the natural analogue of the universal coefficient theorem
for abstract groups (compare, for example, [79], VI, th.15.1).
(5.2.15) Theorem (Universal Coefficient Theorem). Denoting the homology
group Hn (G, O) by Hn (G), the following holds:
(i) Let N be a discrete O-module considered as an element in D with G
acting trivially on N . Then we have a natural cohomological spectral
sequence
E2pq = ExtpO (Hq (G), N ) ⇒ H p+q (G, N ).
In particular, if O is a discrete valuation ring, then we obtain a natural
exact sequence for all n:
0 −→ Ext1O (Hn−1 (G), N ) −→ H n (G, N ) −→ HomO (Hn (G), N ) −→ 0.
(ii) Let M be a compact O-module considered as an element in C with
G acting trivially on M . Then we have a natural homological spectral
sequence
2
Epq
= TorO
p (Hq (G), M ) ⇒ Hp+q (G, M ).
In particular, if O is a discrete valuation ring, then we obtain a natural
exact sequence for all n:
ˆ O M −→ Hn (G, M ) −→ TorO
0 −→ Hn (G)⊗
1 (Hn−1 (G), M ) −→ 0.
.
Proof: Let P → O be a compact O[[G]]-projective resolution of O (concentrated in negative degrees) and let N → I be a discrete O-injective resolution
of N . Consider the spectral sequence associated to the double complex
.
Apq = HomO[[G]] (P −q , I p ) = HomO (PG−q , I p ).
By (2.2.4), its limit term E n is ExtnO[[G]] (O, N ) ∼
= H n (G, N ) (see (5.2.7)). Its
initial terms E2pq are easily computed as ExtpO (Hq (G), N ), which shows the
spectral sequence in (i). If O is a discrete valuation ring, then pd O = 1.
Therefore E2pq = 0 for p ≥ 2, and the spectral sequence induces the asserted
short exact sequences. This proves (i).
The proof of (ii) is similar: one chooses a compact O-projective resolution
Q → M and considers the double complex
ˆ O[[G]] Q−p = PG−q ⊗
ˆ O Q−p .
Apq = P −q ⊗
2
.
Remark: As in the case of abstract groups, one can show that the exact
sequences in (5.2.15) split by an unnatural splitting.
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283
§2. Complete Group Rings
Recall that O/mn is finite for all n. Therefore O[[G]] is an inverse limit of
finite discrete (hence artinian) rings and the ideals
mn O[[G]] + I(U ),
n ∈ IN, U
⊆
G open normal,
are a fundamental system of neighbourhoods of 0 ∈ O[[G]]. We denote by
RadG ⊆ O[[G]] the radical of O[[G]], i.e. the inverse limit of the radicals of
O/mn [G/U ]. ∗) Then RadG is a closed two-sided ideal which is the intersection
of all open left (right) maximal ideals. The powers (RadG )n , n ≥ 1, define a
topology on O[[G]], which we will call the R-topology.
(5.2.16) Proposition. (i) The R-topology is finer than the canonical topology
on O[[G]]; in particular, it is Hausdorff.
(ii) The following assertions are equivalent:
a) RadG ⊆ O[[G]] is open.
b) O[[G]] is a semi-local ring.
c) (G : Gp ) < ∞, where Gp is a p-Sylow subgroup in G.
If assertions (a)–(c) hold, then the finitely many left (right) maximal ideals of
O[[G]] are open.
(iii) O[[G]] is a local ring if and only if G is a pro-p-group. In this case, the
maximal ideal of O[[G]] is equal to mO[[G]] + IG .
Proof: For arbitrary n and U , the radical of O/mn [G/U ] is nilpotent, since
this ring is artinian ([17], chap.8, §6, no. 4, th. 3). This shows (i).
In order to prove a) ⇒ b), assume that a) holds. Then O[[G]]/RadG is
finite, hence there are only finitely many open left maximal ideals in O[[G]],
M1 , . . . , Mr say. Let M be any left maximal ideal. If M were not open,
we could find elements x1 , . . . , xr with xi ∈/ Mi , i = 1, . . . , r. The elements
x1 , . . . , xr generate a left ideal, I say, which is necessarily closed being a
homomorphic image of a free (hence compact) module of finite rank. Applying
(5.2.5) to the module O[[G]]/I, we see that I must be contained in an open
left maximal ideal. But this is obviously not possible, hence M must be open.
This shows a) ⇒ b) and the final assertion of (ii), which obviously also holds
for the right maximal ideals. The implication b) ⇒ a) is trivial.
To show a) ⇒ c) assume that RadG is open and choose n and U such that
n
m O[[G]] + I(U ) ⊆ RadG . We conclude that for every open V ⊆ U which is
normal in G and for every u ∈ U the image of u − 1 in O/m[G/V ] is contained
r
r
in the radical, hence is nilpotent. This implies (u − 1)p ≡ up − 1 ≡ 0, i.e.
r
up ∈ V , for some r. Hence U is a pro-p-group, showing a) ⇒ c). If O[[G]] is
local, then RadG is open and maximal, hence RadG ⊇ m O[[G]] + IG . Now the
∗) The radical of an abstract ring is the intersection of all left (right) maximal ideals.
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Chapter V. Iwasawa Modules
argument above shows that G is a pro-p-group. Thus we proved the ‘only if’
part of (iii).
In order to show c) ⇒ a), let (G : Gp ) < ∞. Then the intersection U of all
(finitely many) conjugates of Gp is an open normal subgroup of G. We will
prove that m O[[G]] + I(U ) is contained in RadG . Let M ⊆ O[[G]] be a left open
maximal ideal. Then N = O[[G]]/M is a simple O[[G]]-module; in particular,
N is finite and p-torsion by (5.2.4). We conclude that mn N = 0 for some n and
thus mN = 0 because N is simple. Hence m ⊆ M, and so mO[[G]] ⊆ RadG .
Let V ⊆ U be any open subgroup which is normal in G. We follow
the argument given in the proof of [32], (5.26): the augmentation ideal of
the artinian ring O/m[U/V ] is the only left maximal ideal, hence equal to
the radical and therefore nilpotent. Using the identity g(u − 1)g 0 (u0 − 1) =
0
gg 0 (ug − 1)(u0 − 1), we see that the image of I(U ) in O/m[G/V ] is a nilpotent
ideal, hence contained in Rad(O/m[G/V ]). Varying V and passing to the
limit, we obtain I(U ) ⊆ RadG , showing the implication c) ⇒ a).
If G is itself a pro-p-group, the arguments above show mO[[G]] + IG ⊆ RadG .
But mO[[G]] + IG is an open maximal ideal, so that O[[G]] is a local ring,
showing the remaining assertion of (iii).
2
Now let M be a compact O[[G]]-module. Then, in addition to the given
topology, there are two other topologies on M :
1. The topology given by the sequence of submodules {mn M + I(U )M }n,U ,
where n ∈ IN, and U runs through the open normal subgroups of G. We
call this topology the (m, I)-topology.
2. The R-topology, which is given by the sequence of submodules
{(RadG )n M }n∈IN .
By (5.2.16), the R-topology is finer than the (m, I)-topology.
(5.2.17) Proposition. (i) The (m, I)-topology is finer than the original topology
on M . In particular, the (m, I)- and the R-topology are Hausdorff.
(ii) If M is finitely generated, then the (m, I)-topology coincides with the
original topology on M .
Proof: Assume that N ⊆ M is an open submodule. Then by continuity, for
every x ∈ M there exists a neighbourhood Vx ⊆ M of x, an nx ∈ IN and an
open Ux ⊆ G, such that (mnx O[[G]] + I(Ux ))Vx ⊆ N . Since M is compact, it
is covered by finitely many Vx1 , . . . , Vxr and therefore
mn O[[G]] + I(U ) M
⊆
N
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§2. Complete Group Rings
for n = max(nx1 , . . . , nxr ) and U = ri=1 Uxi . This shows (i).
If M is finitely generated, then there exists a surjection
T
? r
O[[G]]
M with (m, I)-topology
for some r ∈ IN, which is automatically continuous, as O[[G]] carries the (m, I)topology. This shows that M with the (m, I)-topology is quasi-compact and
Hausdorff by (i). Hence the identity map (M with (m, I)-topology) → M is a
continuous bijection between compact spaces and therefore a homeomorphism.
2
The following lemma is called the Topological Nakayama Lemma for
complete group rings.
(5.2.18) Lemma. (i) If M
∈
C and RadG M = M , then M = 0.
(ii) M ∈ C is generated by x1 , . . . , xr if and only if xi + RadG M , i = 1, . . . , r,
generate M/RadG M as an O[[G]]/RadG -module.
Proof: (i) By (5.2.17), the R-topology is Hausdorff. Hence
0=
∞
\
(RadG )n M = M.
n=1
In order to show the nontrivial implication in (ii), assume that we have
x1 , . . . , xr ∈ M such that xi + RadG M , i = 1, . . . , r, generate M/RadG M .
The map
r
ϕ:
M
i=1
id
O[[G]] ei −→ (M with (m, I)-topology) −→ M ,
ei 7−→ xi ,
is continuous by (5.2.17) (i). Since O[[G]] is compact, the image N of ϕ is
closed and therefore M/N ∈ C . But by construction
RadG (M/N ) = (RadG M + N )/N = M/N ,
hence (i) yields M = N , i.e. ϕ is surjective.
2
(5.2.19) Corollary. Assume that the profinite group G is topologically finitely
generated and that (G : Gp ) < ∞, where Gp is a p-Sylow subgroup in G. Then
the R-topology coincides with the canonical topology on the semi-local ring
O[[G]]. For any compact O[[G]]-module M the R- and the (m, I)-topology
coincide. If M is finitely generated, both coincide with the original topology
on M .
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Chapter V. Iwasawa Modules
Proof: By (5.2.16), the R-topology is finer than the canonical topology, and
O[[G]] is semi-local. Furthermore, RadG is open. In order to show that both
topologies coincide, we have to show that all powers of RadG are open. Choose
n ∈ IN and an open normal subgroup U ⊆ G with
I := mn O[[G]] + I(U ) ⊆ RadG .
It suffices to show that the powers of I are open. As m/m2 ⊆ O/m2 is finite, the
topological Nakayama lemma (5.2.18) (for G = 1) implies that m and hence
also mn is finitely generated. If G is topologically finitely generated, then so
is U and therefore I(U ) is a finitely generated ideal in O[[G]]. We conclude
that I and all its powers are finitely generated; in particular, they are closed
ideals in O[[G]]. It therefore remains to show that I m has finite index in O[[G]]
for all m. To begin with, the ring O[[G]]/I is finite. Furthermore, for all m,
the ideal I m is finitely generated. Hence I m /I m+1 is finitely generated over
O[[G]]/I, and thus finite. Now the assertion follows by induction. By what we
have just seen, the R- and the (m, I)-topology coincide on any O[[G]]-module.
If, finally, M is finitely generated, then, by (5.2.17), the original topology on
M coincides with the (m, I)-topology.
2
(5.2.20) Corollary. Let G be a pro-p-group and let P ∈ C be finitely generated.
Then P is a free O[[G]]-module if and only if P is projective.
Proof: For the nontrivial implication let P be projective and let P/MP ∼
= Kr,
where M is the maximal ideal of the local ring O[[G]] and K = O[[G]]/M.
By (5.2.18) (ii), we get a surjection
r
@
ϕ : O[[G]]
P,
so that O[[G]]r ∼
= P ⊕ ker(ϕ) and ker(ϕ)/M ker(ϕ) = 0. Now (5.2.18) (i)
yields ker(ϕ) = 0.
2
(5.2.21) Corollary. Let G be a finite p-group and let P be a finitely generated
O[G]-module which is free as an O-module and cohomologically trivial as a
G-module. Then P is a free O[G]-module.
Proof: From (1.8.4) it follows that P is O[G]-projective (actually in that
proposition we considered ZZ[G]-modules, but the proof is the same for O[G]modules). Now the result follows from the previous corollary.
2
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§2. Complete Group Rings
We will finish this section with some considerations about abstract O[[G]]modules. We saw in (5.2.17) that a given finitely generated compact O[[G]]module always carries the (m, I)-topology. However, if we consider an abstract
finitely generated O[[G]]-module M , it is not clear whether it can be endowed
with a topology such that it becomes a compact O[[G]]-module. By (5.2.17),
this is possible if and only if the (m, I)-topology on M is Hausdorff. This
is always true if M is projective, because then it is a direct summand in a
free module of finite rank and the (m, I)-topology on O[[G]] is Hausdorff by
definition. We make the following
(5.2.22) Definition. Let A be a ring (with unit) and let M be an A-module
(abstract, resp. compact if A is a topological ring). Then M is called finitely
presented if there exists an exact sequence (a presentation of M )
P1 −→ P0 −→ M −→ 0
with finitely generated (abstract, resp. compact) projective A-modules P1 and
P0 .
If M is a finitely presented abstract O[[G]]-module and P1 → P0 → M → 0 is
a presentation, then P1 and P0 are naturally endowed with the (m, I)-topology
and they are compact. Furthermore, every O[[G]]-homomorphism is automatically continuous for the (m, I)-topology. Hence the image of P1 in P0 is closed
and we can give M the quotient topology. Then, a posteriori by (5.2.17), this
topology is the (m, I)-topology which is therefore Hausdorff. Thus we have
proved the
(5.2.23) Proposition. The forgetful functor from compact O[[G]]-modules to
abstract O[[G]]-modules defines an equivalence of categories:





finitely presented
compact O[[G]]-modules

with continuous



O[[G]]-homomorphisms









1−1
←→





finitely presented
abstract O[[G]]-modules

with abstract



O[[G]]-homomorphisms









.
If we assume that O[[G]] is noetherian (e.g. O = ZZp and G is a compact
Lie group over Qp , cf. [126], V 2.2.4), then every finitely generated module is
finitely presented and the category of finitely generated modules has sufficiently
many projectives. Therefore we can calculate Tor and Ext in either of the
categories (compare exercise 1), and also the topological projective dimension
coincides with the usual one.
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Chapter V. Iwasawa Modules
Remark: Most of the material of this section is contained in the article [19] by
A. BRUMER, where the slightly more general notion of pseudocompact algebras
is considered.
Exercise 1. If one of the compact O[[G]]-modules M and N is finitely presented, then
ˆ O[[G]] N ∼
M⊗
= M ⊗O[[G]] N.
Exercise 2. If M = lim Mi and N = lim Nj are in C, then
←−
i∈I
←−
j ∈J
TorO[[G]]
(M, N ) = lim TorO[[G]]
(Mi , Nj )
n
n
←−
i,j
for all n ∈ IN.
Exercise 3. For a compact right O[[G]]-module M and a compact left O[[G]]-module N ,
ˆ O N by g(m ⊗ n) = mg −1 ⊗ gn. Show that there is a
define a continuous G-action on M ⊗
natural spectral sequence of homological type
2
Ei,j
= Hi G, TorjO (M, N ) ⇒ TorO[[G]]
i+j (M, N ).
Exercise 4. Let H be a closed normal subgroup of G. Show the existence of a spectral
sequence for M ∈ C and N ∈ D of cohomological type
E2i,j = H i G/H, ExtjO[[H]] (M, N ) ⇒ Exti+j
O[[G]] (M, N ).
Exercise 5. Show that
pd O[[G]] = pd O + cdp G,
where p = char(O/m). In particular,
pd O[[G]] = pd O[[Gp ]]
if Gp is a p-Sylow subgroup in G.
Exercise 6. (Topological Nakayama Lemma) Let A be a local ring with maximal ideal m
which is compact in its m-adic topology. Let M be a compact A-module. Then the following
is true.
(i) The m-adic topology of M is finer than the given one. In particular, M is Hausdorff
with respect to the m-adic topology.
(ii) If M is finitely generated, then both topologies coincide.
(iii) If mM = M , then M = 0.
(iv) M is finitely generated if and only if M/mM is a finite dimensional A/m-vector space.
Hint: Imitate the proofs of (5.2.17) and (5.2.18).
Exercise 7. (Generalization of Maschke’s theorem) Let H be a closed, normal subgroup
in G of (not necessarily finite) index prime to p = char(O/m). Then a finitely generated
O[[G]]-module is projective if and only if it is O[[H]]-projective.
Hint: See (2.6.11) and use the spectral sequence in exercise 4.
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§3. Iwasawa Modules
§3. Iwasawa Modules
As in the last section, we assume that O is a commutative noetherian local
ring with maximal ideal m, finite residue field k = O/m and complete in
its m-adic topology. Let O[[T ]] be the power series ring in one variable
over O. Then O[[T ]] is a local ring with maximal ideal (m, T ), residue field
O[[T ]]/(m, T ) = k, noetherian (see [16], chap.III, §2, no.10) and complete
with respect to its (m, T )-adic topology (loc. cit. no.6).
A well-known and useful technical result is the so-called division lemma.
For the proof, we refer the reader to [16], chap.VII, §3, no.8.
(5.3.1) Division Lemma. Let f =
P∞
n=0
an T n
s := inf{n | an
∈
/
∈
O[[T ]] and let
m}
be finite. The number s is called the reduced degree of f . Then every
g ∈ O[[T ]] can be written uniquely as
g = fq + r
with q ∈ O[[T ]] and a polynomial r ∈ O[T ] of degree ≤ s − 1. In particular,
O[[T ]]/(f ) is a free O-module of rank s with basis {T i mod f | i = 0, . . . ,
s − 1}.
(5.3.2) Definition. A polynomial F
mial if it is of the form
∈
O[T ] is called a Weierstraß polyno-
F = T s + as−1 T s−1 + · · · + a1 T + a0
with coefficients a0 , . . . , as−1 contained in m.
(5.3.3) Corollary. Let F be a Weierstraß polynomial. Then the injection
O[T ] ,→ O[[T ]] induces an isomorphism
O[T ]/F O[T ] −→ O[[T ]]/F O[[T ]].
Proof: Let s = deg(F ). Then s is the reduced degree of F . Using (5.3.1),
the commutative diagram
ABC
O[T ]/(F
)
O[[T ]]/(F )
s−1
X
T iO
i=0
gives the result.
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Chapter V. Iwasawa Modules
(5.3.4) Weierstraß Preparation Theorem. Let f ∈ O[[T ]] with finite reduced
degree s. Then there exists a unique decomposition
f =F ·u
into a Weierstraß polynomial F of degree s and a unit u ∈ O[[T ]]. Furthermore,
F is the characteristic polynomial of the endomorphism on the free O-module
O[[T ]]/(f ) given by the multiplication by T .
Proof: We apply the division lemma to f and T s : There exists a unique
P
i
v ∈ O[[T ]] and a unique polynomial G = s−1
i=0 ai T such that
T s = f · v − G.
Since f has reduced degree s and
T s + ās−1 T s−1 + · · · + ā0 = f¯ · v̄
(here ¯ denotes the reduction mod m), it follows that āi = 0 for all i =
0, . . . , s−1, and deg(v̄) = 0. Therefore v ∈ O[[T ]]× and T s +G is a Weierstraß
polynomial. Using corollary (5.3.3), we obtain
O[[T ]]/(f ) = O[[T ]]/(F ) ∼
= O[T ]/(F )
2
thus proving the last assertion.
Now let p = char(O/m) and assume that Γ is a free pro-p-group of rank 1,
i.e. Γ is (noncanonically) isomorphic to the additive group ZZp .
(5.3.5) Proposition. Assume that γ is a topological generator of Γ ∼
= ZZp .
Then the map
∼ O[[Γ ]] ,
O[[T ]] −→
T 7−→ γ − 1 ,
is an isomorphism of topological O-algebras.
Proof: Consider the Weierstraß polynomials
ωn = (T + 1)
pn
−1=T
pn
+
n −1
pX
i=1
pn pn −i
T
,
i
!
n ≥ 0,
and let Γn be the unique subgroup of Γ of index pn . Using corollary (5.3.3),
we see that the map
∼ O[T ]/(ω ) −→ O[Γ/Γ ] ,
O[[T ]]/(ωn ) −→
n
n
T mod ωn 7−→ γ − 1 mod Γn ,
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§3. Iwasawa Modules
is an isomorphism of O-algebras with inverse map γ mod Γn 7→ T +1 mod ωn .
Since
n
n
ωn+1 = ωn ((T + 1)p (p−1) + · · · + (T + 1)p + 1) ,
we obtain a commutative diagram
DEFG
O[[T ]]/(ω
n+1 )
O[Γ/Γn+1 ]
O[[T ]]/(ωn )
and hence an isomorphism
lim
O[[TH ]]/(ωn )
←−
n
O[Γ/Γn ]
lim
O[Γ/Γn ] = O[[Γ ]].
←−
n
Finally, the natural homomorphism
I ]]
O[[T
lim O[[T ]]/(ωn )
←−
n
is an isomorphism: the compactness of O[[T ]] implies its surjectivity and the
inclusions
ωn O[[T ]] ⊆ (m, T )n+1
show that its kernel
T
n
ωn O[[T ]] is zero. Thus we obtain the desired result.
2
Now we specialize to the case O = ZZp .
(5.3.6) Definition. We call the complete group ring Λ = ZZp [[Γ ]] the Iwasawa
algebra and a compact Λ-module an Iwasawa module.
By (5.3.5), the Iwasawa algebra Λ is isomorphic to the power series ring
ZZp [[T ]]. The isomorphism depends on the choice of a topological generator
γ of Γ . In the following we will identify ZZp [[Γ ]] with ZZp [[T ]] using any fixed
generator γ.
(5.3.7) Lemma. The prime ideals of height 1 in Λ are
p = (p) and p = (F ),
where F is an irreducible Weierstraß polynomial over ZZp .
Proof: Since ZZp [[Γ ]] is factorial, the prime ideals of height 1 are of the form
p = (f ) where f is an irreducible element in Λ. Let (f ) =/ (p); then the reduced
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Chapter V. Iwasawa Modules
element f¯ ∈ ZZ/pZZ[[Γ ]] is not trivial, and using the Weierstraß preparation
theorem (5.3.4) we get
(f ) = (F ), F an irreducible Weierstraß polynomial over ZZp .
But a Weierstraß polynomial is irreducible in ZZp [[T ]] if and only if it is
irreducible in ZZp [T ], see [16], chap.VII, §3.8, cor. of prop. 7.
2
Applying the structure theorem (5.1.10) to Λ and using (5.3.5) and remark 4
after (5.1.4), we obtain the
(5.3.8) Structure Theorem for Iwasawa Modules. Let M be a finitely generated Iwasawa module. Then there exist irreducible Weierstraß polynomials
Fj , numbers r, mi , nj , and a homomorphism
≈
r
M −→ Λ ⊕
s
M
Λ/p
mi
⊕
i=1
t
M
n
Λ/Fj j
j=1
with finite kernel and cokernel. The numbers r, mi , nj and the prime ideals
Fj Λ are uniquely determined by M .
(5.3.9) Definition. With the notation of (5.3.8) we call
r(M ) = rankΛ (M ) = r the Λ-rank of M ,
µ(M ) =
s
X
mi
the Iwasawa µ-invariant of M ,
nj deg(Fj )
the Iwasawa λ-invariant of M ,
i=1
λ(M ) =
t
X
j=1
FM,γ =
t
Y
n
Fj j
the characteristic polynomial of M .
j=1
Furthermore, we call a finitely generated Λ-module of the form
E = Λr ⊕
s
M
i=1
Λ/pmi ⊕
t
M
n
Λ/Fj j
j=1
an elementary Λ-module.
Remarks: 1. The invariants defined above depend on M only up to pseudoisomorphism and r(M ), µ(M ) and λ(M ) are independent of the chosen generator γ, in contrast to the characteristic polynomial. Furthermore, observe that
FM,γ = FTΛ (M ),γ , where TΛ (M ) is the Λ-torsion submodule of M .
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2. The invariants µ(M ) and λ(M ) are additive and FM,γ is multiplicative in
short exact sequences of finitely generated Λ-torsion modules. Furthermore, a
finitely generated Λ-torsion module M is finite if and only if λ(M ) = 0 = µ(M ).
3. Let M be a finitely generated Λ-torsion module, then
λ(M ) = dimQp (M ⊗ ZZp Qp )
and FM is the characteristic polynomial of the endomorphism on the Qp -vector
space M ⊗ ZZp Qp given by multiplication by T .
As before, MΓ denotes the module of Γ -coinvariants of M :
MΓ = M/IΓ M ∼
= M/T M,
where IΓ is the augmentation ideal of Λ. The topological Nakayama lemma
(5.2.18) implies the
(5.3.10) Proposition. Let M be an Iwasawa module. Then the following
assertions are equivalent.
(i)
M is a finitely generated Λ-module.
(ii) MΓ is a finitely generated ZZp -module.
(iii) M/mM is a finite-dimensional IFp -vector space.
Very useful is the following
(5.3.11) Lemma. Let
0 −→ M1 −→ M2 −→ M3 −→ 0
be an exact sequence of Λ-modules. Then there is an exact sequence of
ZZp -modules
0 −→ M1Γ −→ M2Γ −→ M3Γ −→(M1 )Γ −→(M2 )Γ −→(M3 )Γ −→ 0 .
Proof: Since the sequences
γ−1
0 −→ MiΓ −→ Mi −→ Mi −→(Mi )Γ −→ 0
are exact for i = 1, 2, 3, the result follows from the snake lemma.
We denote the unique subgroup of Γ of index pn by Γn .
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(5.3.12) Definition. Let M be an Iwasawa module. Then M δ denotes the
maximal Λ-submodule of M on which Γ acts discretely:
Mδ =
[
M Γn .
n
δ
δ
Let M0 = tor ZZp M . If M /M0 =/ 0, then let d = d(M ) be the minimal number
such that Γd acts trivially on M δ /M0 . If this module is zero, we put d = −1.
(5.3.13) Definition. We denote the pn -th cyclotomic polynomial by
ωn
, n ≥ 0,
ξn =
ωn−1
where we put ω−1 = 1 and
!
n −1
pX
pn pn −i
pn
pn
ωn = (T + 1) − 1 = T +
T
, n ≥ 0.
i
i=1
Hence
p−1
X
k−1
ωn = ξ0 · ξ1 · · · ξn and ξ0 = ω0 = T, ξk =
(1 + T )ip
for k
≥
1.
i=0
(5.3.14) Lemma. Let M be a finitely generated Λ-module. Then
(i) M δ is a Λ-torsion module and finitely generated as a ZZp -module, thus
d(M ) < ∞.
(ii) M0 is the maximal finite Λ-submodule of M .
(iii) supp(M δ ) ∩ P (Λ) ⊆ {(ξn ) | n ≥ 0}, i.e. the prime ideals of height 1 in the
support of M δ are principal ideals generated by cyclotomic polynomials.
Moreover, there is a pseudo-isomorphism
M
Mδ ≈
Λ/ξni
i
with ni ≤ d(M ).
(M/M δ )δ is ZZp -torsion-free.
(iv)
(v) Let, in addition, M be Λ-torsion. Then MΓn is finite for all n if and only
if d(M ) = −1.
Proof: Since M is a finitely generated Λ-module and Λ is noetherian, M δ
is also finitely generated. Therefore there exists an n ∈ IN such that Γn acts
trivially on M δ , showing (i) and (ii). Since ωd(M ) (M δ /M0 ) = 0, we get (iii).
From the exact sequence
0 −→ M δ −→ M −→ M/M δ −→ 0
we obtain (using (5.3.11) and passing to the limit) the exact sequence
id
(M δ )Γn .
0 −→ M δ −→ M δ −→(M/M δ )δ −→ lim
−→
n
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The module lim (M δ )Γn is ZZp -torsion-free since the transition maps
−→ n
ωm
ωn
(M δJ )Γn
(M δ )Γm ,
m ≥ n,
coincide with multiplication by pm−n if n is large enough. Hence (M/M δ )δ is
ZZp -torsion-free. This proves (iv).
Finally, (v) follows from the fact that for a finitely generated Λ-torsion
module M , the group MΓn is finite if and only if M Γn is finite.
2
For a finitely generated Λ-module M , we define the Λ-submodule M cycl
of M in the following way. Let M0 be as above, M1 = M δ and we define
inductively
Mi+1 = ker M −→(M/Mi )/(M/Mi )δ for i ≥ 1 .
From the commutative exact diagram
0KLMNOPQRST
Mi+1
M
(M/Mi )/(M/Mi )δ
0
0
(M/Mi )δ
M/Mi
(M/Mi )/(M/Mi )δ
0,
we obtain an exact sequence
0 −→ Mi −→ Mi+1 −→(M/Mi )δ −→ 0
for i ≥ 0 .
The submodules Mi are Λ-torsion modules whose support supp(Mi ) ∩ P (Λ)
is contained in {(ξn ) | n ≥ 0}, and (M/Mi+1 )δ = ((M/Mi )/(M/Mi )δ )δ is ZZp torsion-free by (5.3.14)(iv). Therefore, since M is finitely generated, the
sequence of submodules
M0
⊆
M1
⊆
· · · ⊆ Mi
⊆
Mi+1
⊆
···
stabilizes.
(5.3.15) Definition. Let M be a finitely generated Λ-module. Then
M cycl :=
[
Mi .
i
(5.3.16) Lemma. Let M be a finitely generated Λ-module. Then
(i) M cycl is a Λ-torsion module and finitely generated as a ZZp -module with
M0 ⊆ M δ ⊆ M cycl ,
(ii) supp(M cycl ) ∩ P (Λ) = supp(M δ ) ∩ P (Λ) ⊆ {(ξn ) | 0 ≤ n ≤ d(M )},
supp(M/M cycl ) ∩ P (Λ) is disjoint to {(ξn ) | n ≥ 0},
(iii) (M/M cycl )δ = (M/M cycl )cycl = 0.
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Proof: The first assertion is obvious. If
M ≈ Λr ⊕
Λ/pmi ⊕
M
i
M
n
Λ/Fj j ⊕
j
M
Λ/(ξnk )tk ,
k
where Fj are irreducible Weierstraß polynomials different to ξn for all n
then
M
Mi ≈
Λ/(ξnk )min(tk ,i) for i ≥ 0 ,
≥
0,
k
and
M cycl ≈
M
Λ/(ξnk )tk
k
with nk ≤ d(M ). This proves (ii).
As we have seen above, the Λ-module (M/M cycl )cycl is ZZp -torsion-free.
Furthermore, supp(M/M cycl ) ∩ P (Λ) is disjoint to the set of prime ideals
{(ξn ) | n ≥ 0} by (ii), and so (M/M cycl )δ ⊆ (M/M cycl )cycl = 0.
2
One classical feature of Iwasawa theory is the description of the asymptotic
behaviour of #MΓn when n → ∞, provided these orders are finite. This finiteness is equivalent to d(M ) = −1 as we saw above. The following proposition
covers the case of nonnegative d.
(5.3.17) Proposition. Let M be a finitely generated Λ-torsion module and let
n0 ≥ d(M ) be a fixed number. Then
#(M/ ωωnn0 M ) = pµp
n +λn+ν
for all n large enough, where µ = µ(M ), λ = λ(M ) and ν is a constant not
depending on n.
Proof: Let n ≥ n0 and put
νn :=
ωn
ω n0
= ξn0 +1 · · · ξn
n
= 1 + (1 + T )p 0 + · · · + (1 + T )p
n0 (pn−n0 −1)
.
First, we observe that M/νn M is finite for all n ≥ n0 . Indeed, since νn
νn
is disjoint to supp(M ) ∩ P (Λ) for n ≥ n0 , the homomorphism M −→
M is a
pseudo-isomorphism by (5.1.6). Furthermore, we obtain the
Claim: Multiplication by νn is injective on M/M0 .
Applying the snake lemma to the exact sequence
0 −→ M0 −→ M −→ M/M0 −→ 0,
we obtain the exactness of
0 −→ M0 /νn −→ M/νn −→(M/M0 )/νn −→ 0 ,
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ν
n
since ker(M/M0 −→
M/M0 ) = 0 by the claim, and so
#M/νn = #(M/M0 )/νn · #M0 /νn .
In order to calculate #(M/M0 )/νn , we put N = M/M0 . Using the structure
theorem, we obtain an exact sequence
0 −→ N −→ E −→ C −→ 0,
where C is finite and E is an elementary Λ-module, i.e.
E=
s
M
Λ/pmi ⊕
i=1
t
M
n
Λ/Fj j .
j=1
Since multiplication by νn is injective on E, we have an exact sequence
0 −→ νn C −→ N/νn −→ E/νn −→ C/νn −→ 0,
ν
n
where νn C is defined by the exact sequence 0 → νn C → C →
C → C/νn → 0.
It follows that
#N/νn = #E/νn .
n
Let Ei = Λ/pmi . Obviously, #Ei /ωn = pmi p , and the exact sequence
ν
n
Ei /ωn −→ Ei /νn −→ 0
0 −→ Ei /ωn0 −→
shows that
#Ei /νn = pmi (p
n −pn0 )
.
For the other summands of E, we need the
(5.3.18) Lemma. Let M be a finitely generated Λ-torsion module which is
free of rank λ as a ZZp -module. Then
λ(λ − 1)
ωn+1
M = pM
for n >
.
ωn
2
We proceed with the proof of (5.3.17). If Ej = Λ/Fj (T )nj , then the sequence
ωn+1
ωn
0 −−−→ Ej /νn −−−→ Ej /νn+1 −−−→ Ej /p −−−→ 0
is exact for n 0 by (5.3.18) and the fact that multiplication by ξn+1 =
injective on Ej . Therefore
ωn+1
ωn
#Ej /νn+1 = pnj deg(Fj ) #Ej /νn ,
showing that
#Ej /νn = pnj deg(Fj )(n−n1 ) #Ej /νn1 ,
where n1 > max(n0 , λ(λ − 1)/2). Putting everything together, we obtain
n
#E/νn = pλ(N )n+µ(N )p · const ,
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where the constant does not depend on n if n is large enough. Because
λ(M ) = λ(N ) and µ(M ) = µ(N ), we have finished the proof of the asymptotic
formula.
2
Proof of (5.3.18): The endomorphism on M ⊗ ZZp IFp given by multiplication
by T has the characteristic polynomial X λ . Hence T is nilpotent, or equivalently γ acts unipotently. Choosing a suitable basis of M ⊗ ZZp IFp , the matrix
representing the action of γ is of the form

1


..
∗
.


∈
GL(λ, IFp ),
1
m
hence contained in a p-Sylow subgroup of GL(λ, IFp ). Therefore γ p = 1 on
. Now let n > λ(λ−1)
and let A ∈ M (λ × λ, ZZp ) be
M ⊗ ZZp IFp for m ≥ λ(λ−1)
2
2
the matrix corresponding to the action of γ on M with respect to some basis.
Then
n−1
Ap ≡ I mod p
and so
n
Ap ≡ I mod p2 ,
where I denotes the unit matrix. It follows that
Ap
n (p−1)
n
+ · · · + Ap + I ≡ pI mod p2 ,
so that
Ap
n (p−1)
n
+ · · · + Ap + I = pU
with U
∈
GL(λ, ZZp ) .
Therefore
(γ p
n (p−1)
n
+ · · · + γ p + 1)M = pM .
The following proposition gives a criterion for freeness of a Λ-module.
(5.3.19) Proposition. Let M be a finitely generated Λ-module.
(i) The following assertions are equivalent:
a) pdΛ M ≤ 1.
b) M Γn is ZZp -free for some n (resp. every n).
c) M has no finite nontrivial Λ-submodule.
(ii) M is a free Λ-module if and only if M Γ = 0 and MΓ is ZZp -free.
Proof: First we prove (ii). In order to show the nontrivial implication, let
0 −→ K −→ Λd −→ M −→ 0
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§3. Iwasawa Modules
be a minimal presentation of M by a free module. Using (5.3.11), we obtain
the exact sequence
0 = M Γ −→ KΓ −→ ZZdp −→ MΓ −→ 0.
Since d was chosen minimal and MΓ is ZZp -free, the map on the right is an
isomorphism. Using Nakayama’s lemma, KΓ = 0 implies K = 0.
Now we prove the equivalence of the assertions in (i). Assuming a), there
exists an exact sequence
0 −→ Λd1 −→ Λd0 −→ M −→ 0.
Using (5.3.11), it follows that M Γn ⊆ (Λd1 )Γn showing b). Let M0 be the
maximal finite Λ-module of M . Since M0 = 0 if and only if M0Γn = 0 for
some n, the inclusion M0Γn ⊆ M Γn shows the equivalence between b) and c).
Suppose b) is true and let
0 −→ N −→ Λd0 −→ M −→ 0
be a presentation of M with kernel N . Since (Λd0 )Γ = 0, we obtain by (5.3.11)
the exact sequence
0 −→ M Γ −→ NΓ −→(Λd0 )Γ .
From our assumption it follows that M Γ is ZZp -free, thus NΓ is also ZZp -free.
Hence the Λ-module N is free by (ii) and therefore pdΛ M ≤ 1.
2
For a finitely generated Λ-module M , we define
d0 (M ) = dimIFp H0 (Γ, M )/p,
d1 (M ) = dimIFp p H0 (Γ, M ) + dimIFp H1 (Γ, M )/p,
d2 (M ) = dimIFp p H1 (Γ, M ).
(5.3.20) Proposition. Let M be a finitely generated Λ-module. Then there
exists an exact sequence
0 −→ Λd2 (M ) −→ Λd1 (M ) −→ Λd0 (M ) −→ M −→ 0.
In particular,
rankΛ (M ) = d0 (M ) − d1 (M ) + d2 (M ) = rank ZZp MΓ − rank ZZp M Γ .
Proof: By Nakayama’s lemma (5.2.18), we have a minimal presentation
ϕ : Λd0 (M ) M . Let N = ker ϕ, i.e. we have an exact sequence
0 −→ N −→ Λd0 (M ) −→ M −→ 0 .
Using (5.3.11), we obtain an exact sequence
0 −→ M Γ −→ NΓ −→ ZZdp0 (M ) −→ MΓ −→ 0
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Chapter V. Iwasawa Modules
which induces the exact sequence
0 −→ M Γ /p −→ NΓ /p −→ p (MΓ ) −→ 0
∼ M /p) and an isomorphism
(note that ZZdp0 (M ) /p →
Γ
p (M
Γ
∼
) −→
p (NΓ ) .
It follows that dimIFp NΓ /p = d1 (M ). Hence, again by Nakayama’s lemma, we
get a minimal presentation ψ : Λd1 (M ) N . Let N 0 be the kernel of ψ. Since
N Γ = 0, we find that NΓ0 is ZZp -free and therefore N 0 is Λ-free by (5.3.19)(ii).
Furthermore,
rankΛ (N 0 ) = dimIFp NΓ0 /p = dimIFp p (NΓ ) = dimIFp p (M Γ ) = d2 (M ).
2
∼ ZZ/pZZ. Show that
Exercise 1. Let M be the kernel of the natural projection Λ −→ Λ(p,Γ ) =
M is a torsion-free Λ-module of rank 1 which is not free, and that no pseudo-isomorphism
≈
Λ −→ M exists.
Exercise 2. Let M be a finitely generated Λ-module. Show the following statements.
n
a) For every n = 0, 1, . . . , the ZZp -module M/ωn M , ωn = (1 + T )p − 1, is finitely
generated and
rank ZZ p M/ωn+1 M ≥ rank ZZ p M/ωn M .
b) M is a Λ-torsion module if and only if rank ZZ p M/ωn M is bounded for n → ∞.
c) Let M be Λ-torsion, then the following assertions are equivalent:
(i)
µ(M ) = 0.
(ii) M is a finitely generated ZZp -module.
(iii) dimIFp M ⊗ ZZ p IFp < ∞.
(iv) dimIFp (M/ωn M ⊗ ZZ p IFp ) is bounded for n → ∞.
Exercise 3. Let M be a finitely generated Λ-torsion module. Then the following assertions
are equivalent:
(i)
#M Γn < ∞,
(ii) #MΓn < ∞,
n
(iii) FM,γ (ζ − 1) =/ 0 for all ζ with ζ p = 1.
Note that the statement (iii) is independent of the choice of the generator γ of Γ . If the
conditions above are fulfilled, then
Y
1
#M Γn
= µ(M )pn
| FM,γ (ζ − 1)|p ,
#MΓn
p
pn
ζ
=1
where | · |p denotes the p-adic valuation on Cp normalized by | p |p = p1 .
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§4. Homotopy of Modules
Exercise 4. Let M be a finitely generated Λ-module. Show that the following assertions are
equivalent:
(i) M is a free Λ-module.
(ii) There exist two elements a, b ∈ Λ which generate the maximal ideal of Λ, such that
aM
=0
and
b (M/aM )
= 0,
where for a Λ-module N and an element c ∈ Λ the module c N is defined by {x ∈ N | cx = 0}.
Show that M is Λ-free if and only if M is ZZp -torsion-free and M/pM has no Γ -invariants.
Hint: Use a minimal presentation of M by a free Λ-module and imitate the proof of (5.3.19).
§4. Homotopy of Modules
In this and the following two sections we will follow closely the work of
U. JANNSEN [97]. The reader is strongly advised to consult the original paper
for many more results than are presented here; in particular, classification
theorems of ZZp [[T ]]-modules up to isomorphism.
Let Λ be a ring with unit, not necessarily commutative. Recall that pdΛ M
denotes the projective dimension of a Λ-module M .
(5.4.1) Definition. We denote the full subcategory of Mod(Λ) of modules M
with pdΛ M ≤ 1 by Mod1 (Λ).
(5.4.2) Definition. (i) A homomorphism f : M → N of Λ-modules is homotopic to zero (f ' 0) if it factors through a projective module P
f : M −→ P −→ N.
Two homomorphisms f, g : M → N are homotopic (f ' g) if f − g is homotopic to zero. We denote the homotopy category of Λ-modules by Ho(Λ), i.e.
the category whose objects are Λ-modules and in which the homomorphism
groups are given by HomΛ (M, N )/{f ' 0}.∗) We denote the full subcategory
of Ho(Λ) whose objects are in Mod1 (Λ) by Ho1 (Λ).
(ii) A homomorphism f : M → N of Λ-modules is a homotopy equivalence if
there exists a homomorphism g : N → M such that f g ' idN and gf ' idM ,
i.e. an isomorphism in Ho(Λ). In this case, we say that M and N are homotopy
equivalent (M ' N ).
∗) Observe that the notion of homotopy is compatible with composition of homomorphisms.
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Chapter V. Iwasawa Modules
(5.4.3) Proposition. Let f, g : M → N be homomorphisms of Λ-modules.
(i) The following assertions are equivalent.
a) f ' g,
b) f ∗ , g ∗ : ExtiΛ (N, R) −→ ExtiΛ (M, R) are equal for all Λ-modules R
and all i ≥ 1,
c) f ∗ , g ∗ : Ext1Λ (N, R) −→ Ext1Λ (M, R) are equal for all Λ-modules R.
(ii) The following assertions are equivalent.
a) f is a homotopy equivalence,
b) f ∗ : ExtiΛ (N, R) −→ ExtiΛ (M, R) is an isomorphism for all R and
all i ≥ 1,
c) f ∗ : Ext1Λ (N, R) −→ Ext1Λ (M, R) is an isomorphism for all R,
d) There are projective Λ-modules P and Q and an isomorphism σ such
that f factors as
f : VUW M
can
M ⊕P
σ
N ⊕Q
can
N.
Furthermore, if f : M → N is a homotopy equivalence between finitely
generated Λ-modules M and N , then the projective Λ-modules P and Q
in d) may also be chosen finitely generated.
Proof: (i) Obviously, we may assume that g = 0. Since ExtiΛ (P, R) = 0, i ≥ 1,
for a projective module P , f ' 0 implies f ∗ = 0. This proves a) ⇒ b). The
implication b) ⇒ c) is trivial. In order to show c) ⇒ a), let π : P N be a
surjection with P a projective module and let K = ker π. Taking the pull-back
via f , we obtain the commutative exact diagram
0_`abXYZ[\]^
K
X
M
0
f
0
K
P
π
N
0.
Since f ∗ : Ext1Λ (N, K) → Ext1Λ (M, K) is zero by assumption c), the upper sequence in the diagram splits, thus f factors through the projective Λ-module P .
(ii) Assuming a), there exists g : N → M such that f g ' id and gf ' id.
By (i), it follows that g ∗ f ∗ = id and f ∗ g ∗ = id, and so we obtain b). The
implication b) ⇒ c) is trivial. In order to prove c) ⇒ d), let π : P N be
a surjective Λ-homomorphism with P a projective module and let the module
K be defined by the exactness of the sequence
(∗)
f +π
0 −→ K −→ M ⊕ P −→ N −→ 0 .
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Applying ExtΛ (−, K), we obtain the exact sequence
∼ Ext1 (M ⊕ P, K)
HomΛ (M ⊕ P, K) −→ HomΛ (K, K) −→ Ext1Λ (N, K) −→
Λ
where the right-hand map is an isomorphism by assumption c). A pre-image of
id ∈ HomΛ (K, K) gives us a splitting of the exact sequence (∗) and we obtain
an isomorphism
∼ N ⊕K
σ : M ⊕ P −→
which induces f , using the canonical injection and projection respectively.
Applying ExtΛ (−, R) for an arbitrary Λ-module R and using c) we see that Q :=
K is projective and we obtain d). Obviously, d) implies a), since the injection
M ,→ M ⊕ P and the projection N ⊕ Q N are homotopy equivalences
(having the canonical projection and injection as inverses respectively).
Finally, the proof of the implication c) ⇒ d) shows that we can choose P
finitely generated if N is. The isomorphism σ then implies that Q is also
finitely generated, if M is.
2
(5.4.4) Definition. Let M be a (left) Λ-module. Then
E i (M ) := ExtiΛ (M, Λ) ,
i ≥ 0,
are (right) Λ-modules by functoriality and the right Λ-module structure of the
bimodule Λ. By convention, we set E i (M ) = 0 for i < 0. The Λ-dual E 0 (M )
will also be denoted by M + .
Remarks: 1. By proposition (5.4.3), the functor E i factors through a functor
Ei
Ho(Λ) −→ Mod (Λ)
for i ≥ 1.
2. Clearly, E i can also be viewed as functor from right modules to left modules.
Also several functors defined below will interchange left and right action. In
the case of a group ring, there is a natural equivalence between right and left
modules induced by the involution of the group ring given by passing to the
inverses of the group elements. In general this is not possible, but for the
theory it is not necessary, and in the following we will not specify if we are
talking about left and right Λ-modules or if a functor interchanges left and right
Λ-actions. This would only cause notational complications and it will always
be clear where one has to insert “left” or “right” .
(5.4.5) Definition. We denote the full subcategory of Mod (Λ) of modules M
with M + = 0 by Mod+ (Λ). We denote the full subcategory of Ho(Λ) whose
objects are in Mod+ (Λ) by Ho+ (Λ).
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Chapter V. Iwasawa Modules
(5.4.6) Lemma. The canonical localization functor ho : Mod(Λ) → Ho(Λ)
induces an equivalence of categories
∼ Ho (Λ).
ho : Mod+ (Λ) −→
+
Proof: Recall that for a projective module P , the canonical homomorphism
ϕP : P → P ++ is injective (see [17], chap.II, §2, no.8). If M ∈ Mod+ (Λ) and
f : M → P is any homomorphism from M to a projective module P , then the
commutative diagram
f
defgc
M
P
0
M ++
P ++
shows that f = 0. Therefore a homomorphism in Mod+ (Λ) which is homotopic
to zero is zero. This proves the lemma.
2
Recall that a Λ-module is called finitely presented if there exists an exact
sequence
P1 −→ P0 −→ M −→ 0
with finitely generated projective modules P1 and P0 .
(5.4.7) Definition. We denote the full subcategory of Mod(Λ) whose objects are finitely presented Λ-modules by Modf p (Λ). The notation Modf1 p (Λ),
Hof p (Λ), Hof1 p (Λ), . . . have their obvious meaning.
Now we will construct a contravariant duality functor
D : Hof p (Λ) −→ Hof p (Λ)
as follows:
For every finitely presented module M , we choose a presentation P1 →
P0 → M → 0 of M by finitely generated projectives. Then we define
DM ∈ Ob(Modf p (Λ)) = Ob(Hof p (Λ)) by the exact sequence
0 −→ M + −→ P0+ −→ P1+ −→ DM −→ 0.
If f : M → N is a homomorphism and Q1 → Q0 → N → 0 is the chosen
presentation of N , then we choose α : P0 → Q0 and β : P1 → Q1 to make the
diagram
Ppohijklmn 1
P0
M
0
β
α
f
Q1
Q0
N
0
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305
§4. Homotopy of Modules
commutative (this is possible since P0 and P1 are projective). We define
Df : DN → DM by the commutative diagram
0~}|qrstuvwxyz{
N+
Q+0
f+
0
M+
Q+1
DN
Df
β+
α+
P0+
P1+
0
DM
0.
(5.4.8) Definition. We call the functor
D : Hof p (Λ) −→ Hof p (Λ)
the transpose.
(5.4.9) Proposition.
(i) The functor D is well-defined and (up to canonical functor isomorphism)
independent of the chosen presentations.
(ii) D is a (contravariant) autoduality of Hof p (Λ), i.e. D ◦ D ∼
= id.
(iii) For M
∈
Modf p (Λ), there exists a canonical exact sequence
ϕ
M
M ++ −→ E 2 (DM ) −→ 0 ,
0 −→ E 1 (DM ) −→ M −→
where ϕM is the canonical homomorphism of M to its bidual.
(5.4.10) Definition. For a finitely presented Λ-module M , we set
Ti (M ) := E i (DM ) ,
i = 1, 2.
Note that, by (5.4.3), the groups E i (DM ) are well-defined in Mod(Λ), while
DM is defined only up to homotopy equivalence. Thus Ti (M ) only depends
on the homotopy equivalence class of M .
Proof of (5.4.9): Let R be a Λ-module. Consider the commutative exact
diagram
…†‡„€‚ƒ
P1 ⊗Λ R
P0 ⊗Λ R
M ⊗Λ R
0
ϕP0 ,R
0
Hom(K, R)
Hom(P0+ , R)
ϕM,R
Hom(M + , R)
where K = ker(P1+ DM ) and ϕM,R is the canonical homomorphism
M ⊗ˆ‰ Λ R
ϕM ⊗R
M ++ ⊗Λ R
Hom(M + , R).
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306
Chapter V. Iwasawa Modules
Observe that ϕP0 ,R is an isomorphism, since P0 is finitely generated and projective, and that the dotted arrow factors as
ϕP
1
‹Š R
P1 ⊗
Hom(P1+ , R)
Hom(K, R) .
The snake lemma (1.3.1) implies that
ker ϕM,R ∼
= Ext1 (DM, R) ,
= coker (Hom(P + , R) −→ Hom(K, R)) ∼
,R
Λ
1
coker ϕM,R ∼
= coker (Hom(P0+ , R) −→ Hom(M + , R)) ∼
= Ext1Λ (K, R)
∼
= Ext2 (DM, R).
Λ
Therefore we obtain an exact sequence
ϕM,R
u
v
(∗) 0 → Ext1Λ (DM, R) → M ⊗Λ R −→ Hom(M + , R) → Ext2Λ (DM, R) → 0,
where the homomorphisms u and v a priori depend on the chosen presentation P1 → P0 → M → 0. Now assume that we are given a homomorphism
f : M → N in Modf p (Λ). Let the diagram
P’“”‘ŒŽ 1
P0
M
β
α
f
Q1
Q0
N
0
0
be as in the definition of Df , which we denote for the moment by Df(α,β) .
Then for every Λ-module R, we obtain an exact commutative diagram
0™˜•–š—›œ
Ext1Λ (DM, R)
uP1 ,P0
∗
Df(α,β)
0
Ext1Λ (DN, R)
M ⊗Λ R
ϕM,R
Hom(M + , R)
(f + )∗
f ⊗id
uQ1 ,Q0
N ⊗Λ R
ϕN,R
Hom(N + , R) .
Therefore Df is well-defined, i.e. independent of α, β (use (5.4.3)(i)). Furthermore, setting M = N and f = id, we see that DM does not depend (up to
canonical homotopy equivalence) on the chosen projective presentation. Thus
D is a functor from Modf p (Λ) to Hof p (Λ).
If f : M → N is homotopic to zero, then by definition it factors through
a projective module, P say. But then choose (compare with the construction
above) α : P0 → Q0 also factorizing through P , and β : P1 → Q1 to be zero:
PžŸ ¡¢£¤¥¦§¨©ª«¬ 1
P0
M
0
0
P
P
0
Q1
Q0
N
0.
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307
§4. Homotopy of Modules
This shows Df ' 0. Hence D is well-defined as a functor
D : Hof p (Λ) −→ Hof p (Λ).
The assertion (ii) is now trivial since we can choose the sequence P0+ → P1+
→ DM → 0 as a projective presentation of DM , showing that D(DM ) ' M .
The exact sequence in (iii) follows from (∗) on setting R = Λ and from
the remark that a posteriori the maps u and v do not depend on the chosen
presentation.
2
(5.4.11) Corollary. The transpose D defines an equivalence of categories
Hof1 p­® (Λ)
D
Hof+ p (Λ) ,
D
and D restricted to Hof1 p (Λ) coincides with E 1 .
Proof: If M ∈ Hof+ p (Λ), then the exact sequence 0 → P0+ → P1+ → DM → 0
shows that DM ∈ Hof1 p (Λ). Conversely, if M ∈ Hof1 p (Λ) and we choose the
presentation in the form 0 → P1 → P0 → M → 0, then
DM ∼
= coker (P + → P + ) ∼
= Ext1 (M, Λ) and
0
(DM ) ∼
=
+
ker (P1++
Λ
1
→ P0++ )
∼
= ker (P1 → P0 ) = 0.
2
Together with (5.4.6), the last result implies the
(5.4.12) Corollary. The functor E 1 defines an equivalence of categories
E1
fp
Hof1 p (Λ) −→
∼ Mod + (Λ).
Now we assume that
Λ = ZZp [[G]],
where G is a profinite group. Recall from the discussion in §2 that every
finitely presented Λ-module carries a natural compact topology and that Λhomomorphisms of such modules are continuous. If P is finitely generated
and projective, then so is P + . Suppose that the Λ-module M has a resolution
· · · −→ P2 −→ P1 −→ P0 −→ M −→ 0
by finitely generated projective modules Pi , i = 0, 1, . . . ; in particular, M is
finitely presented. Then the complex of finitely generated projectives
0 −→ P0+ −→ P1+ −→ P2+ −→ · · ·
computes E i (M ). Hence the groups E i (M ) are canonically endowed with a
compact topology. If Λ is noetherian, this applies to every finitely generated
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308
Chapter V. Iwasawa Modules
module M . The following theorem relates the Λ-modules E r (M ) to the
discrete G-modules
Dr (M ∨ ) = lim H r (U, M ∨ )∗ ,
−→
U ⊆G
r
≥
0,
where
M ∨ = Homcts (M, Qp /ZZp ) = lim Hom (MU , Qp /ZZp ) = lim (MU )∗
−→
U
−→
U
is a discrete G-module (see (2.5.1) for the definition of Dr (A) with a discrete
G-module A). We set Dr (M ∨ ) = 0 if r < 0.
(5.4.13) Theorem. Assume that the Λ-module M has a resolution by finitely
generated projective modules.
(i) There exists a functorial exact sequence
0 −→ Dr (M ∨ ) ⊗ ZZp Qp /ZZp −→ E r (M )∨ −→ tor ZZp Dr−1 (M ∨ ) −→ 0
for all r.
If, in addition, tor ZZp M and M/tor ZZp M also have a resolution by finitely
generated projectives, then the following hold for all r:
(ii)
Dr (pm (M ∨ )),
E r (M/tor ZZp M )∨ ∼
= lim
−→
m
(iii)
E r (tor ZZp M )∨ ∼
Dr−1 (M ∨ /pm ).
= lim
−→
m
(iv) There is a long exact sequence
→ E r (M )∨ → lim Dr (pm (M ∨ )) → lim Dr−2 (M ∨ /pm ) → E r−1 (M )∨ →,
−→
m
−→
m
functorial in M and in G.
Proof: Assume for the moment that Λ is noetherian. The functor M p
(M + )∨ from the category of finitely generated Λ-modules to abelian groups is
the composition of the right exact functors M p D0 (M ∨ ) (this functor takes
values in ZZp -modules) and N p N ⊗ ZZp Qp /ZZp because
M + = lim HomΛ (M, ZZp [G/U ]) = lim Hom ZZp [G/U ] (MU , ZZp [G/U ])
←−
U ⊆G
←−
U ⊆G
= lim
Hom ZZp (MU , ZZp ),
←−
U ⊆G
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309
§4. Homotopy of Modules
where the limit is taken over all normal subgroups U ⊆ G with respect to the
duals of the norm maps, so that
(M + )∨ = lim Hom ZZp (MU , ZZp )∨ = lim MU ⊗ ZZp Qp /ZZp .
−→
U ⊆G
−→
U ⊆G
The r-th derived functor of M p D0 (M ∨ ) is M p Dr (M ∨ ) and the first
functor in the composition sends projectives to acyclics for the second functor.
We get a Grothendieck spectral sequence of homological type
ZZ
Eij2 = Tori p (Dj (M ∨ ), Qp /ZZp ) ⇒ Ei+j = E i+j (M )∨ .
Since




N ⊗ Qp /ZZp , i = 0,
=  tor ZZp N ,
i = 1,


0,
i ≥ 2,
we obtain (i) under the assumption that Λ is noetherian.
If Λ is not noetherian, then the category of modules having a resolution by
finitely generated projectives has no good properties. Therefore we cannot use
the general machinery and have to construct the sequence by hand.
Let · · · → P1 → P0 → M → 0 be a resolution of M by finitely generated
projectives. Then the complex
(∗)
· · · −→ D0 (P2∨ ) −→ D0 (P1∨ ) −→ D0 (P0∨ ) −→ 0
calculates D (M ∨ ) and consists of torsion-free, hence flat, ZZp -modules.
Choosing a Cartan-Eilenberg resolution of the complex (∗) by a ZZp -projective
double complex, we obtain (i) from the spectral sequence associated to the
double complex tensored by Qp /ZZp .
The exact sequence
ZZ
Tori p (N, Qp /ZZp )
.
(∗∗)
0 −→(M/tor ZZp M )∨ −→ M ∨ −→(tor ZZp M )∨ −→ 0
and the fact that (M/tor ZZp M )∨ /pm = 0 imply that
∨
m
∼ lim D
lim Dr−1 ((tor ZZp M )∨ /pm ) −→
r−1 (M /p ).
−→
m
−→
m
This shows that it suffices to show (iii) for the case that M is ZZp -torsion. Next
we show that we may assume M to be ZZp -torsion-free in (ii).
Consider, for every m, the pm -torsion sequence associated to (∗∗)
0 −→ pm (M/tor ZZp M )∨ −→ pm M ∨ −→ pm (tor ZZp M )∨ −→ 0.
Since tor ZZp M is finitely generated, there exists an n such that the pn -multiplication map
n
∨ p
∨
pm+n (tor ZZp M ) −→ pm (tor ZZp M )
is the zero map for all m. Hence lim
lim
Dr (pm (M ∨ ))
−→
m
−→ m
∼ lim
−→
−→
m
Dr (pm ((tor ZZp M )∨ )) = 0 and thus
Dr (pm ((M/tor ZZp M )∨ )).
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Chapter V. Iwasawa Modules
In order to prove (ii), let
.
P :
· · · −→ P1 −→ P0 −→ M −→ 0
be a resolution of the ZZp -torsion-free module M by finitely generated projectives. Then
0 −→(M/pm )∨ −→(P0 /pm )∨ −→(P1 /pm )∨ −→ · · ·
is a resolution of the discrete G-module (M/pm )∨ by cohomologically trivial
G-modules. Therefore
.
E r (M )∨ = H r ((P + )∨ )
.
.
= H (lim lim (((P /pm )∨ )U )∗ )
r
= lim
−→
m
= lim
−→
m
= lim
−→
m
−→ −→
m U ⊆G
lim H r (((P /pm )∨ )U )∗
−→
U ⊆G
lim H r (U, (M/pm )∨ )∗
−→
U ⊆G
Dr (pm (M ∨ )).
This shows (ii).
In order to prove (iii), we may assume that M = tor ZZp M . Then M is already
annihilated by some power of p and hence the same is true for Dr−1 (M ∨ ).
Therefore Dr−1 (M ∨ ) ⊗ Qp /ZZp = 0 and (i) implies that
E r (M )∨ ∼
= tor ZZp Dr−1 (M ∨ )
= Dr−1 (M ∨ )
Dr−1 (M ∨ /pm ).
= lim
−→
m
(The last limit becomes stationary.) This shows (iii).
Finally, (iv) follows from (ii),(iii) and the long exact Ext-sequence
· · · → E r (M )∨ → E r (M/tor ZZp M )∨ → E r−1 (tor ZZp M )∨ → · · · .
2
(5.4.14) Corollary. If cdp G = n is finite and if M has a resolution by finitely
generated projectives, then E r (M ) = 0 for r > n + 1.
Proof: Obviously, we have Dr (A) = 0 for r > n and for all p-primary
discrete G-modules A. Using (5.4.13)(i), we obtain the result.
2
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311
§4. Homotopy of Modules
(5.4.15) Corollary. Assume that G is a duality group ∗) at p of dimension n
Dn (ZZ/pi ZZ). Then the following hold for
with dualizing module Dn(p) = lim
−→ i
every Λ-module M which has a resolution by finitely generated projectives:
(i) If M is free of finite rank as a ZZp -module, then


= M ⊗ ZZp Dn(p) if r = n,
 lim Dn ((M/pm )∨ ) ∼
−→
r
∨ ∼
E (M ) =
m

 0
otherwise.
(ii) If M is a finite p-primary G-module, then


Hom ZZp (M ∨ , Dn(p) ) if r = n + 1,
E r (M )∨ ∼
=
0
otherwise.
Proof: By (5.4.13)(ii), we have in case (i)
E r (M )∨ = lim Dr ((M/pm )∨ ) ,
−→
m
which is zero for r =/ n by (3.4.6), and
lim H 0 (U, Hom ZZp ((M/pm )∨ , Dn(p) ))
E n (M )∨ ∼
= lim
−→ −→
m U ⊆G
∼
Hom ZZp ((M/pm )∨ , Dn(p) ) ∼
= M ⊗ ZZp Dn(p) .
= lim
−→
m
In order to prove (ii), we use (5.4.13)(iii):
E r (M )∨ ∼
= Dr−1 (M ∨ ) ,
and hence E n+1 (M )∨ ∼
= Dn (M ∨ ) ∼
= Hom ZZp (M ∨ , Dn(p) ) and E r (M ) = 0 for
r =/ n + 1.
2
We finish this section with some remarks concerning the change of the
group. Let H be an open subgroup in G. We consider the forgetful functor
from abstract (resp. compact) ZZp [[G]]-modules to abstract (resp. compact)
ZZp [[H]]-modules.
(5.4.16) Lemma. The forgetful functor sends projectives to projectives, i.e.
an abstract (resp. compact) ZZp [[G]]-module which is projective as an abstract
(resp. compact) ZZp [[G]]-module is also projective as an abstract (resp. compact)
ZZp [[H]]-module. If H is normal and of prime-to-p index in G, then also the
converse statement is true.
∗) see III §4.
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Chapter V. Iwasawa Modules
Proof: A projective module is a direct summand in a free module. It thus
suffices to show that ZZp [[G]] is a free ZZp [[H]]-module. But this is clear because
Zp [[H]]).
ZZp [[G]] ∼
= ZZp [[G]] ⊗ ZZp [[H]] ZZp [[H]] ∼
= IndH
G (Z
Now suppose that H is normal in G and that p - (G : H). Since #(G/H)−1 ∈
ZZp , the functor M p M G/H is exact on ZZp [G/H]-modules. Therefore the
equality
Hom ZZp [[G]] (M, N ) = Hom ZZp [[H]] (M, N )G/H
implies isomorphisms
ExtiZZp [[G]] (M, N ) ∼
= ExtiZZp [[H]] (M, N )G/H
for all abstract (compact) ZZp [[G]]-modules M, N and all i
the remaining statement.
≥
0. This implies
2
The following proposition shows that the functors E i commute with the
forgetful functor:
(5.4.17) Proposition. Suppose M is a ZZp [[G]]-module which has a resolution
by finitely generated projectives. Let H be an open subgroup in G. Then we
have natural isomorphisms of (right) ZZp [[H]]-modules
ExtiZZp [[G]] (M, ZZp [[G]]) ∼
= ExtiZZp [[H]] (M, ZZp [[H]])
for all i ≥ 0.
Proof: The isomorphism ZZp [[G]] ∼
Zp [[H]]) (see the proof of (5.4.16)),
= IndH
G (Z
together with Frobenius reciprocity, shows that
Hom ZZ [[G]] (M, ZZp [[G]]) ∼
= Hom ZZ [[H]] (M, ZZp [[H]]).
p
p
By (5.4.16) and since the index of H in G is finite, the forgetful functor sends
finitely generated projectives to finitely generated projectives. Therefore the
above isomorphism extends to Exti for all i ≥ 0.
2
§5. Homotopy Invariants of Iwasawa Modules
In this section, we specialize again to the case G = Γ ∼
= ZZp , so that
Λ = ZZp [[Γ ]] is the Iwasawa algebra. Then Λ is a commutative 2-dimensional
regular local ring and therefore its projective dimension pd(Λ) is equal to 2
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§5. Homotopy Invariants of Iwasawa Modules
313
(see [129], th. 19.2; this also follows from §2, ex.5). Thus for every Λ-module
M the projective dimension pdΛ M is equal or less than 2, i.e. there exists a
projective resolution
0 −→ P2 −→ P1 −→ P0 −→ M −→ 0
of M of length 2 (or smaller). This implies that E i (M ) = 0 for i ≥ 3. By
(5.4.9)(iii), the following definition agrees with the one given in (5.4.10).
(5.5.1) Definition. For a finitely generated Λ-module M we set
T0 (M ) := the maximal finite submodule of M ,
T1 (M ) := ker (ϕM : M → M ++ ) ,
T2 (M ) := coker (ϕM : M → M ++ ) .
(5.5.2) Lemma. For a Λ-module M the following assertions are equivalent.
(i) M is projective,
(ii) M ' 0.
If M is finitely generated, then M is free if these hold.
Proof: Obviously, (i) ⇒ (ii). If M ' 0, then the identity map id : M → M
factors through a projective module, hence M is itself projective, being a direct
summand of a projective. The last assertion follows from (5.2.20).
2
(5.5.3) Proposition. Let M be a finitely generated Λ-module. Then
(i) T1 (M ) is the Λ-torsion submodule of M .
(ii) E 1 (M ) is a Λ-torsion module. If M is Λ-torsion, then E 1 (M ) has no
nontrivial finite submodules, i.e. T0 E 1 (M ) = 0. If M is finite, then
E 1 (M ) = 0.
(iii) T2 (M ) is finite, and T2 (M ) = 0 if and only if M/T1 (M ) is free (hence
M∼
= Λr ⊕ T1 (M ) for some r in this case).
In particular, T2 (M ) = 0 for a Λ-torsion module M .
(iv) E 2 (M ) is finite. One has
E 2 (M ) ∼
= E 2 (T0 (M )) ∼
= T0 (M )∨
and the following assertions are equivalent:
a) E 2 (M )= 0,
b) T0 (M ) = 0,
c) pdΛ M ≤ 1.
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Chapter V. Iwasawa Modules
Proof: Let K be the field of fractions of Λ. By definition, see (5.5.1), we
have the exact sequence
∼ M ++ ⊗ K −→ T (M ) ⊗ K −→ 0.
0 −→ T1 (M ) ⊗Λ K −→ M ⊗Λ K −→
Λ
2
Λ
Thus T1 (M ) and T2 (M ) are Λ-torsion modules. Since M ++ is torsion-free, we
have proved (i) and the first statement of (ii) because E 1 (M ) ∼
= E 1 (DDM )
∼
= T1 (DM ). Now let M be a Λ-torsion module and let
P1 −→ P0 −→ M −→ 0
be a presentation of M by free Λ-modules. We obtain the exact sequence
0 = M + −→ P0+ −→ P1+ −→ DM −→ 0
showing that pdΛ DM ≤ 1, and so DM has no finite nontrivial Λ-submodules
by (5.3.19)(i) .∗) Thus E 1 (M ) = E 1 (DDM ) = T1 (DM ) has this property.
Finally, if M is finite, then pn and ωn annihilate M for some n large enough
and the same is true for E 1 (M ). Thus E 1 (M ) is finite and therefore zero.
In order to prove (iii), we observe that (M/T1 (M ))p is a free Λp -module of
finite rank by (i). Hence
∼ (M/T (M ))++ = M ++
(M/T1 (M ))p −→
1
p
p
for all prime ideals p of Λ of height ≤ 1. Thus T2 (M ) is pseudo-null, i.e. finite.
Furthermore, M ++ is free and the exact sequence
0 −→ M/T1 (M ) −→ M ++ −→ T2 (M ) −→ 0
shows that (M/T1 (M ))Γ
⊆
(M ++ )Γ = 0 and proves the exactness of
0 −→ T2 (M )Γ −→(M/T1 (M ))Γ −→(M ++ )Γ .
Therefore (M/T1 (M ))Γ is ZZp -free if and only if T2 (M )Γ = 0 or equivalently
T2 (M ) = 0. Now (5.3.19) (ii) implies assertion (iii).
Since E 2 (M ) = T2 (DM ), this module is finite by (iii). From the structure
theorem for Λ-modules, we obtain an exact sequence
f
0 −→ T0 (M ) −→ M −→ E −→ C −→ 0,
where C is finite and
E∼
= Λr ⊕
s
M
i=1
Λ/p
mi
⊕
t
M
n
Λ/Fj j .
j=1
The long exact Ext-sequence implies the exactness of
E 2 (imf ) −→ E 2 (M ) −→ E 2 (T0 (M )) −→ 0
and that
¯
E 2 (E)
E 2 (imf )
∗) In fact, DM is only defined up to homotopy equivalence, but we see that DM has no
nontrivial finite submodules for every choice of DM .
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315
§5. Homotopy Invariants of Iwasawa Modules
is surjective. Since pdΛ E ≤ 1, we get E 2 (E) = 0, and so E 2 (M ) ∼
= E 2 (T0 (M )).
Now we recall that Γ is a Poincaré group of dimension 1 with dualizing module
Qp /ZZp (trivial action). Therefore (5.4.15)(ii) implies E 2 (T0 (M )) = T0 (M )∨ .
The equivalence between a) and b) is now trivial and the equivalence between
b) and c) is contained in (5.3.19)(i).
2
(5.5.4) Corollary. Let M be a finitely generated Λ-module. Then
∼ E 1 (M ).
(i) E 1 (M/T0 (M )) −→
(ii) E 1 (M ) = 0 if and only if M/T0 (M ) is free
(hence M ∼
= Λr ⊕ T0 (M ) for some r ≥ 0 in this case).
Proof: The exact sequence
0 = E 0 (T0 (M )) −→ E 1 (M/T0 (M )) −→ E 1 (M ) −→ E 1 (T0 (M )),
together with (5.5.3)(ii), implies assertion (i). By (5.5.3)(iv), we have
pdΛ (M/T0 (M )) ≤ 1. Therefore the following assertions are equivalent:
M/T0 (M ) is free ⇐⇒ M/T0 (M ) ' 0
(by lemma (5.5.2))
⇐⇒ E 1 (M/T0 (M )) = 0
(by corollary (5.4.12))
⇐⇒ E 1 (M ) = 0
(by (i)).
2
(5.5.5) Definition. Let M be a finitely generated Λ-torsion module and let
{πn } be a sequence of non-zero elements of Λ such that
π0 ∈ m , πn+1 ∈ πn m ,
and such that the set of prime ideals dividing the principal ideals πn Λ, n ≥ 0,
is disjoint to the set of prime ideals of height 1 in supp(M ). Let
α(M ) := lim
Hom(M/πn M, Qp /ZZp )
←−
n
with respect to the inductive system
M/πn −→ M/πm
for m ≥ n ≥ 0 ,
x mod πn 7−→ ππmn x mod πm .
The Λ-module α(M ) is called the Iwasawa-adjoint of M .
(5.5.6) Proposition. For a finitely generated Λ-torsion module M one has a
canonical isomorphism
α(M ) ∼
= E 1 (M ).
In particular, α(M ) is independent of the choice of the sequence {πn }.
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316
Chapter V. Iwasawa Modules
Proof: From the exact sequence
T0 (M )/πn −→ M/πn −→(M/T0 (M ))/πn −→ 0
and lim
T0 (M )/πn = 0, we obtain α(M/T0 (M )) ∼
= α(M ). Thus using
−→ n
(5.5.4)(i), we may assume T0 (M ) = 0. Then the sequence
π
n
0 −→ M −→
M −→ M/πn −→ 0
is exact and M/πn is finite for all n ≥ 0, because the multiplication on M by πn
is a pseudo-isomorphism by (5.1.6). Using (5.5.3)(iv), we get isomorphisms
∼ E 2 (M/π ) ∼ Hom
E 1 (M )/πn −→
Zp ).
n =
ZZ (M/πn , Qp /Z
p
1
Since πn → 0 in Λ when n → ∞ and E (M ) is finitely generated, it follows
that
E 1 (M ) = lim E 1 (M )/πn ∼
= lim Hom ZZp (M/πn , Qp /ZZp ) = α(M ). 2
←−
n
←−
n
(5.5.7) Corollary. Let M be a finitely generated Λ-torsion module. Assume
that µ(M ) = 0. Then
E 1 (M ) ∼
= Hom ZZ (M, ZZp ).
p
Proof: Since µ(M ) = 0, we can take πn = pn+1 . From (5.5.6) we obtain
E 1 (M ) ∼
Hom ZZp (M/pn , Qp /ZZp )
= lim
←− n
= lim Hom ZZp (M, ZZ/pn ) = Hom ZZp (M, ZZp ).
←− n
(5.5.8) Proposition. Let M be a finitely generated Λ-module.
(i) There exists an exact sequence
0 −→ E 2 (T2 (M )) −→ E 1 (M ) −→ E 1 (T1 (M )) −→ 0
inducing isomorphisms
(ii) E 2 (T2 (M )) ∼
= E 1 (M/T1 (M )) ∼
= T0 (E 1 (M )),
(iii) E 1 (T1 (M )) ∼
= E 1 (M )/T0 (E 1 (M )).
Furthermore, there are canonical isomorphisms
(iv) E 1 (E 1 (M )) ∼
= T1 (M )/T0 (M ),
(v) E 2 (E 1 (M )) ∼
= T2 (M ),
(vi) E 2 (E 2 (M )) ∼
= T0 (M ).
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317
§5. Homotopy Invariants of Iwasawa Modules
Proof: From the exact sequence
ϕ
M
0 −→ T1 (M ) −→ M −→
M ++ −→ T2 (M ) −→ 0,
we obtain exact sequences
∼ E 2 (T (M )) → E 2 (M ++ ) = 0,
0 = E 1 (M ++ ) → E 1 (im ϕM ) →
2
0 = E 0 (T1 (M )) → E 1 (im ϕM ) → E 1 (M ) → E 1 (T1 (M )) → E 2 (im ϕM ) = 0.
This implies (i) and the first isomorphism of (ii). Using T0 (E 1 (T1 (M ))) = 0 (by
(5.5.3)(ii)) and the finiteness of E 2 (T2 (M )), we obtain the second isomorphism
in (ii) and assertion (iii).
In order to prove (iv), we first observe that by (5.5.4)(i) and (iii) above,
E 1 (E 1 (M )) ∼
= E 1 (E 1 (M )/T0 (E 1 (M )))
∼
= E 1 (E 1 (T1 (M ))) ∼
= E 1 (E 1 (T1 (M )/T0 (M ))).
f
In the proof of (5.4.11) we saw that E 1 coincides with D on Ho1p (Λ). Since
pdΛ (T1 (M )/T0 (M )) ≤ 1 by (5.5.3)(iv), we therefore obtain
E 1 (E 1 (T1 (M )/T0 (M ))) = E 1 (D(T1 (M )/T0 (M )))
= T1 (T1 (M )/T0 (M )) = T1 (M )/T0 (M ).
Finally, from (i) we get the isomorphism
∼ E 2 (E 2 (T (M ))) −→ 0
0 = E 2 (E 1 (T1 (M ))) −→ E 2 (E 1 (M )) −→
2
and using (5.5.3)(iv),
E 2 (E 2 (T2 (M ))) ∼
= T2 (M )∗∗ = T2 (M ) ,
thus proving (v). Again by (5.5.3)(iv) we obtain (vi).
(5.5.9) Corollary. We have the following equivalences:
E 1 (M ) is finite ⇐⇒ T1 (M ) is finite ⇐⇒ E 1 (E 1 (M )) = 0.
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318
Chapter V. Iwasawa Modules
(5.5.10) Proposition. Let M be a finitely generated Λ-module.
(i) E 0 (M )
∼
= lim
←−
pm (M
∨ Γn
)
is free of the same rank as M ,
n,m
(ii) E 1 (tor ZZp M )
∼
(M ∨ /pm )Γn ,
= lim
←−
n,m
(iii) E 1 (M/tor ZZp M ) ∼
(pm (M ∨ ))Γn
= lim
←−
n,m
(∼
= Hom ZZp (M, ZZp ) if M is a Λ-torsion module),
(iv) E 1 (M δ )
∼
= lim
←−
pm ((M
∨
)Γn ) ∼
= Hom ZZp (M δ , ZZp ),
n,m
(v) E 1 (M/M δ )
∼
((M ∨ )Γn )/pm ,
= lim
←−
n,m
(vi) E 2 (M )
∼
(M ∨ )/(pm , Γn ) ,
= lim
←−
n,m
where the transition maps are the obvious ones. Recall that M δ =
[
M Γn is
n
the maximal Λ-submodule of M on which Γ acts discretely.
Proof: Since H 0 (Γn , A) = AΓn and H 1 (Γn , A) = AΓn for a discrete Γ -module
A, the assertions (i), (ii), (iii) and (vi) follow immediately from (5.4.13)(ii)-(iv).
If M is Λ-torsion, then M/tor ZZp M is a torsion module with trivial µ-invariant
and we obtain from (5.5.7) that
E 1 (M/tor ZZp M ) = Hom ZZp (M/tor ZZp M, ZZp ) = Hom ZZp (M, ZZp ) ,
showing the additional statement in (iii). Using (5.4.13)(i), we obtain an exact
sequence
(M ∨ )Γn /pm −→ E 1 (M ) −→ lim
0 −→ lim
←−
←−
n,m
pm ((M
∨
)Γn ) −→ 0 ,
n,m
where the cokernel is isomorphic to Hom ZZp (M δ , ZZp ) while the kernel vanishes
for M = M δ . By (5.5.3)(iv) we have E 2 (M/M δ ) = 0, thus we obtain an exact
sequence
0 −→ E 1 (M/M δ ) −→ E 1 (M ) −→ E 1 (M δ ) −→ 0.
Since the first exact sequence is functorial in M , it must be isomorphic to the
second one, and we get (iv) and (v).
2
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319
§5. Homotopy Invariants of Iwasawa Modules
For the rest of this section we consider Λ-modules up to pseudo-isomorphism
(denoted by ≈).
(5.5.11) Proposition. Let M and M 0 be finitely generated Λ-torsion modules.
Then
M ≈ M0
if and only if E 1 (M 0 ) ≈ E 1 (M ).
Proof: We may assume T0 (M ) = 0 = T0 (M 0 ). From the exact sequence
0 −→ M −→ M 0 −→ C −→ 0
with a finite Λ-module C, we obtain
0 = E 1 (C) −→ E 1 (M 0 ) −→ E 1 (M ) −→ E 2 (C) ,
and hence E 1 (M 0 ) ≈ E 1 (M ). Conversely, since E 1 (M 0 ) ≈ E 1 (M ), (5.5.8)(iv)
implies the pseudo-isomorphisms M ≈ E 1 (E 1 (M )) ≈ E 1 (E 1 (M 0 )) ≈ M 0 .
2
(5.5.12) Definition. Let M be a Λ-module. We define the Λ-module M ◦ to be
the Λ-module M with the inverse Γ -action:
if x ∈ M ◦ then γ ◦ x := γ −1 x for γ
Remark: If F
∈
∈
Γ.
Λ is a Weierstraß polynomial, then
(Λ/F )◦ ∼
= Hom (Λ/F, ZZp ).
We have (Λ/F )◦ ∼
= Λ/F 0 for the Weierstraß polynomial F 0 which is obtained
from F by substituting (1 + T )−1 − 1 for T and multiplying by (1 + T )deg F .
(5.5.13) Proposition. Let M be a finitely generated Λ-module. Then there
exists a pseudo-isomorphism
E 1 (M ) ≈ T1 (M )◦ .
Proof: From the exact sequence
0 −→ tor ZZp M −→ M −→ M/tor ZZp M −→ 0,
we obtain the exact sequence
0 → E 1 (M/tor ZZp M ) → E 1 (M ) → E 1 (tor ZZp M ) → E 2 (M/tor ZZp M ).
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320
Chapter V. Iwasawa Modules
Using (5.5.8)(i) and (5.5.7), we see that
E 1 (M/tor ZZp M ) ≈ E 1 (T1 (M )/tor ZZp M )
∼
= Hom ZZp (T1 (M )/tor ZZp M, ZZp )
∼
= (T1 (M )/tor ZZ M )◦ .
p
Furthermore,
s
M
E 1 (tor ZZp M ) ≈ E 1 (
∼
=
s
M
Λ/pmi )
for some mi
≥
0
i=1
E 1 (Λ/pmi )
i=1
∼
=
s
M
(Λ/pmi )◦ ≈ (tor ZZp M )◦ ,
i=1
so that
E 1 (M ) ≈ E 1 (M/tor ZZp M ) ⊕ E 1 (tor ZZp M )
≈ (T1 (M )/tor ZZp M )◦ ⊕ (tor ZZp M )◦ ≈ T1 (M )◦ .
2
Exercise: Let M be a finitely generated Λ-torsion module and let Y be defined by the exact
sequence
Y
0 −→ T0 (M ) −→ M −→
Mp −→ Y −→ 0.
ht(p)=1
Then there is a canonical isomorphism
α(M ) ∼
= Homcts (Y, Qp /ZZp ).
This was the original definition of the adjoint due to Iwasawa.
Hint: Let {πn } be a sequence of elements of Λ as considered in (5.5.5) and let a be the
annihilator ideal of M . Let S be the multiplicatively closed subset in Λ̄ := Λ/a which is
generated by the images of π0 , π1 , . . .. Prove the isomorphism
A := S −1 Λ̄ ∼
= lim Λ̄(i) ,
−→ i
where the inductive system is given by Λ̄(i) = Λ̄, Λ̄(i) → Λ̄(j) , λ 7→
on π0 , π1 , . . . , the ring A is artinian, and thus
Y
∼
A=
Λp /aΛp .
πj
πi λ.
By the assumptions
p∈ supp(M )
ht(p)=1
Conclude that
lim M ⊗Λ Λ̄(i) ∼
= M ⊗Λ A ∼
=
−→
i
πj
πi
ker(πj )
Mp .
ht(p)=1
Finally, consider the exact commutative diagram
πi
0½¸¹º»¼³´µ¶·°±²
ker(πi )
M
M
0
Y
M
πj
M
M/πi M
0
πj
πi
M/πj M
0.
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§6. Differential Modules and Presentations
321
§6. Differential Modules and Presentations
R. FOX, in 1953, established a theory of derivations in the group ring of
a free discrete group, the “free differential calculus”. In the following we
consider a topological version of this theory for the completed group ring of a
profinite group.
Throughout this section, let p be a prime number, c a class of finite groups
containing ZZ/pZZ and closed under taking subgroups, homomorphic images
and group extensions. Let G be a pro-c-group and let α : ZZp [[G]] → ZZp denote
the augmentation map (cf. §2).
(5.6.1) Definition. A derivation of ZZp [[G]] into a compact ZZp [[G]]-module A
is a continuous ZZp -linear map
D : ZZp [[G]] −→ A
satisfying
D(f g) = D(f )·α(g) + f ·D(g)
for all f, g
∈
ZZp [[G]].
Remark: A derivation is clearly determined by its restriction to G ⊆ ZZp [[G]]
and this restriction is a continuous inhomogeneous 1-cocycle of G with values
in the compact G-module A (cf. II §7). On the other hand, one easily verifies
that every continuous inhomogeneous 1-cocycle of G with values in A extends
to a derivation of ZZp [[G]] into A. Note that
(i) D(a) = 0 for a ∈ ZZp ,
(ii) D(σ −1 ) = −σ −1 D(σ) for σ ∈ G.
Now we construct a ZZp [[G]]-module Ω ZZp [[G]] as follows: take the free compact ZZp [[G]]-module which is (topologically) generated by the symbols df ,
f ∈ ZZp [[G]], and then take the quotient by the closed submodule generated by
the relations
d(af + bg) − adf − bdg
d(f g) − df ·α(g) − f ·dg
where f, g ∈ ZZp [[G]], a, b ∈ ZZp . (In particular, these relations imply that d1 = 0
for the unit element 1 ∈ G.) The map
d : ZZp [[G]] −→ Ω ZZp [[G]] , f 7−→ df ,
is a derivation and the pair (Ω ZZp [[G]] , d) satisfies the universal property:
For every derivation D : ZZp [[G]] → A, there exists a unique continuous
ZZp [[G]]-homomorphism ϕ : Ω ZZp [[G]] → A with D = ϕ ◦ d.
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322
Chapter V. Iwasawa Modules
(5.6.2) Definition. We call Ω ZZp [[G]] the module of (noncommutative) differential forms of ZZp [[G]].
Recall that the kernel of the augmentation map: α : ZZp [[G]] −→ ZZp is called
the augmentation ideal and denoted by IG ⊆ ZZp [[G]].
(5.6.3) Proposition. There is a canonical isomorphism
∼ I ,
ϕ : Ω ZZp [[G]] −→
G
satisfying ϕ(dσ) = σ − 1 for every σ
Proof: For σ, τ
∈
G ⊆ ZZp [[G]].
G, we have the identity
στ − 1 = (σ − 1) + σ(τ − 1)
in IG , so ϕ defines a homomorphism. Furthermore, the inverse map ϕ−1 :
IG −→ Ω ZZp [[G]] , σ − 1 7−→ dσ, is well-defined.
2
∈
(5.6.4) Proposition. If F is a free pro-c-group of rank r on the generators
x1 , . . . , xr , then Ω ZZp [[F ]] is a free ZZp [[F ]]-module of rank r on the generators
dx1 , . . . , dxr .
Proof: Let G be a pro-c-group and let A be a compact ZZp [[G]]-module. We
have a split exact sequence
π
1 −→ A −→ E −→ G −→ 1
where E is the semi-direct product of G by A and the action of G on A is the
natural one.∗) Since A is a pro-p-group, E is a pro-c-group. Therefore we
have a 1−1-correspondence between the continuous homomorphic sections to
π and the derivations from ZZp [[G]] to A.∗∗)
Now let G = F be a free pro-c-group. Then, by the observation above, a
derivation from ZZp [[F ]] to A is uniquely determined by the arbitrarily chosen
images of a set of free generators of F . This proves the proposition.
2
Returning to the general case, assume that we are given an exact sequence
1 −→ H −→ G −→ G −→ 1
of pro-c-groups with the corresponding exact sequence
0 −→ I −→ ZZp [[G ]] −→ ZZp [[G]] −→ 0,
∗) The sequence corresponds to the zero element in H 2 (G, A), cf. I §2.
∗∗) cf. ex.1 in I §2.
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323
§6. Differential Modules and Presentations
where
I = IH ZZp [[G ]].
The following proposition is the profinite analogue of what is called the
second fundamental sequence in commutative algebra.
(5.6.5) Proposition. There exists an exact sequence of ZZp [[G]]-modules
θ
I/I 2 −→ ZZp [[G]] ⊗ ZZp [[G ]] Ω ZZp [[G ]] −→ Ω ZZp [[G]] −→ 0
and the image of θ is canonically isomorphic to H ab (p) = H1 (H , ZZp ).
Proof: Recall that for a compact ZZp [[H ]]-module M ,
MH := M/IH M = ZZp ⊗ ZZp [[H ]] M.
Furthermore, ZZp [[G ]] is a (compactly) induced H -module and therefore homologically trivial.∗) Now we apply H (H , −) = Tor ZZp [[G ]] (ZZp [[G]], −) to
the exact sequence
.
.
aug
0 −→ IG −→ ZZp [[G ]] −→ ZZp −→ 0.
Via the identification ZZp [[G]] = ZZp [[G ]]/I and by (5.6.3), we obtain the commutative diagram with exact rows:
0¾¿ÀÁÂÃÄÅÆÇÈÉÊË
IH ZZp [[G ]]/IH IG
IG /IH IG
0
H1 (H , ZZp )
ZZp [[G]] ⊗ ZZp [[G ]] Ω ZZp [[G ]]
The inclusion (IH ZZp [[G ]])2
⊆
ZZp [[G ]]/IH ZZp [[G ]]
ZZp
0
ZZp [[G]]
ZZp
0.
IH IG then finally implies the existence of
θ : I/I 2 −→ IH ZZp [[G ]]/IH IG .
2
Now assume that G is a finitely generated pro-c-group. Then we have an
exact sequence
1 −→ R −→ Fd −→ G −→ 1
for some d ∈ IN, where Fd is a free pro-c-group of rank d and R is the normal
subgroup of Fd generated by the relations of G (with respect to the chosen
∗) In other words, the Pontryagin dual of ZZ [[G]] is an induced discrete G-module, hence a
p
cohomologically trivial G-module.
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324
Chapter V. Iwasawa Modules
generators of G, i.e. the images of a basis of Fd in G). The abelian pro-pgroup Rab (p) is a ZZp [[G]]-module in a natural way and we call it the p-relation
module of G with respect to the given presentation of G. The following
theorem is a profinite analogue of a theorem of Lyndon for discrete groups.
(5.6.6) Theorem. There is a canonical exact sequence
0 −→ Rab (p) −→ ZZp [[G]]d −→ ZZp [[G]] −→ ZZp −→ 0.
In particular, if cdp G ≤ 2, then Rab (p) is a projective ZZp [[G]]-module.
Proof: We apply (5.6.5) to the exact sequence 1 → R → Fd → G → 1 and
observe that Ω ZZp [[G]] = IG by (5.6.3) and that Ω ZZp [[Fd ]] is a free ZZp [[Fd ]]-module
of rank d by (5.6.4). The last assertion follows from (5.2.13).
2
We now consider the following general problem: Given an exact sequence
of pro-c-groups
1 −→ H −→ G −→ G −→ 1,
what can we say about the structure of H ab (p) as a ZZp [[G]]-module?
Theorem (5.6.6) gives us information in the case that G is a free pro-cgroup. In the general case, assume that G is finitely generated and choose a
presentation F G of G by a free pro-c-group F of rank d. Then we obtain
a commutative diagram
ÌÍÎÏÐÑÒÓÔÕÖ×ØÙÚÛÜÝ
1
1
N
N
1
R
F
G
1
1
H
G
G
1
1
1
where R and N are defined by the exactness of the corresponding sequences.
In addition, we set
X : = H ab (p),
Y : = ZZp [[G]] ⊗ ZZp [[G ]] Ω ZZp [[G ]] = IG /IH IG .
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§6. Differential Modules and Presentations
(5.6.7) Proposition. With the notation above, we have a commutative exact
diagram
Þßàáâãäåæçèéêëìíîïðñò
0
0
0
H2 (H , ZZp )
ab
(p)
NH
Rab (p)
X
0
0
H2 (H , ZZp )
ab
(p)
NH
ZZp [[G]]d
Y
0
IG
IG
0
0.
In particular, there is an exact sequence
0 −→ Rab (p) −→ X ⊕ ZZp [[G]]d −→ Y −→ 0.
Furthermore, if cdp G ≤ 2, then N ab (p) is a projective ZZp [[G ]]-module and
ab
NH
(p) is a projective ZZp [[G]]-module. If cdp G ≤ 2 and cdp G ≤ 1, then
H2 (H , ZZp ) is a projective ZZp [[G]]-module.
Proof: The upper horizontal sequence is the homological form of the five
term exact sequence (1.6.7) for the group extension 1 → N → R → H → 1
and the module ZZp . The zero on the left follows because cdp R ≤ cdp F = 1.
We obtain the lower horizontal sequence by taking H -homology of the exact
sequence
(∗)
0 −→ N ab (p) −→ ZZp [[G ]]d −→ IG −→ 0
(apply (5.6.6) to 1 → N → F → G → 1), using the homological H -triviality
of ZZp [[G ]] and the resulting isomorphism H1 (H , IG ) ∼
= H2 (H , ZZp ).
The right-hand vertical sequence is the H -homology sequence associated to
0 → IG → ZZp [[G ]] → ZZp → 0 (using the same argument as above). Finally, the
left-hand vertical sequence is the sequence from theorem (5.6.6). The diagram
commutes since all arrows are the natural ones and we use the compatible
identifications
ZZp [[G]]d = ZZp [[G]] ⊗ ZZp [[F ]] Ω ZZp [[F ]]
= ZZp [[G]] ⊗ ZZp [[G ]] ZZp [[G ]] ⊗ ZZp [[F ]] Ω ZZp [[F ]] = H0 (H , ZZp [[G ]]d ).
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Chapter V. Iwasawa Modules
Considering the pull back extension X 0 of
0óôõö÷øù
Rab (p)
ZZp [[G]]d
IG
0
X0
Y,
0 ∼
d
we see that X = ZZp [[G]] ⊕ X, which gives us the exact sequence asserted in
the proposition.
Finally, we assume cdp G ≤ 2. Then by (5.2.13) and the exact sequence (∗),
ab
(p) is a
we conclude that N ab (p) is a projective ZZp [[G ]]-module, so that NH
projective ZZp [[G]] -module. If, in addition, cdp G ≤ 1, then pd ZZp [[G]] ≤ 2 (see
§2 ex.5 and (5.2.13)), and so pd ZZp [[G]] H2 (H , ZZp ) = pd ZZp [[G]] X − 2 = 0.
2
(5.6.8) Corollary. With the notation above, we assume in addition that G is a
finitely generated pro-p-group with finitely many defining relations. Let
1 −→ N −→ F −→ G −→ 1
ab
be a minimal presentation of G by a free pro-p-group F . Then NH
is a finitely
generated free ZZp [[G]]-module and
ab
rank ZZp [[G]] NH
= dimIFp H 2 (G , IFp ).
Proof: From the exact sequence
∼
0 −→ H 1 (G , IFp ) −→
H 1 (F, IFp ) −→ H 1 (N, IFp )G −→ H 2 (G , IFp ) −→ 0
it follows that
ab
)/G = dimIFp H 1 (N, IFp )G = dimIFp H 2 (G , IFp ),
dimIFp (NH
ab
and so NH
is finitely generated as a ZZp [[G]]-module. By (5.6.7) the ZZp [[G]]ab
module NH
(p) is projective, and so it follows from (5.2.20) that it is free.
2
Now we give a description of the ZZp [[G]]-module Y = IG /IH IG up to homotopy equivalence in terms of the p-dualizing module D2(p) = lim
D2 (ZZ/pm ZZ)
−→ m
of G .
(5.6.9) Proposition. Assume that cdp G = 2 and that N ab (p) is a finitely
generated ZZp [[G ]]-module. Let Z = (D2(p)H )∨ . Then
Y ' DZ,
so Y is determined by Z up to projective summands. If, in addition, the group
H2 (H , ZZp ) vanishes, then
E 1 (Y ) ∼
=Z.
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327
Proof: Applying (5.6.6) to the exact sequence 1 → N → F → G → 1, we
see that the finitely generated ZZp [[G ]]-module N ab (p) is projective. Thus ZZp
possesses a resolution by finitely generated projective ZZp [[G ]]-modules and we
get an exact sequence
(ZZp [[G ]]d )+ −→ N ab (p)+ −→ E 1 (IG ) −→ 0.
Applying (5.4.13)(ii) to ZZp , we obtain
E 1 (IG ) ∼
D2 (ZZ/pm ZZ))∨ = (D2(p) )∨ .
= E 2 (ZZp ) = (lim
−→
m
Taking H -coinvariants of the sequence above, yields the exact sequence
ab
(p)+ −→ Z −→ 0
(ZZp [[G]]d )+ −→ NH
(here we use Hom ZZp [[G ]] (M, ZZp [[G ]])H = Hom ZZp [[G]] (MH , ZZp [[G]]) for a
finitely generated projective ZZp [[G ]]-module M ). From the exact sequence
ab
NH
(p) −→ ZZp [[G]]d −→ Y −→ 0
ab
and the fact that NH
(p) is projective, we now obtain DY ' Z and Y ' DZ.
If H2 (H , ZZp ) = 0, then we have the exact sequence
ab
0 −→ NH
(p) −→ ZZp [[G]]d −→ Y −→ 0
inducing the exact sequence
ab
(ZZp [[G]]d )+ −→ NH
(p)+ −→ E 1 (Y ) −→ 0 .
Thus E 1 (Y ) ∼
= Z.
2
In the following, we will make use of some well-known facts about group
algebras of finite groups, which we briefly recall.
(5.6.10) Theorem. Let R be a complete discrete valuation ring with quotient
field K of characteristic zero and residue field k of positive characteristic,
and let G be a finite group. Furthermore, let L, M, N be finitely generated
R[G]-modules. Then the following hold:
(i) If M ⊕ L ∼
= N ⊕ L, then M ∼
= N.
(ii) If M and N are projective and M ⊗ K ∼
= N ⊗ K as K[G]-modules, then
∼
M = N.
(iii) If M and N are projective and M ⊗ k ∼
= N ⊗ k as k[G]-modules, then
∼
M = N.
Assertion (i) is a consequence of the Krull-Schmidt-Azuyama theorem, see
[32], §6, cor.6.15. For (ii) we refer the reader to [214], chap.16.1, cor. 2 of
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Chapter V. Iwasawa Modules
thm. 34. Finally, (iii) follows in a straightforward manner from Nakayama’s
lemma.
Returning to the profinite situation, one often wants to determine the ZZp [[G]]module structure of Y = IG /IH IG not only up to homotopy equivalence but
up to isomorphism. The following proposition is useful in the applications.
(5.6.11) Proposition. Let G be a profinite group. Let M and N be finitely
generated ZZp [[G]]-modules such that
(i) M ' N ,
(ii) M ⊗ Qp ∼
= (N ⊕ ZZp [[G]]m ) ⊗ Qp for some m ∈ IN.
Then
M∼
= N ⊕ ZZp [[G]]m .
In particular, a finitely generated projective ZZp [[G]]-module P is free if and
only if P ⊗ Qp is (ZZp [[G]] ⊗ Qp )-free. Furthermore, instead of (ii), it suffices
to assume the weaker condition
(ii)0 MU ⊗ Qp ∼
= NU ⊗ Qp ⊕ Qp [G/U ]m
for all open normal subgroups U of G.
Proof: Since M ' N ' N ⊕ ZZp [[G]]m , it follows from (5.4.3)(ii) that there
are finitely generated projective ZZp [[G]]-modules P1 and P2 such that
M ⊕ P1 ∼
= N ⊕ ZZp [[G]]m ⊕ P2 .
From the second assumption (ii) (or from (ii)0 ) and by (5.6.10)(i), we get
(P1 )U ⊗ Qp ∼
= (P2 )U ⊗ Qp
for all open normal subgroups U of G. Hence the projective ZZp [G/U ]-modules
(P1 )U and (P2 )U are isomorphic by (5.6.10)(ii). Again by (5.6.10)(i), we obtain
MU ∼
= NU ⊕ ZZp [G/U ]m .
In particular, we have, for every n ∈ IN and every open normal subgroup
U ⊆ G, an isomorphism of finite ZZ/pn ZZ[G/U ]-modules
(M/pn )U ∼
= (N/pn )U ⊕ ZZ/pn ZZ[G/U ]m .
Passing to the projective limit, the result follows by the usual compactness
argument.
2
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329
§6. Differential Modules and Presentations
(5.6.12) Proposition. Let
1 −→ H −→ G −→ ZZp −→ 1
be an exact sequence of a profinite groups such that H is of order prime to p.
Then the augmentation ideal IG of ZZp [[G]] is a free ZZp [[G]]-module of rank 1,
i.e.
IG ∼
= ZZp [[G]] .
Proof: Since cdp G = 1, the ZZp [[G]]-module IG is projective by (5.2.13). Let
U ⊆ G be an open normal subgroup. From the exact sequence
0 −→ H1 (U, ZZp ) −→(IG )U −→ ZZp [G/U ] −→ ZZp −→ 0 ,
using Maschke’s theorem (2.6.12) and observing that H1 (U, ZZp ) = U ab (p) is
isomorphic to ZZp , we obtain
Qp ⊕ (IG )U ⊗ Qp ∼
= Qp [G/U ] ⊕ Qp ,
and by (5.6.10)(i), we get (IG )U ⊗ Qp ∼
= Qp [G/U ]. The result now follows
from (5.6.11).
2
(5.6.13) Proposition. Let
1 −→ H −→ G −→ G −→ 1
be an exact sequence of profinite groups, where G is finitely generated of
cdp G ≤ 2 and G is a pro-p-group of cdp G ≤ 2 with finite relation rank
r(G) = dimIFp H 2 (G, ZZ/pZZ) < ∞. Furthermore, assume that
H2 (G , ZZp ) = 0 = H2 (H , ZZp ).
In particular, this is fulfilled if scdp G
≤
2.
Then X = H ab (p) is a finitely generated ZZp [[G]]-module with pd ZZp [[G]] X
and the following assertions are equivalent:
≤
1
(i) X is free as a ZZp [[G]]-module.
(ii) XG is free as a ZZp -module.
(iii) The map p G ab −→ p Gab is injective.
Proof: Since cdp G ≤ 2, we have an exact sequence
p
0 −→ H2 (G, ZZp ) −→ H2 (G, ZZp ) −→ H2 (G, ZZ/pZZ) ,
and from our assumption, we know that dimIFp H2 (G, ZZ/pZZ) = r(G) is finite.
Thus H2 (G, ZZp ) is free of finite rank as a ZZp -module. The Hochschild-Serre
sequence
0 −→ H2 (G, ZZp ) −→ XG −→ G ab (p) −→ Gab −→ 0
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Chapter V. Iwasawa Modules
(using H2 (G , ZZp ) = 0) shows that XG is finitely generated as a ZZp -module and
that XG is ZZp -free if and only if ker (G ab (p) → Gab ) is ZZp -torsion-free. This
proves the equivalence (ii) ⇐⇒ (iii).
Since H2 (H , ZZp ) = 0 and noting that cdp G , cdp G
shows that
≤
2, proposition (5.6.7)
ab
(p) −→ Rab (p) −→ X −→ 0
0 −→ NH
is a projective resolution of X of length 1, i.e. pd ZZp [[G]] X ≤ 1.
Recall that ZZp [[G]] is a local ring since G is a pro-p-group by (5.2.16)(iii),
and therefore finitely generated projective ZZp [[G]]-modules are free by (5.2.20).
Since XG is finitely generated as a ZZp -module, Nakayama’s lemma (5.2.18)
implies that X is a finitely generated ZZp [[G]]-module.
Now assume that XG is ZZp -free and let
0 −→ P −→ ZZp [[G]]r −→ X −→ 0
∼ X . We obtain an
be a minimal resolution of X, i.e. (ZZp [[G]]r )G = ZZrp →
G
∼
≤
isomorphism H1 (G, X) = PG . Since cdp G
2 and H2 (G , ZZp ) = 0, the
Hochschild-Serre spectral sequence shows that
2
∞
H1 (G, H1 (H , ZZp )) = E1,1
= E1,1
= 0.
Thus PG = 0 and therefore the compact module P is trivial by Nakayama’s
lemma . This proves the nontrivial implication (ii) ⇒ (i).
2
(5.6.14) Corollary. Assertions (i) – (iii) of (5.6.13) are true if cdp G
in this case
≤
1, and
rank ZZp [[G]] X = dimIFp H 1 (G , IFp ) − d(G) + r(G) ,
where d(G) = dimIFp H 1 (G, IFp ) and r(G) = dimIFp H 2 (G, IFp ).
Proof: Since cdp G
≤
1, we have scdp H , scdp G
pG
ab
≤
2. Furthermore,
= (H 2 (G , IFp ))∨ = 0
showing that (iii) is true. The Hochschild-Serre sequence
0 −→ H 1 (G, IFp ) −→ H 1 (G , IFp ) −→ H 1 (H , IFp )G −→ H 2 (G, IFp ) −→ 0
implies that
dimIFp XG /p = dimIFp H 1 (H , IFp )G
= dimIFp H 1 (G , IFp ) − d(G) + r(G).
Since X is ZZp [[G]]-free, we have dimIFp XG /p = rank ZZp [[G]] X. This finishes
the proof.
2
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§6. Differential Modules and Presentations
We now consider the following situation, which is related to Iwasawa theory.
Let
1 −→ H −→ G −→ Γ −→ 1
be an exact sequence of profinite groups, where G is finitely generated and
Γ ∼
= ZZp . Let Λ = ZZp [[Γ ]]. Then X = H ab (p) = H1 (H , ZZp ) is a finitely
generated Iwasawa module, since
dimIFp (X/p)Γ = dimIFp H 1 (G , ZZ/pZZ) − 1
is finite by (5.3.10). Furthermore, we assume that the dimensions
hi = dimIFp H i (G , ZZ/pZZ) , i ≤ 2 ,
are finite, and we set
χ2 (G ) =
2
X
(−1)i hi .
i=0
(5.6.15) Lemma.
(i) pdΛ X
≤
1 if and only if H 1 (Γ, H 1 (H , Qp /ZZp )) is p-divisible.
(ii) If cdp G ≤ 2, then H2 (H , ZZp ) = H 2 (H , Qp /ZZp )∨ is a free Λ-module of
finite rank.
Proof: From (5.3.19)(i), we know that pdΛ X ≤ 1 is equivalent to the statement that X Γ is ZZp -free, or dually that H 1 (Γ, H 1 (H , Qp /ZZp )) is p-divisible.
This proves (i).
The surjection H 2 (G , Qp /ZZp ) H 2 (H , Qp /ZZp )Γ (which is obtained from
the Hochschild-Serre spectral sequence noting that cdp Γ = 1) shows that the
ZZp -module H2 (H , ZZp )Γ is finitely generated, and so H2 (H , ZZp ) is a finitely
generated Λ-module by (5.3.10). It follows from (5.6.7) that H2 (H , ZZp ) is
projective, hence Λ-free by (5.2.20).
2
(5.6.16) Proposition. Suppose pdΛ X
≤
1. Then there exists an exact sequence
0 → Λh2 −t → Λh1 −1 → X → 0 ,
where t = dimIFp (H2 (H , ZZp )/p)Γ . In particular,
rankΛ (X) = −χ2 (G ) + t.
If cdp G
≤
2, then t = rankΛ H2 (H , ZZp ).
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Chapter V. Iwasawa Modules
Proof: This follows from (5.3.20). Indeed, we have
d0 (X) = dimIFp (X/p)Γ = h1 − 1,
d1 (X) = dimIFp p (XΓ ) + dimIFp X Γ /p
= dimIFp H 1 (H , Qp /ZZp )Γ /p + dimIFp p H 1 (Γ, H 1 (H , Qp /ZZp ))
= dimIFp H 1 (G , Qp /ZZp )/p + dimIFp p H 1 (Γ, H 1 (H , Qp /ZZp )) ,
d2 (X) = dimIFp p X Γ = 0 ,
where the last equality follows from pdΛ X
sequence
≤
1 and (5.3.19)(i). The exact
0 → H 1 (Γ, H 1 (H , Qp /ZZp )) → H 2 (G , Qp /ZZp ) → H 2 (H , Qp /ZZp )Γ → 0 ,
which is induced by the Hochschild-Serre spectral sequence noting that
cdp Γ = 1, and the fact that H 1 (Γ, H 1 (H , Qp /ZZp ))/p = 0 by (5.6.15), shows
that
d1 (X) = dimIFp H 1 (G , Qp /ZZp )/p + dimIFp p H 2 (G , Qp /ZZp )
−dimIFp p H 2 (H , Qp /ZZp )Γ .
The exact cohomology sequence
0 → H 1 (G , Qp /ZZp )/p → H 2 (G , ZZ/pZZ) → p H 2 (G , Qp /ZZp ) → 0
p
obtained from 0 → ZZ/pZZ → Qp /ZZp → Qp /ZZp → 0 now implies that
d1 (X) = dimIFp H 2 (G , ZZ/pZZ) − dimIFp p H 2 (H , Qp /ZZp ))Γ
= h2 − t .
Finally, t is equal to rankΛ H2 (H , ZZp ) if cdp G
≤
2 by (5.6.15).
2
(5.6.17) Theorem. With the notation above, the following assertions are
equivalent:
(i) X contains no finite nontrivial Λ-submodule and rankΛ X = −χ2 (G ).
(ii) H 2 (H , Qp /ZZp ) = 0 and H 2 (G , Qp /ZZp ) is p-divisible.
Proof: By (5.3.19)(i) and (5.6.16), assertion (i) is equivalent to pdΛ X ≤ 1
and t = 0, hence to the fact that H 1 (Γ, H 1 (H , Qp /ZZp )) is p-divisible and that
H 2 (H , Qp /ZZp ) = 0. But this is precisely statement (ii), which can be seen
using the exact sequence
0 → H 1 (Γ, H 1 (H , Qp /ZZp )) → H 2 (G , Qp /ZZp ) → H 2 (H , Qp /ZZp )Γ → 0 .
2
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§6. Differential Modules and Presentations
333
(5.6.18) Theorem. Assume that G (and hence also H ) is a pro-p-group.
Then the following assertions are equivalent:
(i) H is a free pro-p-group.
(ii) H 2 (G , Qp /ZZp ) is p-divisible, H 2 (H , Qp /ZZp ) = 0 and the µ-invariant of
X is zero.
Proof: A pro-p-group G is free if and only if H 2 (G, ZZ/pZZ) = 0 (see (3.5.17)
and (3.3.2)(ii)), thus if and only if Gab is ZZp -torsion-free and H 2 (G, Qp /ZZp )
= 0. Hence (i) is equivalent to the statement that X is ZZp -torsion-free and
H 2 (H , Qp /ZZp ) = 0. The latter occurs precisely when µ(X) = 0 and X Γ =
H 1 (Γ, H 1 (H , Qp /ZZp ))∨ is ZZp -torsion-free. Again using the exact sequence
in the proof of (5.6.17), we obtain the result.
2
(5.6.19) Corollary. In the situation of theorem (5.6.18), assume, in addition,
that cdp G ≤ 2. Let U be an open subgroup of G , V = H ∩ U and
Γ 0 = U /V ⊆ Γ . Then Y = V ab is a finitely generated Λ0 = ZZp [[Γ 0 ]]-module
and the following assertions are equivalent:
(i) µ(X) = 0 and H 2 (H , Qp /ZZp ) = 0.
(ii) µ(Y ) = 0 and H 2 ( V , Qp /ZZp ) = 0.
Proof: Since cdp U = cdp G ≤ 2, the cohomology groups H 2 (G , Qp /ZZp ) and
H 2 (U , Qp /ZZp ) are p-divisible. Thus (i) (resp. (ii)) is equivalent to the freeness
of H (resp. V ) by (5.6.18). Since V is open in H and cdp H < ∞, we have
cdp V = cdp H by (3.3.5)(ii). Now (3.5.17) implies the result.
2
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Arithmetic Theory
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Chapter VI
Galois Cohomology
§1. Cohomology of the Additive Group
The Galois groups G = G(L|K) of Galois extensions L|K are profinite
groups and we may such use cohomology theory which we have developed in
the preceding chapters. We are particularly interested in the meaning of the
cohomology groups H n (G, A) for extensions of local and global fields, but we
first study their properties for general Galois extensions L|K.
Of particular importance is the absolute Galois group GK = G(K̄|K) of a
field K. It depends on the choice of a separable closure K|K, and is therefore
unique only up to inner automorphisms (we will return to this point in chapter
XII). By (1.6.3), however, its cohomology is independent of the choice and we
write
H n (K, A) := H n (GK , A).
In the following we will assume that all extension fields are contained in a
fixed separable closure.
The first G(L|K)-module which comes to mind is the additive group of a
Galois extension L of K. It is cohomologically trivial. In order to show this,
we may assume that G = G(L|K) is finite, see (1.2.5). Then the assertion is
trivial if char(K) = 0, because in this case L is uniquely divisible and the result
follows from (1.6.2)(c). In general, one argues as follows: the G-module L is
induced, because of the existence of a normal basis, i.e. of an element θ ∈ L
such that
L=
M
Kσθ.
σ ∈G
Because of (1.3.7) and (1.2.5), we obtain the
(6.1.1) Proposition. If L|K is an arbitrary Galois extension with Galois
group G, then
H q (G, L) = 0 for all q > 0.
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Chapter VI. Galois Cohomology
We call a field L p-closed if it has no Galois extensions of degree p. For
example, the separable closure K|K is p-closed and also the maximal pextension K(p)|K, i.e. the composite of all Galois extensions of p-power
order.
Let us assume now that the characteristic p = char(K) is positive. If the field
L is p-closed then the homomorphism
℘ : L −→ L, ℘(x) = xp − x,
is surjective, which can be seen as follows. Consider, for a ∈ L, the separable
polynomial f (x) = xp − x − a. If α is a root of f , then α + 1, . . . , α + p − 1 are
the other roots. Therefore, if a were not in the image of ℘, then the splitting
field of this polynomial would be a cyclic extension of L of degree p. But we
assumed L to be p-closed. We thus obtain the exact sequence
℘
0 −→ ZZ/pZZ −→ L −→ L −→ 0,
and the associated exact cohomology sequence yields, together with (6.1.1),
the
(6.1.2) Corollary. Let p = char(K) > 0. If L|K is a p-closed extension, then
H n (G(L|K), ZZ/pZZ) =



K/℘K
0
for
for
n = 1,
n ≥ 2.
Applying the last corollary to the fixed field of L with respect to a p-Sylow
group of G(L|K), (3.3.6) implies the
(6.1.3) Corollary. Let p = char(K) > 0. If L|K is a p-closed extension, then
cdp G(L|K) ≤ 1 .
From the above corollaries and (3.9.1), (3.9.5), we obtain the following
theorem, which in a sense gives a complete survey of the Galois extensions of
K of p-power degree.
(6.1.4) Theorem. If p = char(K) > 0, then the Galois group G of the maximal
p-extension K(p)|K is a free pro-p-group of rank
rk(G) = dimIFp K/℘K.
(6.1.5) Corollary. If K is a countable field of characteristic p > 0 then
G(K(p)|K) is a free pro-p-group of finite or countable infinite rank.
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§1. Cohomology of the Additive Group
Next we determine the Galois groups of the maximal p-extensions of local
and global fields of characteristic p.
(6.1.6) Lemma. Let K = k((t)) be the field of Laurent series over a perfect
field k with p = char(k) > 0. Then tk[[t]] ⊆ ℘K, in particular, ℘K is
open in the topology of K as a discrete valuation field. If R is a system of
representatives of k modulo ℘k, then the set L of Laurent polynomials,
L={
X
ai ti + a0 , ai
∈
k for i < 0 and a0
∈
R}
(i,p)=1
i<0
is a system of representatives of K modulo ℘K.
Proof: An easy computation shows that all power series of the form
are in the image of ℘, and for a = bp and i < 0 we have
P∞
i=1
ai ti
atpi − bti = ℘(bti ).
With these observations in mind, one immediately verifies that the set L is a
system of representatives of K modulo ℘K.
2
We obtain the following
(6.1.7) Proposition. Let K = k((t)) be the field of Laurent series over a field k
with p = char(k) > 0. If k is finite or countable, then G(K(p)|K) is a free
pro-p-group of countable infinite rank. If k is uncountable, then G(K(p)|K) is
free of uncountable rank.
Proof: In order to prove the statement, we may assume that k is perfect:
otherwise we replace k by its perfect hull without changing the absolute Galois
group and the cardinalities in question. Now the result follows from (6.1.6)
and (6.1.4).
2
In the global case we obtain the
(6.1.8) Proposition. Let K be a global field of positive characteristic p > 0.
Then G(K(p)|K) is a free pro-p-group of countable infinite rank.
Proof: By (6.1.5), the group in question is a free pro-p-group of at most
countable rank. In order to prove that the rank is infinite, we may restrict
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340
Chapter VI. Galois Cohomology
to the case K = IF(t), where IF is a finite field. All the infinitely many
representatives of IF((t)) modulo ℘IF((t)) given in (6.1.6) are in K and obviously
also independent modulo ℘K. Therefore, by (6.1.4), the rank of G(K(p)|K)
is infinite.
2
Let Kp |K denote the maximal abelian extension of exponent p = char(K),
i.e. the composite of all cyclic extensions of degree p. Its Galois group is
G(Kp |K) = G/Gp [G, G],
where [G, G] is the closure of the commutator subgroup of G = G(K(p)|K).
For the Pontryagin dual of this Galois group we obtain the isomorphism
G(Kp |K)∨ ∼
= H 1 (G, ZZ/pZZ) ∼
= K/℘K
The isomorphism associates to a ∈ K the character χa : G(Kp |K) → IFp , given
by χa (σ) = σα − α, where α is a root of the equation xp − x = a. Dually, we
obtain the isomorphism of Artin-Schreier theory (cf. [160], chap.IV, §3)
G(Kp |K) ∼
= Hom(K/℘K, Q/ZZ).
This explicit description of Kp |K has the following generalization to the
maximal abelian extension Kpn |K of exponent pn , whose Galois group is
n
G(Kpn |K) = G/Gp [G, G].
For every n ≥ 1, there exists a unique functor Wn from the category of rings
to itself, such that for every commutative ring R with unit the underlying set
of Wn (R) is Rn and the map
gh : Wn (R) −→ Rn ,
gh(a0 , . . . , an−1 ) =
(a0 , ap0 + pa1 , . . . , a0p
n−1
+ pa1p
n−2
+ p2 ap2
n−3
+ · · · + pn−1 an−1 ),
is a homomorphism of rings (see [212], chap.II, §6 or [160], chap.II, §4,
ex. 2-4). The elements (a0 , . . . , an−1 ) ∈ Wn (R) are called the Witt vectors of
n−1
length n and the elements a0 , ap0 + pa1 , . . . , ap0 + · · · + pn−1 an−1 the ghost
components of (a0 , . . . , an−1 ). For R = IFp , we have a canonical isomorphism
∼ Z
Wn (IFp ) −→
Z/pn ZZ,
(a0 , . . . , an−1 ) 7−→ ã0 + ã1 p + · · · + ãn−1 pn−1 mod pn ,
where ãi ∈ ZZ is the unique representative of ai with 0 ≤ ãi < p. The Witt
vectors with respect to a general ring R should be seen as an abstract version
of this p-adic expansions.
For an arbitrary ring R we have W1 (R) = R, and for each n
sequence of abelian groups
≥
1 an exact
V
0 −→ Wn (R) −→ Wn+1 (R) −→ R −→ 0,
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341
§1. Cohomology of the Additive Group
where V is given by (a0 , . . . , an−1 ) 7→ (0, a0 , . . . , an−1 ) and is called the
Verschiebung. If R is an integral domain of characteristic p, i.e. pR = 0, then
we have an exact sequence
℘
0 −→ Wn (IFp ) −→ Wn (R) −→ Wn (R),
where ℘ = F − id, F ((a0 . . . , an−1 )) = (ap0 , . . . , apn−1 ), so that ker(℘) = Wn (IFp )
∼
= ZZ/pn ZZ. We obtain the following generalization of Artin-Schreier theory.
(6.1.9) Theorem (ARTIN-SCHREIER-WITT). Let K be a field of characteristic
p > 0. Then for the Galois group of the maximal abelian extension Kpn |K of
exponent pn , we have a canonical isomorphism
G(Kpn |K) ∼
= Hom(Wn (K)/℘Wn (K), Q/ZZ).
Proof: Consider the separable closure K|K and the exact sequence of GK modules
℘
0 −→ ZZ/pn ZZ −→ Wn (K) −→ Wn (K̄) −→ 0 ;
(∗)
here the surjectivity of ℘ is obvious for n = 1, and for n > 1 it follows
recursively from the commutative and exact diagrams
0úûüýþÿ
Wn (K)
V
Wn+1 (K)
℘
0
Wn (K)
The exact sequence
℘
V
Wn+1 (K)
W1 (K)
0
℘
W1 (K)
0.
V
0 −→ Wn (K) −→ Wn+1 (K) −→ K̄ −→ 0
shows by induction on n that the GK -module Wn (K) is cohomologically trivial,
since K is cohomologically trivial by (6.1.1). The cohomology sequence
associated to (∗) is therefore an exact sequence
℘
0 −→ ZZ/pn ZZ −→ Wn (K) −→ Wn (K) −→ H 1 (GK , ZZ/pn ZZ) −→ 0
which yields the isomorphism
H 1 (GK , ZZ/pn ZZ) = Hom(G(Kpn |K), Q/ZZ) ∼
= Wn (K)/℘Wn (K),
and, dually, the isomorphism
G(Kpn |K) ∼
= Hom(Wn (K)/℘Wn (K), Q/ZZ).
2
If K|k is a Galois extension of local or global fields, it is natural to consider
also the ring of integers OK as a Galois module. We formulate the next results
in such generality that they apply to the local and the global case as well.
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342
Chapter VI. Galois Cohomology
Recall that the normalization B of a Dedekind domain A in a finite Galois
extension K of its quotient field k is called tamely ramified if for each prime
ideal p of A the unique prime factorization of pB in B has the form
pB = (P1 · · · Pg )e ,
with e prime to the characteristic of A/p, and such that the residue field
extensions (B/Pi ) | (A/p), i = 1, . . . , g, are separable. If K|k is an infinite
Galois extension, we say that B|A is tamely ramified if the normalization of A
in every finite normal subextension of K|k is tamely ramified.
(6.1.10) Theorem. Let A be a Dedekind domain with quotient field k, K|k
a Galois extension and B the integral closure of A in K. Then the following
conditions are equivalent
(i) B is a cohomologically trivial G(K|k)-module.
(ii) B|A is tamely ramified.
Proof: By (1.8.2), we may assume that K|k is finite. We consider the trace
map
X
TrB|A : B −→ A, x 7→
σx .
σ ∈G(K|k)
Claim: TrB|A is surjective if and only if B|A is tamely ramified.
Proof of the claim: We start with the case that A, and hence also B, is a
complete discrete valuation ring and denote the residue fields of A and B
by κ and κ0 , respectively. By Nakayama’s Lemma, the homomorphism of
finitely generated A-modules TrB|A is surjective if and only the induced map
Tr: κ0 → κ is surjective. As the inertia group acts trivially on κ0 , we have
Tr = e · Trκ0 |κ , where e is the ramification index. As Trκ0 |κ is surjective if κ0 |κ is
a separable field extension and zero otherwise, we see that TrB|A is surjective
if and only if κ0 |κ is separable and p - e, where p = char(κ).
In the general situation, TrB|A is surjective if and only if for every prime
ideal p of A the induced map
TrB|A ⊗ Âp : B ⊗A Âp −→ Âp
is surjective. As
Y
B ⊗A Âp =
B̂P ,
P|p
the general case reduces to the case of complete discrete valuation rings. This
proves the claim.
We put G = G(K|k) and start by showing the implication (i)⇒(ii). If B
is cohomologically trivial, then Ĥ 0 (G, B) = 0, hence TrB|A is surjective. By
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343
§2. Hilbert’s Satz 90
the claim, B|A is tamely ramified. It remains to show (ii)⇒(i). As the trace
is surjective, we have Ĥ 0 (G, B) = 0. Next we prove that H 1 (G, B) = 0. Let
a(σ) ∈ B be a 1-cocycle and let x ∈ B be such that TrB|A (x) = 1. Setting
b :=
X
a(σ)σx ,
σ ∈G
we obtain for τ
τb =
X
∈
G,
τ a(σ)(τ σx) =
σ ∈G
X
(a(τ σ) − a(τ ))(τ σx) = b − a(τ )TrB|A (x) .
σ ∈G
Therefore a(τ ) = (1 − τ )b , hence a(τ ) is a 1-coboundary.
Since the arguments above hold in the same way for all subgroups of G(K|k),
the result follows from (1.8.4).
2
In the case of a finite, cyclic Galois group, it is easy to calculate the Herbrand
index.
(6.1.11) Proposition. Let A be a Dedekind domain with quotient field k, K|k
a finite, cyclic Galois extension and B the integral closure of A in K. If A/m
is finite for all maximal ideals m ⊆ A, then the Herbrand index h(G(K|k), B)
is trivial, i.e.
#Ĥ −1 (G(K|k), B) = #Ĥ 0 (G(K|k), B),
and both groups are finite.
Proof: Let G(K|k) = {σ1 , . . . , σn }. We consider a normal basis of K|k, i.e.
we choose an α ∈ K such that σ1 α, . . . , σn α is a k-basis of K. For suitably
chosen a ∈ k × , we have an inclusion
B
⊆
Aaσ1 α + . . . + Aaσn α
of B into a free A[G(K|k)]-module. The quotient has finite A-length and is
therefore finite. By (1.7.5) and (1.7.6), we obtain h(G(K|k), B) = 1.
2
§2. Hilbert’s Satz 90
Again let L|K be a Galois extension and G its Galois group. The multiplicative group L× is also a G-module. It is only cohomologically trivial in
exceptional cases, but we always have
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Chapter VI. Galois Cohomology
(6.2.1) Theorem (Hilbert’s Satz 90).
H 1 (G, L× ) = 1.
Proof: By (1.2.5), we may assume that L|K is finite. Let a : G → L× be an
inhomogeneous 1-cocycle. For c ∈ L× we put
X
b=
a(σ)σc.
σ ∈G
Since the automorphisms σ ∈ G are linearly independent (see [17], chap.5, §7,
No.5), we may choose c ∈ L× in such a way that b is non-zero. For τ in G we
obtain
X
X
τ (b) =
τ (a(σ))τ σc =
a(τ )−1 a(τ σ)τ σc = a(τ )−1 b.
σ ∈G
−1
Thus a(τ ) = bτ (b)
for all τ
∈
2
G, i.e. a is a coboundary.
In the preceding section we obtained from the equality H 1 (G, L) = 0 the
theorem of Artin-Schreier-Witt for abelian extensions of exponent pn , where
p = char(K). From the vanishing of H 1 (G, L× ) we obtain a similar result,
called Kummer theory.
Let n be a natural number, not divisible by the characteristic of K. We
denote the group of n-th (resp. of all) roots of unity in the separable closure K
of K by µn (resp. µ). As H 1 (GK ,K × ) = 1, we obtain from the exact sequence
n
1 −→ µn −→K × −→K × −→ 1
the exact sequence
n
δ
K × −→ K × −→ H 1 (GK , µn ) −→ 0,
and hence an isomorphism
H 1 (GK , µn ) ∼
= K × /K ×n .
Let Kn |K be the maximal abelian extension of exponent n, i.e. the composite
of all finite abelian extensions L|K in K of degree dividing n. If µn ⊆ K, we
obtain isomorphisms
Homcts (G(Kn |K), µn ) ∼
= H 1 (GK , µn ) ∼
= K × /K ×n .
Dually, we obtain the
(6.2.2) Theorem. If n ≥ 1 is prime to the characteristic of K and µn
then
G(Kn |K) ∼
= Hom(K × /K ×n , µn ).
⊆
K,
Hilbert’s Satz 90 may be extended from the multiplicative group L× of a
Galois extension L|K to the G-group GLn (L) of all invertible n × n-matrices
over L in terms of non-abelian cohomology (see I §2).
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345
§2. Hilbert’s Satz 90
(6.2.3) Theorem. For every Galois extension L|K with Galois group
G = G(L|K) we have
H 1 (G, GLn (L)) = 1.
Proof: By proposition (1.2.5) (which also holds by the same arguments for
non-abelian cohomology), we may assume that L|K is finite. Let a be an
1-cocycle of G with values in GLn (L) and consider for a vector x ∈ Ln the
X
vector
b(x) =
a(σ)σx.
σ ∈G
The set {b(x), x ∈ Ln } generates the L-vector space Ln . In fact, if f is a linear
form on Ln which vanishes on the b(x), then for all λ ∈ L,
X
X
0 = f (b(λx)) =
f (a(σ)σλσx) =
f (a(σ)σx)σλ,
σ ∈G
σ ∈G
i.e. we have a linear relation between the σλ. But the automorphisms σ are
linearly independent, so that f (a(σ)σx) = 0, and since the matrices a(σ) are
invertible, we find f = 0.
From this observation, we can find vectors x1 , . . . , xn such that the yi = b(xi )
are linearly independent. If c is the matrix with columns x1 , . . . , xn , then
b := b(c) = (b(x1 ), . . . , b(xn )) = (y1 , . . . , yn ) is an invertible matrix and
b=
X
a(σ)σc.
σ ∈G
As in the case n = 1, we conclude that a(σ) = b(σb)−1 , i.e. a is a coboundary.
2
We denote the G-group of all invertible n × n-matrices over L with determinant equal to 1 by SLn (L).
(6.2.4) Corollary.
H 1 (G, SLn (L)) = 1.
Proof: From the exact sequence
det
1 −→ SLn (L) −→ GLn (L) −→ L× −→ 1,
we obtain the exact sequence of pointed sets
det
GLn (K) −→ K × −→ H 1 (G, SLn (L)) −→ H 1 (G, GLn (L)) = 1
(see I §3 ex.8), in which det is surjective.
2
The non-abelian cohomology groups H 1 have various interesting meanings,
all of which arise from the bijection
H 1 (G, A) ∼
= TORS (A)
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Chapter VI. Galois Cohomology
(see (1.2.3)). We mention the following application to the theory of quadratic
forms.
Let V be a vector space over K equipped with a quadratic form ϕ. Denote
by ϕL the canonical extension of ϕ to the L-vector space VL = V ⊗K L. The
orthogonal group O(ϕL ) of ϕL is the group of automorphisms of the pair
(VL , ϕL ), i.e. the L-automorphisms f : VL → VL with ϕL (f (v)) = ϕL (v) for all
v ∈ VL . The Galois group G = G(L|K) acts on VL by σ(v ⊗ λ) = v ⊗ σλ and
on O(ϕL ) by σ(f ) = σ ◦ f ◦ σ −1 for σ ∈ G and f : VL → VL in O(ϕL ). Thus
O(ϕL ) is a G-group.
We say that (V, ϕ) and (V 0 , ϕ0 ) are isomorphic over L if the pairs (VL , ϕL )
and (VL0 , ϕ0L ) are isomorphic. We denote by Eϕ (L|K) the set of isomorphism
classes of pairs (V 0 , ϕ0 ) which become isomorphic to (V, ϕ) over L.
(6.2.5) Proposition. We have a canonical bijection of pointed sets
Eϕ (L|K) ∼
= H 1 (G, O(ϕL )).
Proof: Let (V 0 , ϕ0 ) be a representative of a class in Eϕ (L|K) and let X(V 0 , ϕ0 )
be the nonempty set of isomorphisms f : (VL , ϕL ) →(VL0 , ϕ0L ). This is a G-set
by σ(f ) = σ ◦ f ◦ σ −1 and is equipped with a simply transitive right action of
the G-group O(ϕL ) compatible with the G-action. In other words, X(V 0 , ϕ0 )
is an O(ϕL )-torsor in the sense of I §2 and we obtain a map
(∗)
Eϕ (L|K) −→ TORS (O(ϕL )).
We prove the bijectivity of this map by constructing an inverse as follows.
Let X be an O(ϕL )-torsor and let x ∈ X. Then for every σ ∈ G we have a
unique Aσ ∈ O(ϕL ) such that σx = xAσ . The function σ 7→ Aσ is a 1-cocycle.
Since O(ϕL ) ⊆ GL(VL ) and H 1 (G, GL(VL )) = {1} by (6.2.3), there exists an
L-automorphism f of VL such that
Aσ = f −1 ◦ σ(f )
for all σ
∈
G.
For the quadratic form ϕ0 = f (ϕ) we have
σ(ϕ0 ) = σ(f )(σ(ϕ)) = σ(f )(ϕ) = (f ◦ Aσ )(ϕ) = f (ϕ) = ϕ0 .
Hence ϕ0 is rational over K, i.e. a quadratic form on V . Associating to X the
isomorphism class of the pair (V, ϕ0 ), we get a map TORS (O(ϕL )) → Eϕ (L|K).
A straightforward check, which we leave as an exercise for the reader, shows
that this map is inverse to (∗). The bijection
Eϕ (L|K) ∼
= H 1 (G, O(ϕL ))
is now the composite of (∗) with the bijection TORS (O(ϕL )) ∼
= H 1 (G, O(ϕL ))
of (1.2.4).
2
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347
§2. Hilbert’s Satz 90
Remark: The proof shows that the bijection Eϕ (L|K) ∼
= H 1 (G, O(ϕL )) is
0
0
explicitly given by associating to a pair (V , ϕ ) the class of the 1-cocycle Aσ
given by σ(f ) = f Aσ , where f is an isomorphism (VL , ϕL ) →(VL0 , ϕ0L ).
Another application of non-abelian cohomology is to the G-group P GLn (L)
which may be defined by the exact sequence
1 −→ L× −→ GLn (L) −→ P GLn (L) −→ 1.
Let IPn−1
K be the (n − 1)-dimensional projective space over K, considered as a
K-variety. The group P GLn (L) is the automorphism group
P GLn (L) = AutL (IPLn−1 ),
where IPn−1
= IPn−1
⊗K L. The Galois group G = G(L|K) acts on IPLn−1 via
L
K
the action on L and the action on P GLn (L) is induced by the action on IPLn−1 :
for σ ∈ G and A ∈ P GLn (L), we have σA = σ ◦ A ◦ σ −1 .
A Brauer-Severi variety over K of dimension n−1 is a K-variety X which
becomes isomorphic to IPn−1 over a Galois extension L|K. This means that
X ⊗K L and IPn−1
are isomorphic as L-varieties. We then say that X splits
L
over L. Let us denote by BSn (L|K) the pointed set of isomorphism classes
[X] of Brauer-Severi varieties over K of dimension n − 1 which split over L.
We define a canonical map
BSn (L|K) −→ H 1 (G, P GLn (L))
as follows. Let X be an (n − 1)-dimensional Brauer-Severi variety over K
which splits over L. Then the set T (X) of all isomorphisms of L-schemes
f : IPLn−1 −→ X ⊗K L
is a P GLn (L)-torsor. In fact, on the one hand G acts on X ⊗K L via the second
factor, and hence on T (X) by σ(f ) = σ ◦ f ◦ σ −1 , and on the other hand T (X)
is equipped with a simply transitive right action of P GLn (L), compatible with
the G-action:
σ(f ◦ A) = σ ◦ f ◦ A ◦ σ −1 = σf σ −1 σAσ −1 = σ(f ) σA.
Therefore we obtain a canonical map
BSn (L|K) −→ TORS (P GLn (L)),
(1)
X 7−→ T (X).
The composite with the bijection (1.2.4)
TORS (P GLn (L)) ∼
= H 1 (G, P GLn (L))
gives a map
(2)
BSn (L|K) −→ H 1 (G, P GLn (L)).
It is explicitly given by associating to a Brauer-Severi variety X the class of
the cocycle Aσ = f −1 ◦ σ(f ), where f is an element of T (X).
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Chapter VI. Galois Cohomology
(6.2.6) Theorem.
H 1 (G, P GLn (L)) ∼
= BSn (L|K).
Proof: We construct an inverse of the map (1) as follows. Let T be any
P GLn (L)-torsor and let f ∈ T . For every σ ∈ G there is a unique Aσ ∈
P GLn (L) such that σ(f ) = f Aσ . Aσ is a 1-cocycle of G with values in
P GLn (L). We change the action of σ ∈ G on IPLn−1 by σ ∗ = Aσ ◦ σ (note
that (στ )∗ = Aστ στ = Aσ σAτ στ = Aσ σAτ σ −1 στ = Aσ σAτ τ = σ ∗ τ ∗ ). In
order to indicate the changed G-action on IPLn−1 , we write Y (T ) instead. The
action of G on Y (T ) is a so-called “descent datum” and by descent theory
of algebraic geometry, there exists a K-variety X(T ) = Y (T )/G, unique up
to isomorphism, such that we have an isomorphism of L-varieties X(T ) ⊗K
L ∼
= Y (T ) compatible with the G-action (see [67], chap.VIII, 7.8). This
isomorphism induces an isomorphism of P GLn (L)-torsors
∼
T (X(T )) = IsomL (IPn−1
L , Y (T )) = T,
which is given by A 7→ f A. If we start with another f ∈ T , then we obtain
a Brauer-Severi variety over K isomorphic to X(T ). If, conversely, we start
with a Brauer-Severi variety X which splits over L, then X(T (X)) ∼
= X, since
we have an isomorphism
X(T (X)) ⊗K L ∼
= X ⊗K L
of L-varieties compatible with the G-action, and so X(T (X)) and X are both
solutions of the same descent problem.
2
Exercise 1. Let M |K be a Galois extension and L|K a finite subextension. Denote by c 7→ c̄
a section G(M |L)\G(M |K) → G(M |K) of G(M |K) → G(M |L)\G(M |K), i.e. a system of
right representatives. Associate to every inhomogeneous 1-cocycle x : G(M |L) → P GLn (L)
the function cor x : G(M |K) → P GLn (M ) given by
Y
(cor x)(σ) =
c̄ −1 x(c̄σcσ −1 ).
c
Show that cor x is a 1-cocycle and that we obtain a canonical map
1
1
cor L
K : H (G(M |L), P GLn (M )) −→ H (G(M |K), P GLn (M )), [x] 7−→ [cor x],
which does not depend on the choice of the section c 7→ c̄. P GLn may be replaced by any
other algebraic group over K.
Exercise 2. Let M ⊇ L ⊇ K be as above and let 1 → A → B → C → 1 be an exact sequence
of algebraic groups over K with A in the center of B. Show that we have a commutative
diagram
C(M )) δ H 2 (G(M |L), A(M ))
H 1 (G(M |L),
cor
H 1 (G(M |K), C(M ))
cor
δ
H 2 (G(M |K), A(M )).
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§3. The Brauer Group
§3. The Brauer Group
Again let L|K be a Galois extension with group G = G(L|K). We now
study the group H 2 (G, L× ). It is linked with a classical theory which began
with the discovery of the quaternion algebra
H = IR + IRi + IRj + IRk
by the Irish mathematician W. R. HAMILTON (1805–1865). The multiplication
in H is given by i2 = j 2 = k 2 = −1, ij = −ji = k (implying
o
njk =−kj = i and
ki = −ik = j). H is also given as the matrix algebra H = −zū uz̄ | z, u ∈ C .
H has center IR and is a skew field, i.e. has no nontrivial two-sided ideal,
and it is in essence the only IR-algebra with these properties. This example
raised the question of central simple algebras over arbitrary fields K, i.e.
finite dimensional K-algebras with center K and without nontrivial two-sided
ideals. For these algebras we have the following equivalent characterizations.
(6.3.1) Proposition. For a finite dimensional K-algebra A the following
conditions are equivalent:
(i) A is a central simple algebra.
(ii) If K|K is the separable closure of K, then the K-algebra AK = A ⊗K K
is isomorphic to a full matrix algebra Mn (K).
(iii) There exists a finite Galois extension L|K such that AL = A ⊗K L is a
full matrix algebra Mn (L).
(iv) A ∼
= Mm (D), where D is a skew field over K of finite degree.
For the proof of this proposition and for the basic properties of central
simple algebras, we refer to [102], [42] and [14], chap.VIII, §5, §10. We say
that the central simple K-algebra A splits over the extension L|K, or that L is
a splitting field for A, if A ⊗K L ∼
n. A is called a cyclic
= Mn (L) for some√
algebra if it has a cyclic splitting field L|K, of degree dimK A.
Two central simple K-algebras A and B are called similar if
A ⊗K Mr (K) ∼
= B ⊗K Ms (K)
for some r, s and we write A ∼ B. This is the same to say that the skew fields
associated with A and B are isomorphic. The tensor product A ⊗K B of two
central simple K-algebras is again central simple. This leads to the
(6.3.2) Definition. The Brauer group Br(K) of a field K is the set of
all similarity classes [A] of central simple K-algebras A, endowed with the
multiplication
[A][B] = [A ⊗K B].
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Chapter VI. Galois Cohomology
The product is well-defined since
[A ⊗K Mr (K) ⊗K B ⊗K Ms (K)] = [A ⊗K B ⊗K Mrs (K)] = [A ⊗K B].
The identity of Br(K) is 1 = [K] and the inverse of [A] is [A]−1 = [Aop ],
where Aop is A as a K-vector space and the multiplication of Aop is given by
Aop × Aop → Aop , (a, b) 7→ ba.
If L|K is an extension of fields, then we have the homomorphism
res K
L : Br(K) −→ Br(L), A 7−→ [A ⊗K L],
and we denote the kernel by Br(L|K). This is the group of central simple Kalgebras which split over L. If L|K runs through the finite Galois subextensions
of K|K, then by (6.3.1)(iii)
Br(K) =
[
Br(L|K).
L
In the finite Galois case the classes of Br(L|K) are explicitly represented by the
following K-algebras. Let G = G(L|K) and n = [L : K]. Let x : G×G → L×
be a normalized (i.e. x(σ, 1) = x(1, σ) = 1) inhomogeneous 2-cocycle. On the
n2 -dimensional K-vector space
A=
M
Leσ
(eσ formal symbols)
σ ∈G
we define a multiplication by
X
(
σ
X
xσ eσ )(
τ
yτ eτ ) =
X
xσ σyτ x(σ, τ )eστ .
σ,τ
This multiplication is associative because of the cocycle relation
x(σ, τ )x(στ, ρ) = σx(τ, ρ)x(σ, τ ρ),
and makes A a K-algebra with the identity 1 = e1 . This K-algebra is called
the crossed product of L and G by x and is denoted by
A = C(L, G, x).
The crossed products have the following properties
(6.3.3) Proposition.
(i) C(L, G, x) is a central simple K-algebra which splits over L.
(ii) The normalized cocycles x and y are cohomologous if and only if
C(L, G, x) ∼
= C(L, G, y).
(iii) C(L, G, xy) ∼ C(L, G, x) ⊗K C(L, G, y).
(iv) Every simple central K-algebra which splits over L is similar to a crossed
product.
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§3. The Brauer Group
For the proof of this proposition we refer to [102]. Associating now to a
cohomology class [x] ∈ H 2 (G, L× ) the class [C(L, G, x)] (x normalized), we
obtain a map
H 2 (G, L× ) −→ Br(L|K),
which is well-defined and injective by (ii), multiplicative by (iii) and surjective
by (iv), hence an isomorphism of groups. If M ⊇ L ⊇ K are two finite Galois
extensions, then it is evident that the diagram
H 2 (G(M |K),
M ×)
Br(M |K)
inf
incl
H 2 (G(L|K), L× )
Br(L|K)
is commutative. Taking direct limits, we obtain the
(6.3.4) Theorem. For every Galois extension L|K we have a canonical isomorphism
H 2 (G(L|K), L× ) ∼
= Br(L|K).
In particular,
H 2 (K,K × ) ∼
= Br(K),
showing that Br(K) is a torsion group.
In general, Brauer groups are difficult to calculate. We will determine them
for local and global fields in later chapters. For finite fields, we have the
following result.
(6.3.5) Proposition. Let K be a finite field. Then Br(K) = 0.
Proof: Let L|K be a finite extension. Then G(L|K) is a cyclic group, and
h(G(L|K), L× ) = 1 by (1.7.6). Therefore
#Br(L|K) = #H 2 (G(L|K), L× ) = #H 1 (G(L|K), L× ) = 1
by Hilbert’s Satz 90. Passing to the limit over all L, we obtain Br(K) = 0.
2
For a natural number n prime to the characteristic of K, we consider the
exact sequence
n
1 −→ µn −→K × −→K × −→ 1
of GK -modules. The associated exact cohomology sequence, Hilbert’s Satz 90
and the above theorem yield an isomorphism
H 2 (K, µn ) ∼
= n Br(K).
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Chapter VI. Galois Cohomology
0
Taking direct limits over n and denoting by Br(K)(p ) the subgroup of elements
of order prime to p, we obtain the
(6.3.6) Corollary. We have
H 2 (K, µ) ∼
= Br(K)
or H 2 (K, µ) ∼
= Br(K)(p )
0
according to whether char(K) = 0 or char(K) = p > 0.
The Brauer group Br(K) is functorial in K in a twofold sense. If ρ : K →
K is a homomorphism of fields, then K 0 may also be viewed as a K-algebra
and we obtain for every K-algebra A, a K-algebra
0
Aρ = A ⊗K,ρ K 0
together with a homomorphism ρ : A → Aρ , a 7→ a⊗10 . Aρ is also a K 0 -algebra
and we obtain a homomorphism
ρ∗ : Br(K) −→ Br(K 0 ),
If ρ is an inclusion K
⊆
[A] 7−→ [Aρ ].
L, then ρ∗ is the restriction map
res K
L : Br(K) −→ Br(L).
If L|K is a finite separable extension, we have on the other hand also a
corestriction homomorphism
cor LK : Br(L) −→ Br(K),
which is obtained as follows. Let ρ : L →K run through the K-embeddings of
L into the separable closure K of K. If A is any L-algebra, then we form the
tensor product of the L-algebras Aρ ,
T (A) =
O
Aρ .
ρ
For every σ ∈ GK we have the K-isomorphism σ : Aσ−1 ρ → Aρ and we get a
GK -action on T (A), viewed as a K-algebra, given by
σ( ⊗ρ aρ ) = ⊗ρ bρ
with bρ = σaσ−1 ρ .
We now take the fixed ring and obtain a K-algebra
cor LK (A) = T (A)GK .
If A is a central simple L-algebra, then cor LK (A) is a central simple K-algebra
(see [102]). It is obvious that cor LK (A ⊗L B) ∼
= cor LK (A) ⊗K cor LK (B) and
cor LK (Mn (L)) = Mnd (K) with d = [L : K]. Therefore we obtain a canonical
homomorphism
cor LK : Br(L) −→ Br(K),
[A] 7−→ [cor LK (A)],
the corestriction for Brauer groups.
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353
§3. The Brauer Group
(6.3.7) Proposition. Let L|K be a finite subextension of the separable closure
K̄|K and let σ ∈ GK = G(K|K). We then have the commutative diagrams
×
H 2 (L,K
)
∼
res cor
H 2 (K,K × )
Br(L)
res cor
∼
×
H 2 (L,K
)
∼
Br(L)
σ∗
σ∗
Br(K), H 2 (σL,K × )
∼
Br(σL).
One has to show that, for a finite Galois extension M |K containing L, the
diagrams
H 2 (G(M |L), M × )
Br(M |L)
res cor
res cor
H 2 (G(M |K), M × )
H 2 (G(M |L), M × )
σ∗
Br(M |K), H 2 (G(M |σL), M × )
Br(M |L)
σ∗
Br(M |σL)
are commutative, where the horizontal maps are given by forming crossed
products. Let x be a normalized 2-cocycle of G(M |K), resp. of G(M |L), with
values in M × . Then by a straightforward argument
C(M, G(M |K), x) ⊗K L = C(M, G(M |L), res x),
resp.
C(M, G(M |L), x) ⊗K,σ σL = C(M, G(M |σL), σ∗ x),
giving the commutativity for the maps res and σ∗ . For cor the proof is more
involved and needs a more comprehensive study of non-abelian cohomology.
We refer to [181], th. 11 or [91], th. 3.13.20.
In I §5 we have seen that the cohomology functor
H 2 (K × ) : L 7−→ H 2 (L,K × )
may be interpreted as a G-modulation for the absolute Galois group G = GK
with respect to the maps res , cor , σ∗ in the sense of (1.5.12). The isomorphism
(6.3.6) and the above proposition (6.3.7) show that also the Brauer group may
be interpreted as a G-modulation and that we have the
(6.3.8) Theorem. The functor
Br : L 7−→ Br(L)
is a G-modulation and the family of isomorphisms H 2 (L,K × ) ∼
= Br(L) is an
isomorphism
∼ Br
H 2 (K × ) −→
of G-modulations.
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354
Chapter VI. Galois Cohomology
The main assertion of this theorem is the double coset formula (1.5.11) for
the classes [A] of central simple algebras, i.e. the similarity
res
K
M
◦ cor LK (A) ∼
Y
σL
◦ σ∗ ◦ res
cor M
M
L
Lσ −1 M (A),
σ ∈R
for two finite separable extensions M, L ⊇ K and a central simple L-algebra
A. Here R is a system of representatives of GM \GK /GL . When L = M one
Q
N
may replace by and the similarity ∼ by an isomorphism
cor LK (A) ⊗K L ∼
=
O
σ∗ (A) =
σ
O
(A ⊗K,σ σL).
σ
For a K-algebra A, we have
cor LK (A ⊗K L) ∼
= A⊗d ,
d = [L : K].
We finish this section by giving the following geometric interpretation for the
Brauer group. In the preceding section we defined the Brauer-Severi varieties
over K as the K-varieties X which become isomorphic to IPn−1 for some n
over some Galois extension L, i.e.
X ⊗K L ∼
= IPn−1
L
as L-varieties. We denote by BS(K) the set of isomorphism classes of all
Brauer-Severi varieties over K, by BS(L|K) the subset of classes which
split over L and by BSn (L|K) the subset of BS(L|K), consisting of the
isomorphism classes of K-varieties which become isomorphic to IPn−1 over
L. Then
BS(L|K) =
[
BSn (L|K).
n∈IN
(6.3.9) Theorem. For every Galois extension L|K and every n ≥ 1 there exists
a natural injective map
BSn (L|K) ,→ Br(L|K)
whose image is the set of classes in Br(L|K) which are represented by a central
simple algebra of dimension n2 over K. If [L : K] is finite and divides n, then
the above map is bijective.
Proof: The exact sequence of G(L|K)-groups
1 −→ L× −→ GLn (L) −→ P GLn (L) −→ 1
induces a map (see I §3 ex.8)
δ : BSn (L|K) ∼
= H 1 (G(L|K), P GLn (L)) −→ H 2 (G(L|K), L× ).
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355
§3. The Brauer Group
A K-algebra A is central simple of dimension n2 and splits over L if and
only if A ⊗K L ∼
= Mn (L). Since AutL (Mn (L)) = P GLn (L), the same arguments as in the case of Brauer-Severi varieties show that the set CSAn (L|K)
of K-isomorphism classes of such algebras is naturally isomorphic to
H 1 (G(L|K), P GLn (L)). Thus we obtain a diagram
H 1 (G(L|K),"#!$ P GLn (L))
δ
can
CSAn (L|K)
H 2 (G(L|K), L× )
Br(L|K)
where the map can sends a central simple algebra to its similarity class in
Br(L|K). Two similar central simple algebras of the same dimension are
isomorphic and therefore can is injective. The statement of the theorem now
follows from the fact (see [102] V §30) that the above diagram commutes.
Finally, if n = m · [L : K], then the class in H 2 (G(L|K), L× ) of a 2-cocycle α
is represented by the algebra C(L, G(L|K), α) ⊗K Mm (K).
2
Exercise 1. If M ⊇ L ⊇ K are two finite separable extensions, then the corestriction of Brauer
M
M
groups obeys the rule cor L
K ◦ cor L = cor K .
Exercise 2. Let L|K be a cyclic extension of degree n, σ a generator of the Galois group G
and a ∈ K × . Show that the ring
n−1
M
(a, L|K, σ) :=
L ei
i=0
n
with multiplication e = a, eλ = (σλ)e (λ
∈
L), is a cyclic central simple algebra.
Exercise 3. Under the assumptions of ex.2 show that (a, L|K, σ) ∼
= (b, L|K, σ) provided
a/b ∈ NL|K (L× ), and
∼ Mn ((ab, L|K, σ)).
(a, L|K, σ) ⊗K (b, L|K, σ) =
Exercise 4. Keeping the assumptions of ex.2, show that the assignment a 7→ (a, L|K, σ)
induces an isomorphism
∼
θσ : K × /NL|K L× −→
Br(L|K).
Exercise 5. Deduce from ex.4 that the quaternion algebra H is the only central skew field
over IR different from IR (Theorem of Frobenius).
Exercise 6. (Theorem of Albert-Hochschild). If K|k is a totally inseparable extension, then
the restriction map Br(k) → Br(K) is surjective.
Exercise 7. If K is a field of characteristic p > 0, then the Brauer group Br(K) is p-divisible.
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Chapter VI. Galois Cohomology
§4. The Milnor K -Groups
The Hilbert symbol is the main ingredient in the formulation of the general
reciprocity law of n-th power residues (see [160], chap.V, §3). It is defined for
a p-adic local field Kp , and is a map
, : Kp× × Kp× −→ µn
p
which is multiplicative in both arguments a and b and satisfies the relation
( a,1−a
) = 1 for all a =/ 0, 1.∗) This example leads naturally to a general notion
p
of a symbol for any field F as a multi-multiplicative map
F × × ·{z
· · × F ×} −→ A,
|
(a1 , . . . , an ) 7−→ [a1 , . . . , an ],
n times
into a (multiplicatively written) abelian group A such that
[a1 , . . . , an ] = 1 whenever ai + aj = 1 for some i =/ j.
Every such symbol factors through a group KnM (F ), which is the universal
target of symbols and is defined as follows.
(6.4.1) Definition. The n-th Milnor K-group of a field F is the quotient
KnM (F ) = (F × ⊗ ZZ · · · ⊗ ZZ F × )/In ,
where In is the subgroup of F × ⊗ ZZ · · · ⊗ ZZ F × generated by the elements
a1 ⊗ · · · ⊗ an such that ai + aj = 1 for some i =/ j.
We have the canonical symbol
F × × · · · × F × −→ KnM (F ),
(a1 , . . . , an ) 7−→ {a1 , . . . , an },
where {a1 , . . . , an } = a1 ⊗ · · · ⊗ an mod In . Every other symbol of n arguments is obtained from this by composition with a homomorphism of the
group KnM (F ).
It is convenient to put K0M (F ) = ZZ. The multiplication in KnM (F ) will be
written additively, i.e.
{. . . , ai bi , . . .} = {. . . , ai , . . .} + {. . . , bi , . . .},
although for
= F × the multiplicative notation will also be used. For
×
×
⊗ ·{z
· · ⊗ F ×} ⊗ Im in
n, m ∈ IN the images of In ⊗ F
⊗ ·{z
· · ⊗ F ×} and F
|
|
K1M (F )
m times
∗) The reciprocity law is the product formula
n times
a,b
= 1 for two numbers a, b ∈ K × in a
number field K with the completions Kp (infinite primes included).
Q
p
p
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§4. The Milnor K-Groups
F × ⊗ ·{z
· · ⊗ F ×} belong to In+m . Therefore we have a homomorphism
|
n+m times
M
M
KnM (F ) × Km
(F ) −→ Kn+m
(F ),
({a1 , . . . , an }, {b1 , . . . , bm }) 7−→ {a1 , . . . , an , b1 , . . . , bm },
and we obtain a graded ring
M
K (F ) =
∞
M
KnM (F ).
n=0
One knows that KnM (IF) = 0 for a finite field IF and n ≥ 2, and that
×
×
×
K2M (Q) = µ2 ⊕ IF×
3 ⊕ IF5 ⊕ IF7 ⊕ IF11 ⊕ . . .
and KnM (Q) ∼
= µ2 for n ≥ 3. If F is algebraically closed, then it is not difficult
to show that the groups KnM (F ) are uniquely divisible for n ≥ 2. In general it is
a difficult and usually most cumbersome problem to compute the K-groups or
investigate their properties. On the other hand K-theory has nowadays attained
a most prominent position in algebra, number theory and arithmetic geometry.
But it is beyond the scope of this book to go into the K-groups more closely
(we refer to [142] and [232] for further literature). We mention them here
because of their close relation to Galois cohomology. This connection arises
as follows.
Let N be a natural number prime to the characteristic of F . From the exact
sequence
N
1 −→ µN −→ F̄ × −→ F̄ × −→ 1,
we obtain a surjective homomorphism
δF : F × −→ H 1 (F, µN )
with kernel F ×N . On the other hand, we have for every n ≥ 1 the cup-product
∪
∗)
H 1 (F, µN ) × · · · × H 1 (F, µN ) −→ H n (F, µ⊗n
N ),
hence a map
(∗)
F × × ·{z
· · × F ×} −→ H n (F, µ⊗n
N ).
|
n times
Let us denote the image of (a1 , . . . , an ) ∈ F × × · · · × F × by
(a1 , . . . , an )F := δF a1 ∪ · · · ∪ δF an .
∗) Note that µ⊗n is isomorphic to µ as an abelian group, but as a G -module it is different:
N
F
N
if χ : GF →(ZZ/N ZZ)× is the character that gives the action of GF on µN , σ : ζ 7→ ζ χ(σ) , than
χn is the character for µ⊗n
N .
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Chapter VI. Galois Cohomology
(6.4.2) Theorem (TATE). The map (∗) induces a homomorphism
hF : KnM (F ) −→ H n (F, µ⊗n
N ),
called the Galois symbol.
Proof: The multiplicativity in each argument is clear from the definition
and we only have to show that (a1 , . . . , an )F = 1 if ai + aj = 1 for i =/ j. It
suffices to consider the case n = 2: if n > 2 and, say i = 1, j = 2, then
(a1 , . . . , an )F = (a1 , a2 )F ∪ (a3 , . . . , an )F .
Q
So let n = 2 and a ∈ F × , a =/ 1. Let X N − a = i fi (X) with fi (X) monic
and irreducible in F [X]. For each i let ai be a root of fi (X) and let Fi = F (ai ).
Then
Y
Y
1−a=
fi (1) =
NFi |F (1 − ai ).
i
i
Hence
Y
(1 − a, a)F = (
i
NFi |F (1 − ai ), a)F =
Y
(NFi |F (1 − ai ), a)F .
i
Because of the formula cor (α ∪ res β) = (cor α) ∪ β (see (1.5.3) (iv)), and
because cor is the norm on H 0 and commutes with δ, we have
(NFi |F (1 − ai ), a)F = cor FFi (1 − ai , a)Fi = cor FFi (1 − ai , aN
i )Fi
Fi
N
= cor F (1 − ai , ai )Fi = 1,
hence (1 − a, a)F = 1.
2
For the Galois symbol, we have the fundamental
(6.4.3) Theorem (VOEVODSKY-ROST). For every field F and every N ∈ IN
prime to the characteristic of F , the Galois symbol yields an isomorphism
∼ H n (F, µ⊗n ).
hF : KnM (F )/N KnM (F ) −→
N
For N = 2 this was first conjectured by J. MILNOR in [142] (in the language
of quadratic forms) and the general form was stated by S. BLOCH and K. KATO
in [13]. For many years it was known as the Bloch-Kato conjecture.
Theorem (6.4.3) is trivial for n = 0 and arbitrary N ∈ IN, as well as for
N = 1 and arbitrary n. For n = 1 it is just a reformulation of Kummer theory.
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§4. The Milnor K-Groups
For a proof of (6.4.3), which lies far beyond the scope of this book, we
refer to V. VOEVODSKY’s paper [Vo]. The special case n = 2 had been settled
already in 1982 by A. S. MERKUR’EV and A. A. SUSLIN, see [138], [239] or
[137]. The next steps towards the general conjecture was the case n = 3 and
N a power of 2 which was proved by Merkur’ev and Suslin (see [MS]) and,
independently, by M. ROST (see [Ro]) in 1986. In 2001, Voevodsky proved
the case with arbitrary n and N a power of 2, see [242], [243]. Finally, using
results of Rost, Voevodsky proved the general statement in 2010. There also
exists a version of (6.4.3) for N a power of the characteristic of F , which had
been proven already by Bloch and Kato themselves, see [13].
For local and global fields and n = 2 the Bloch-Kato Conjecture was proven
earlier by J. TATE (see [231] or [232]), who also constructed an `-adic variant
of the Galois symbol. Let ZZ` (n) = lim µ⊗n
as a compact GF -module. Then
←− m `m
there exists an `-adic Galois symbol
n
hF : KnM (F ) −→ Hcts
(F, ZZ` (n)).
Tate showed that the following theorem follows from (6.4.3) (for n = 2).
(6.4.4) Theorem. For every prime number `, the `-primary part of K2M (F ) is
the direct sum of its maximal divisible subgroup, which is killed by hF , and a
subgroup which is mapped isomorphically by hF onto the torsion subgroup of
2
(F, ZZ` (2)).
Hcts
If F is a global field, then the group K2M (F ) is a torsion group with no
non-zero divisible subgroup; so, in this case, its `-primary part is mapped
2
isomorphically onto the torsion subgroup of Hcts
(F, ZZ` (2)).
Exercise 1. Let K be a field with a normalized discrete valuation v. Let O be its valuation
ring, p the maximal ideal, π a prime element and κ(v) = O/p the residue field. Define a map
M
∂v : K × × · · · × K × −→ Kn−1
(κ(v))
{z
}
|
n times
×
as follows. Let α1 , . . . , αn ∈ K , αi = π ai εi , ai = v(αi ), εi ∈ O× , and put ε̄i = εi
mod p ∈ κ(v).
For n = 1, put ∂(α1 ) = a1 . For n > 1 and k1 , . . . , km with 1 ≤ k1 < . . . < km ≤ n, m ≤ n,
put
∂ k1 ,...,km (α1 , . . . , αn ) = ak1 . . . akm xy,
M
where x is the symbol {ε̄1 , . . . , ε̄n } ∈ Kn−m
(κ(v)) with omitted elements at the k1 , . . . , km -th
M
places if m < n, and equal to 1 if m = n; y is equal to {−1, . . . , −1} ∈ Km−1
(κ(v))
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360
Chapter VI. Galois Cohomology
if m > 1, and equal to 1 otherwise. Now put
X
(−1)n−k1 −···−km ∂ k1 ,...,km (α1 , . . . , αn ),
∂v (α1 , . . . , αn ) =
k1 ,...,km
1≤m≤n
and show that this function induces a homomorphism
M
∂v : KnM (K) −→ Kn−1
(κ(v)),
which is called the tame symbol.
Hint: [50], chap. IV, 2.
Exercise 2. Keeping the assumptions of ex.1, let {α, β} ∈ K2M (K). Then
∂v (α, β) = (−1)v(α)v(β) αv(β) β −v(α) mod p.
Exercise 3. Let K = F (X) be a rational function field in one variable and let v run through the
normalized discrete valuations of K which are trivial on F and which are different from the
valuation v∞ given by v∞ (f (X)/g(X)) = deg (g(X)) − deg (f (X)) for f (X), g(X) ∈ F [X].
The sequence
M
⊕∂v
M
0 −→ KnM (F )% −→ KnM (K)
(κ(v)) −→ 0
Kn−1
/ v∞
v=
is exact and splits (Theorem of Tate-Milnor, cf. [142]).
Hint: Study the proof in [142] or in [50] chap. IX, 2.4.
§5. Dimension of Fields
An important invariant of a field k is the cohomological dimension cd Gk of
its absolute Galois group. In order to study its properties we have to make use
of a remarkable principle, which roughly says that a homogeneous polynomial
equation f (x1 , . . . , xn ) = 0 over a field k has automatically a non-zero solution
in k if the number of variables is large compared to its degree. This principle
gives rise to another notion of dimension, which is directly attached to a field
and which we are now going to introduce.
By an n-form in k we mean a homogeneous polynomial f ∈ k[x1 , . . . , xn ].
A nontrivial zero in k is an n-tuple (α1 , . . . , αn ) ∈ k n , different from (0, . . . , 0),
such that f (α1 , . . . , αn ) = 0.
We obtain special n-forms as follows. Let K|k be a finite extension of degree n and let x1 , . . . , xn be indeterminates. Then the extension K(x1 , . . . , xn )
of k(x1 , . . . , xn ) is also finite and we may consider the norm N of this extension;
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361
§5. Dimension of Fields
N (ξ) is the determinant of the transformation a 7→ ξa of the k(x)-vector space
K(x). Choosing a basis ω1 , . . . , ωn of K|k,
N (x1 , . . . , xn ) := N
n
X
xi ωi
i=1
defines an n-form in k of degree n which induces the norm map from K to k:
NK|k : K −→ k ,
(a1 , . . . , an ) 7→ N (a1 , . . . , an ) .
These forms are called norm forms. They have the special property of having
only the trivial zero in k (since NK|k (α) =/ 0 for α =/ 0). More generally, we
call an n-form f of degree d a normic form of order i if n = di and if f has
only the trivial zero in k.
(6.5.1) Lemma. If k is not algebraically closed, then k admits normic forms
of arbitrarily large degree.
Proof: Since k is not algebraically closed, there exists a finite extension K|k
of degree n > 1 for some n ∈ IN, thus a normic n-form of order 1, as we saw
above.
Let f, g be forms in n1 , n2 variables of degrees d1 , d2 respectively. We denote
by f (g | . . . | g) the form which is obtained by inserting g for each variable
and using new variables after each occurrence of |. We obtain in this way
an n1 n2 -form of degree d1 d2 , which is normic of order i if both f and g are
normic of order i. From this observation we obtain for each normic n-form f
of degree d and order i and each ν ∈ IN a normic nν -form f (ν) of degree dν and
order i by setting
f (1) = f
and
f (ν+1) = f (ν) (f | . . . | f ).
2
(6.5.2) Definition. The diophantine dimension dd(k) of a field k is the
smallest integer r ≥ 0 such that any n-form f of degree d ≥ 1 has a nontrivial
zero in k, whenever n > dr . We set dd(k) = ∞ if no such number r exists.
Clearly, the fields with dd(k) = 0 are the algebraically closed fields. Fields
with dd(k) ≤ 1 are called quasi-algebraically closed and fields with dd(k) ≤ i
are called Ci -fields. Thus a field is a Ci -field if every n-form of degree d with
n > di has a nontrivial zero.
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362
Chapter VI. Galois Cohomology
(6.5.3) Theorem (ARTIN, LANG, NAGATA). Let dd(k) = r and let f1 , . . . , fs be
n-forms of degree d. If n > sdr , then these forms have a common nontrivial
zero in k.
Proof:∗) If k is algebraically closed, i.e. r = 0, then the equations f1 =
0, . . . , fs = 0 define a projective variety in the projective space IPn−1 (k) of
dimension greater than or equal to (n − 1) − s ≥ 0. The result follows from
this.
Assume then that k is not algebraically closed. Let Φ be a normic form of
order 1 and of degree e ≥ s, which exists by (6.5.1) . Consider the sequence of
forms
Φ(1) = Φ(1) (f ) = Φ(f1 , . . . , fs | f1 , . . . , fs | . . . | f1 , . . . , fs | 0, . . . , 0)
Φ(2) = Φ(2) (f ) = Φ(1) (f1 , . . . , fs | f1 , . . . , fs | . . . | f1 , . . . , fs | 0, . . . , 0)
etc., where after each vertical line we use new variables, and we insert as many
complete sets of f ’s as possible. We put Φ(0) = Φ.
Thus Φ(1) has n[ es ] variables and has degree de ≤ ds([ es ] + 1). (If x is a real
number, the symbol [x] will always mean the largest integer less than or equal
to x.) If r = 1, we want
n[ es ] > ds([ es ] + 1)
or
(n − ds)[ es ] > ds.
Since n − ds > 0, this can be arranged by taking e large. Then Φ(1) has a
nontrivial zero, which, since Φ is normic, is a common zero of all the fi .
If r > 1, we have to use the higher Φ(m) . Now Φ(m) has degree Dm = dm e,
and if Nm is the number of variables in Φ(m) , then N0 = e and
Nm+1 = n[ Nsm ] .
We want to choose m so large that Nm > (Dm )r . Now [ Nsm ] =
0 ≤ tm < s. Hence
Nm
s
− tm
, where
s
n[ Nsm ]
Nm+1
n Nm
n
tm
=
=
−
(Dm+1 )r dr (Dm )r sdr (Dm )r sdr er (dr )m
n Nm
n
s
≥
− r r r m.
r
r
sd (Dm )
sd e (d )
Using the same inequality for m, m − 1, . . . , 2, 1, we get
∗) The proof is taken from [60].
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363
§5. Dimension of Fields
Nm+1
(Dm+1 )r
!
n
sdr
2
n
sdr
m+1
e
s n
1
− r
r
r
e
e s (d )m+1
n
sdr
m+1
1
s2
(e
).
−
er
n−s
!
n
Nm−1
s
s
≥
− r r m−1 −
r
r
r
(Dm−1 )
e (d )
sd
e (dr )m
..
..
..
.
.
.
!
m+1
m m−1
n
N0
1
n
s n
n
≥
+
+ ··· + 1
−
sdr
(D0 )r er s (dr )m+1
s
s
m
=
≥
n
−1
s .
n
−1
s
Since e can be chosen large enough that (n − s)e − s2 > 0, and
see that (DNmm)r → ∞ as m → ∞ and we are done.
n
sdr
> 1, we
2
Note: Lang [121] generalized this theorem to the case where the fi have
different degrees d1 , . . . , ds and n > di1 + · · · + dis , but only under the extra
hypothesis that k has a normic form of order i of every degree. It would be
interesting to remove this hypothesis if possible.
We are now able to prove the following theorem, which in this generality is
due to S. LANG and in the essential case to C. TSEN.
(6.5.4) Theorem. If K|k is an extension of finite transcendence degree n, then
dd(K) ≤ dd(k) + n.
Proof: Suppose first that K|k is algebraic, i.e. n = 0. Let dd(k) = r < ∞
and let f (x1 , . . . , xm ) be an m-form in K of degree d with m > dr . We have
to show that f has a nontrivial zero in K. Since the coefficients of f lie in a
finite extension of k, we may assume that K|k is finite. Let ω1 , . . . , ωs be a
basis of K|k. Introduce new variables yij with
xi = ω1 yi1 + · · · + ωs yis ,
i = 1, . . . , m. Then f (x) = f1 (y)ω1 + · · · + fs (y)ωs , where f1 , . . . , fs are smforms of degree d in k. Finding a nontrivial zero of f is equivalent to finding a
common nontrivial zero of f1 , . . . , fs in k. But this can be done by the previous
theorem, since sm > sdr .
Now let n > 0. Then K is an algebraic extension of a purely transcendental
extension k(t1 , . . . , tn ). By the above proof and by induction, we are reduced
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Chapter VI. Galois Cohomology
to the case K = k(t) with an indeterminate t. By homogeneity, it suffices to
consider forms with coefficients in the polynomial ring k[t].
Suppose f (x1 , . . . , xm ) is an m-form of degree d with m > dr+1 and with
coefficients in k[t]. We have to show that it has a nontrivial zero in K.
Introduce new variables yij with
xi = yi0 + yi1 t + yi2 t2 + · · · + yis ts ,
i = 1, . . . , m, where s will be specified later. If ` is the highest degree of the
coefficients of f , we get
f (x) = f0 (y) + f1 (y)t + · · · + fds+` (y)tds+` ,
where f0 , . . . , fds+` are m(s + 1)-forms of degree d in k. We can apply theorem
(6.5.3) to these forms provided that
m(s + 1) > (ds + ` + 1)dr
or
(m − dr+1 )s > (` + 1)dr − m.
This can be satisfied by taking s large. The common nontrivial zero of the fµ ’s
in k gives a nontrivial zero of f in K = k(t).
2
(6.5.5) Corollary (TSEN). If K is a function field in one variable over an
algebraically closed field k, then dd(K) = 1.
In fact, K|k is of transcendence degree 1, hence dd(K) ≤ dd(k) + 1 = 1, and
thus dd(K) = 1, since K is not algebraically closed.
We mention the following two further results without giving the proofs.
Both may be found in [60], 6.25 and 2.3, and the second one can also be found
in [213], chap.1, §2, th.3.
(6.5.6) Theorem (LANG). A field K which is complete with respect to a discrete
valuation with algebraically closed residue field has dd(K) = 1.
(6.5.7) Theorem (CHEVALLEY - WARNING). A finite field IF is of diophantine
dimension one, i.e. dd(IF) = 1.
For the cohomological applications the fields K with dd(K) = 1 play a
particularly important role.
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365
§5. Dimension of Fields
(6.5.8) Proposition. Let K be a field with dd(K) = 1. Then the Brauer group
Br(K) is zero, and for every finite Galois extension L|K the norm map
NL|K : L× −→ K ×
is surjective.
Proof: Let L|K be a Galois extension of degree n.
. . . , ωn be a basis
PLet ω1 , n
×
and α ∈ K . The norm form N (x1 , . . . , xn ) = N i=1 xi ωi is an n-form of
degree n, and
f (x1 , . . . , xn , x) = N (x1 , . . . , xn ) − αxn
is an (n+1)-form of degree n. It has therefore a nontrivial zero (a1 , . . . , an , a) ∈
K n . We have a =/ 0, since otherwise (a1 , . . . , an ) would be a nontrivial zero of
the norm form. Setting bi = ai /a we obtain
N (b1 ω1 + · · · + bn ωn ) = N (b1 , . . . , bn ) = α.
This shows that the norm is surjective.
We now have Ĥ n (G(L|K), L× ) = 1 for n = 0, and for n = 1 this is true by
Hilbert’s Satz 90. Since every intermediate field K 0 of L|K has dd(K 0 ) ≤ 1,
this holds also for the extension L|K 0 . From this it follows by (1.8.4) that L×
is a cohomologically trivial G-module, i.e. H 2 (G(L|K), L× ) = 0. Therefore
H 2 (G(L|K), L× ) = 0.
Br(K) ∼
= H 2 (K,K × ) = lim
2
−→
L
We now turn to the cohomological dimension of fields.
(6.5.9) Definition. The cohomological dimensions cdp (k), cd(k), scdp (k) and
scd(k) of a field k are defined as the cohomological dimensions cdp G, cd G,
scdp G, scd G of its absolute Galois group G = Gk (see (3.3.1)).
The cohomological dimension cdp (k) for a field of char(k) = p > 0 plays an
exceptional role:
(6.5.10) Proposition. If k is a field of characteristic p > 0, then cdp (k) ≤ 1.
Proof: This is a special case of (6.1.3).
2
We now consider a prime number p different from the characteristic of the
field in question.
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366
Chapter VI. Galois Cohomology
(6.5.11) Proposition. For a field k such that char(k) =/ p, and for a natural
number n ∈ IN, the following conditions are equivalent.
(i) cdp (k) ≤ n,
(ii) H n+1 (K,K × )(p) = 0 and H n (K,K × ) is p-divisible for every algebraic
extension K|k,
(iii) H n+1 (K,K × )(p) = 0 and H n (K,K × ) is p-divisible for every finite separable extension K|k.
Proof: Let K|k be algebraic. The exact cohomology sequence associated to
the exact Kummer sequence
p
1 −→ µ −→K × −→K × −→ 1 ∗)
p
says that the condition
H n+1 (K,K × )(p) = 0 and H n (K,K × ) is p-divisible
is equivalent to H n+1 (K, µp ) = 0. Assume cdp (k) ≤ n. Since GK is isomorphic to a closed subgroup of Gk , we have by (3.3.5) cdp GK ≤ n, hence
H n+1 (K, µp ) = 0. This yields (i) ⇒ (ii). The implication (ii) ⇒ (iii) is trivial.
Assume that (iii) holds. Let K be the fixed field of a p-Sylow subgroup
Gp . The degree [K(µp ) : K] divides p − 1 as well as p. This means that
µp ⊆ K, i.e. µp ∼
= ZZ/pZZ as a GK -module. Writing K as a union of finite
separable extension of k which contain µp , we obtain H n+1 (K, ZZ/pZZ) = 0,
hence cdp k = cdp K ≤ n by (3.3.6) and (3.3.2)(iii) .
2
(6.5.12) Corollary. If dd(K) ≤ 2 and cd(K) = 2, then dd(K) = 2.
Proof: If dd(K) ≤ 1, then dd(K 0 ) ≤ 1 for every finite extension by (6.5.4),
hence Br(K 0 ) = 0 by (6.5.8) and thus cd(K) ≤ 1 by (6.5.11).
2
If K|k is an algebraic field extension, then by (3.3.5)
cdp (K) ≤ cdp (k),
since GK is isomorphic to a closed subgroup of Gk . If K|k is a finite extension,
then the inequality cdp (K) =/ cdp (k) is an exceptional case, as in the example
0 = cd2 (C) < cd2 (IR) = ∞.
Namely, from (3.3.5) we obtain the
∗) As before, K denotes always the separable closure of K.
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§5. Dimension of Fields
(6.5.13) Proposition. If K|k is a finite extension such that cdp (k) < ∞ or
p - [K : k], then cdp (K) = cdp (k) and also scdp (K) = scdp (k).
The main result on cohomological dimension of general fields is the following
(6.5.14) Theorem. Let k be a field such that cdp (k) < ∞ for a prime number
p =/ char(k). If K|k is a finitely generated extension of transcendence degree n,
then
cdp (K) = cdp (k) + n.
Proof:∗) The field K is a finite extension of a purely transcendental extension
k(t1 , . . . , tn ) of k. By (6.5.13) and induction on n, we are reduced to the case
K = k(t). Consider the diagram of fields
+*&')(
Gk̄(t)
k(t)
k(t)
Gk
k
Gk
Gk(t)
k(t)
k
where k (resp. k(t)) denotes the separable closure of k (resp. k(t)). We have
G(k(t)|k(t)) = G(k|k) = Gk and we obtain a group extension
(∗)
1 −→ Gk(t) −→ Gk(t) −→ Gk −→ 1.
From (3.3.8) it follows that
cdp (k(t)) ≤ cdp (k) + cdp (k(t)).
Every finite extension K of k(t) has dd(K) = 1 by (6.5.5) and thus has a trivial
Brauer group Br(K) ∼
= H 2 (K,K × ) = 0. Since H 1 (K,K × ) = 0, we obtain
from (6.5.11) cdp (k(t)) = 1, hence
cdp (k(t)) ≤ cdp (k) + 1.
For the proof of the equality we replace k by the fixed field of a p-Sylow
subgroup of Gk . The dimensions cdp (k) and cdp (k(t)) do not change by (3.3.6).
Hence we may assume that Gk is a pro-p-group. It follows that the Gk -module
µp of p-th roots of unity is contained in k, and is hence isomorphic to ZZ/pZZ.
∗) The proof is taken from [210], chap.II, § 4.
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Chapter VI. Galois Cohomology
Let d = cdp (k). We have to show H d+1 (k(t), µp ) =/ 0. To this end we consider
the Hochschild-Serre spectral sequence
E2ij = H i (Gk , H j (H, µp )) ⇒ H i+j (Gk0 , µp ),
associated to the exact sequence (∗), where k 0 = k(t) and H = G(k(t) | k(t)).
We have proved already that cdp H = 1, i.e. H j (H, µp ) = 0 for j > 1.
Therefore, by (2.1.4),
H d (Gk , H 1 (H, µp )) ∼
= H d+1 (Gk0 , µp ).
p
From the exact sequence 1 → µp → k(t) × → k(t) × → 1 we obtain an isomorphism of Gk -modules H 1 (H, µp ) = k(t)× /k(t)×p , and hence an isomorphism
H d+1 (Gk0 , µp ) ∼
= H d (Gk , k(t)× /k(t)×p ).
Now let vp : k(t)× → ZZ be the p-adic valuation associated to the prime
ideal p = (t) of k[t]. The valuation vp induces a surjective homomorphism
k(t)× /k(t)×p → ZZ/pZZ of Gk -modules, hence a homomorphism
H d (Gk , k(t)× /k(t)×p ) −→ H d (Gk , ZZ/pZZ),
which is again surjective because cdp Gk = d. Since Gk is a nontrivial pro-pgroup, H d (Gk , ZZ/pZZ) =/ 0 by (3.3.2)(iii). This implies H d (Gk , k(t)× /k(t)×p )
=/ 0, and so H d+1 (Gk , µp ) =/ 0.
2
Remark. The formula cdp (k(t1 , . . . , tn )) = cdp (k) + n holds true if cdp (k) = ∞
(see [8]).
Using Lang’s theorem (6.5.6), we obtain in a similar way the
(6.5.15) Theorem. Let K be a field, complete with respect to a discrete
valuation with perfect residue field k. If p =/ char(K) is a prime number and
cdp (k) < ∞, then
cdp (K) = cdp (k) + 1.
The proof is completely analogous to the above proof. One uses the exact
sequence 1 → GK̃ → GK → Gk → 1, where K̃|K is the maximal unramified
extension with Galois group G(K̃|K) ∼
= Gk .
From the last two theorems, we obtain C(X, Y ), IFq (X), Qp as examples
of fields K with cd(K) = 2. The first two fields have also diophantine
dimension 2, which follows from (6.5.8) and (6.5.12). E. ARTIN conjectured
that dd(Qp ) = 2 also. But this is not true as was shown by G. TERJANIAN
[234]. However, Qp comes close to the C2 -property in the following sense. A
field k is said to have property Ci (d) if every form of degree d in more than
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§5. Dimension of Fields
di variables has a nontrivial zero in k. J. AX and S. KOCHEN [9] proved the
following result:
For every positive integer d there exists a finite set of primes, A(d),
such that Qp has property C2 (d) for all p ∈/ A(d).
One knows that Qp has the properties C2 (2) and C2 (3) for all p. One also knows
that for given p and d there exists an integer i ≥ 2 such that Qp has property
Ci (d). But Qp is not Ci for any i (see [AK]). In contrast, notice that the fields
Fp ((X)) are C2 , see [60], cor. 4.9. For general fields k it is conjectured by
J.-P. SERRE, cf. [210], 5th edition, chap. II §4.5, that the inequality
cd(k) ≤ dd(k)
should hold. In this direction Serre showed (in an exercise) that cd2 (k) ≤ dd(k)
if the Bloch-Kato conjecture holds for N = 2. As mentioned before, this
conjecture has been proven in the meantime (cf. (6.4.3)). Furthermore, we
have the following result.
(6.5.16) Theorem. If k is a C2 -field, then cd(k) ≤ dd(k).
If dd(k) ≤ 1, then dd(K) ≤ 1 for every finite extension K|k by (6.5.4), hence
Br(K) = 0 by (6.5.8), and therefore cd(k) ≤ 1 by (6.5.11). The implication
dd(k) ≤ 2 ⇒ cd(k) ≤ 2 is a result of A. S. MERKUR’EV and A. A. SUSLIN, see
[223], cor. 24.9.
Exercise: Assume k is a C2 -field, i.e. dd(k) ≤ 2.
(i) Every quadratic form of 5 variables has a nontrivial zero.
(ii) If D is a skew field with center k and finite over k, then the reduced norm Nrd : D× → k ×
is surjective.
det
(The reduced norm Nrd is the composite of D −→ D ⊗k K ∼
= Mn (K) −→ K × , where
K|k is a splitting field of D and n2 = dimk D.)
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Chapter VII
Cohomology of Local Fields
§1. Cohomology of the Multiplicative Group
We now begin the development of cohomology in number theory. As a
ground field we take a nonarchimedean local field k, i.e. a field which is
complete with respect to a discrete valuation and has a finite residue field.
This covers two cases, namely p-adic local fields, i.e. finite extensions of Qp
for some prime number p, and fields of formal Laurent series IF((t)) in one
variable over a finite field. For the basic properties of local fields we refer to
[160], chapters II and V. As always, k̄|k denotes a separable closure of k and
K|k the subextensions of k̄|k. vk denotes the valuation of k, normalized by
vk (k × ) = ZZ, and κ the residue field. For every Galois extension K|k we set
H i (K|k) := H i (G(K|k), K × ),
i ≥ 0.
If K|k is finite, we also set Ĥ i (K|k) = Ĥ i (G(K|k), K × ) for i ∈ ZZ. The basis
for the results in this chapter is the following theorem.
(7.1.1) Theorem. (Class Field Axiom). For a finite cyclic extension K|k we
have

#Ĥ i (K|k) =


[K : k]
for
i = 0,
1
for
i = 1.
Proof: We first show that for any finite Galois extension K|k there exists
a submodule V1 ⊆ UK1 which has finite index and is cohomologically trivial.
Let G = G(K|k) = {σ1 , . . . , σn }. We consider a normal basis of K|k, i.e. we
choose an α ∈ K such that σ1 α, . . . , σn α is a k-basis of K. For suitably chosen
a ∈ k × , we find an inclusion of G-modules
M := Ok aσ1 α + . . . + Ok aσn α ⊆ OK .
We have an isomorphism M ∼
= Ok [G] of G-modules. Furthermore, M has
m
finite index in OK , hence π OK ⊆ M for some m ≥ 1, where π ∈ Ok
is a uniformizer. This implies (π m+i M ) · (π m+i M ) ⊆ π 2m+2i OK ⊆ π m+2i M .
Therefore the subsets
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372
Chapter VII. Cohomology of Local Fields
Vi := 1 + π m+i M
⊆
UK1
are G-submodules of finite index and we have isomorphisms of G-modules
∼ M/πM,
Vi /Vi+1 −→
vi 7−→ π −m−i (vi − 1) mod πM
for all i ≥ 1. Since Vi /Vi+1 ∼
= (Ok /π)[G] is cohomologically trivial, the finite
G-modules V1 /Vi are cohomologically trivial for all i ≥ 1, and so is their
inverse limit V1 ∼
V1 /Vi .
= lim
←− i
Now assume that G is cyclic. For the Herbrand index of UK , we obtain
h(G, UK ) = h(G, V1 ) · h(G, UK /V1 ) = 1
by (1.7.5) and (1.7.6). The exact sequence 0 → UK → K × → ZZ → 0 implies
h(G, K × ) = h(G, ZZ) = #G. Finally, H 1 (G, K × ) = 0 by Hilbert’s Satz 90, and
so #Ĥ 0 (G, K × ) = #G = [K : k].
2
Next we show the vanishing of the cohomology of the unit group in unramified extensions.
(7.1.2) Proposition. Let K|k be a Galois extension. Then the following holds.
(i) If K|k is unramified, then the group of units UK and the group of principal
units UK1 are cohomologically trivial G(K|k)-modules.
(ii) If K|k is tamely ramified, then the group of principal units UK1 is a
cohomologically trivial G(K|k)-module.
Proof: By a direct limit argument, we may assume that K|k is finite. Let
K|k be unramified and let λ|κ be the associated residue extension. Then
G(K|k) ∼
= G(λ|κ), and
UKi /UKi+1 ∼
=λ
is a cohomologically trivial G(K|k)-module for all i ≥ 1 by (6.1.1). It follows
by induction that the finite G-module UK1 /UKi is cohomologically trivial for all
i ≥ 1. Since the cohomology of a finite group with values in an inverse limit of
finite modules is the inverse limit of the cohomology groups, the isomorphism
UK1 ∼
UK1 /UKi implies that
= lim
←−
i
n
H (G(K|k), UK1 ) = lim H n (G(K|k), UK1 /UKi ) = 0 for all n ≥ 1.
←− i
UK1
Hence
is cohomologically trivial. Finally, by Hilbert’s Satz 90 and the
vanishing of the Brauer group of a finite field (6.3.5), λ× is cohomologically
trivial, and the exact sequence
0 −→ UK1 −→ UK −→ λ× −→ 0
shows that UK is also cohomologically trivial.
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373
§1. Cohomology of the Multiplicative Group
If K|k is tamely ramified, let K 0 be the maximal unramified extension of k
inside K. Since UK1 is a ZZp -module and the order of G(K|K 0 ) is prime to p,
we have H j (K|K 0 , UK1 ) = 0 for j > 0, and the Hochschild-Serre spectral
sequence
E2ij = H i (K 0 |k, H j (K|K 0 , UK1 )) ⇒ H i+j (K|k, UK1 )
induces natural isomorphisms
∼ H i (K|k, U 1 )
H i (K 0 |k, UK1 0 ) →
K
2
for all i ≥ 0. Therefore (i) implies (ii).
We now consider the maximal unramified extension k̃|k. Its Galois group
Γk = G(k̃|k) is topologically generated by the Frobenius automorphism ϕk and
is canonically isomorphic to ẐZ; ϕk corresponds to 1 ∈ ẐZ. From the two exact
sequences
vk̃
ZZ −→ 0,
0 −→ Uk̃ −→ k̃ × −→
0 −→ ZZ −→ Q −→ Q/ZZ −→ 0,
in which Uk̃ and Q are cohomologically trivial Γk -modules, we obtain the
isomorphisms
δ −1
v
ϕ
k̃
2
1
Z) −→
Z,
H 2 (k̃|k) −→
Z) −→
∼ Q/Z
∼ H (Γk , Z
∼ H (Γk , Q/Z
where ϕ is given by ϕ(χ) = χ(ϕk ). We denote the composition by
∼ Q/Z
Z.
invk̃|k : H 2 (k̃|k) −→
For a finite separable extension K|k we have a commutative diagram
/.-,
H 2 (K̃|K)
inv
res
Q/ZZ
[K:k]
inv
H 2 (k̃|k)
Q/ZZ,
where the map res is induced by the compatible pair ΓK → Γk , k̃ × ,→ K̃ × .
Indeed, this follows directly from the definition of the map inv and the diagram
6978423501
H 2 (K̃|K)
res
H 2 (k̃|k)
H 2 (ΓK , ZZ)
δ −1
eK|k ·res
H 2 (Γk , ZZ)
H 1 (ΓK , Q/ZZ)
eK|k ·res
δ −1
H 1 (Γk , Q/ZZ)
Q/ZZ
eK|k ·fK|k
Q/ZZ,
which is commutative since the Frobenius automorphism ϕK ∈ ΓK is mapped
onto the fK|k -th power of the Frobenius automorphism ϕk ∈ Γk . Further
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Chapter VII. Cohomology of Local Fields
observe that eK|k · fK|k = [K : k]. This, and what follows, was already seen
in III §2 in a general setting.
If K|k is a Galois extension, then, because H 1 (K|k) = 0, the sequence
(∗)
inf
res
0 −→ H 2 (K|k) −→ H 2 (k̄|k) −→ H 2 (k̄|K)
is exact.∗) We identify H 2 (K|k) with its image in H 2 (k̄|k). Of crucial
importance is the following
(7.1.3) Theorem.
H 2 (k̄|k) = H 2 (k̃|k).
Proof: First, we claim that
#H 2 (K|k) [K : k]
for a finite Galois extension K of k. In fact, this is true if K|k is cyclic because
of H 2 ∼
= Ĥ 0 . If G(K|k) is a p-group, it follows inductively from the exact
sequence
0 −→ H 2 (L|k) −→ H 2 (K|k) −→ H 2 (K|L),
where G(K|L) is a normal subgroup of G(K|k) of order p . In the general
case, let Σp be a p-Sylow subfield of K|k. Since the restriction map
M
H 2 (K|Σp )
res : H 2 (K|k) ,→
p
is injective by (1.6.10), we obtain
Y
#H 2 (K|Σp )
#H 2 (K|k) p
Y
[K
: Σp ] = [K : k] .
p
Let n = [K : k] and let kn be the unramified extension of k of degree n.
Then
H 2 (K|k) = H 2 (kn |k) ,
where we identify H 2 (K|k) and H 2 (kn |k) with their images in H 2 (k̄|k). In
fact, because
#H 2 (K|k) [K : k] = [kn : k] = #H 2 (kn |k),
it suffices to show the inclusion "⊇". But this follows from the exact commutative diagram
0@:;<=>?AB
H 2 (K|k)
H 2 (k̄|k) res H 2 (k̄|K)
H 2 (k̃|k)
res
invk̃|k
Q/ZZ
H 2 (K̃|K)
invK̃|K
[K:k]
Q/ZZ,
∗) The group H 2 (k̄|k) is often called the “Brauer group”, because it is isomorphic to the
group Br(k) of central simple algebras by (6.3.4).
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§1. Cohomology of the Multiplicative Group
in which invk̃|k and invK̃|K are isomorphisms as shown above. Since H 2 (kn |k) ⊆
H 2 (k̃|k) has order n = [K : k], it is mapped by the middle arrow res , and thus
by the upper arrow res , to zero, hence
H 2 (kn |k) ⊆ H 2 (K|k).
We therefore obtain
H 2 (k̄|k) =
[
H 2 (K|k) =
[
H 2 (kn |k) = H 2 (k̃|k).
n
K
2
(7.1.4) Corollary. We have a canonical isomorphism
∼ Q/Z
Z,
invk : H 2 (k̄|k) −→
called the invariant map. For every finite separable extension K|k we have
the commutative diagrams
HGFEDC
H 2 (k̄|K)
res
inv
cor
H 2 (k̄|k)
Q/ZZ
[K:k]
inv
id
Q/ZZ.
The commutativity for cor follows from cor ◦ res = [K : k]. If K|k is Galois,
then the exact sequence (∗) shows that invk induces an isomorphism
∼
invK|k : H 2 (K|k) −→
1
ZZ/ZZ.
[K:k]
In other words (see (3.1.8)):
(7.1.5) Corollary. The pair (Gk , k̄ × ) is a class formation.
As a first application, we obtain (by (6.3.4)) the
(7.1.6) Corollary. For the Brauer group Br(k) of central simple k-algebras
we have a canonical isomorphism
∼ Q/Z
invk : Br(k) −→
Z,
and Br(L)(p) = 0 for any extension L|k of degree divisible by p∞ .
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Chapter VII. Cohomology of Local Fields
The last assertion follows from (7.1.4): if Lα |k runs through the finite
subextensions of L|k, then
H 2 (k̄|L)(p) = lim H 2 (k̄|Lα )(p) ∼
= lim (Q/ZZ)(p) = 0.
−→
res
−→
[Lα :k]
From (3.1.4) and III §1 ex.4, we obtain the
(7.1.7) Corollary. Let K|k be a finite Galois extension with Galois group G.
The cup-product with the fundamental class uK|k ∈ H 2 (K|k) yields isomorphisms for i ≥ 0 :
∼ Ĥ i+2 (G, K × ),
uK|k ∪ : Ĥ i (G, ZZ) −→
∼ Ĥ i+2 (G, Q/Z
uK|k ∪ : Ĥ i (G, Hom(K × , Q/ZZ)) −→
Z).
(7.1.8) Theorem. The following assertions hold:
(i) The cohomological dimension of Gk is
(
cdp (k) =
2,
1,
if p =/ char(k),
if p = char(k),
and cdp (L) ≤ 1 for every extension L|k of degree divisible by p∞ .
(ii) Suppose the characteristic of k does not divide n ∈ IN. Then we have
i
H (k, µn ) =
 × ×n

 k /k

1
ZZ/ZZ
n



0
for
i = 1,
for
i = 2,
for
i ≥ 3.
(iii) If A is a finite Gk -module of order prime to char(k), then the groups
H i (k, A) are finite for all i ≥ 0.
(iv) Let A be a Gk -module which is finitely generated as a ZZ-module and
assume that the order of tor(A) is prime to char(k). Then H 1 (k, A) is
finite.
Proof: (i) Since H 1 (L, k̄ × ) = 0 and H 2 (L, k̄ × )(p) = 0 as we have just seen, we
obtain from (6.5.11) that cdp (L) ≤ 1. We apply this to the maximal unramified
extension k̃|k. Let Γ = G(k̃|k) and let A be a p-torsion Gk -module. We have
cd (k̃) ≤ 1, hence H j (k̃, A) = 0 for j > 1. Therefore the Hochschild-Serre
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377
§1. Cohomology of the Multiplicative Group
spectral sequence
E2ij = H i (Γ, H j (k̃, A)) ⇒ H i+j (k, A)
together with cdp Γ = 1 yields by (2.1.4)
H i+1 (k, A) ∼
= H i (Γ, H 1 (k̃, A)) = 0 for i ≥ 2,
hence cdp (k) ≤ 2. If p =/ char(k), the equality cdp (k) = 2 follows from
H 2 (k, µp ) ∼
= ZZ/pZZ =/ 0, see below. If p = char(k), then cdp (k) = 1 by (6.1.3).
(ii) Recalling that H 0 (k, k̄ × ) = k × , H 1 (k, k̄ × ) = 1, H 2 (k, k̄ × ) ∼
= Q/ZZ, the
Kummer sequence (for n prime to char(k))
n
1 −→ µn −→ k̄ × −→ k̄ × −→ 1
yields H 1 (k, µn ) ∼
= k × /k ×n and H 2 (k, µn ) ∼
= n1 ZZ/ZZ. H i (k, µn ) = 0 for i > 2
follows from (i).
(iii) Let A be a finite Gk -module of order m prime to char(k). Then H i (k, A) =
0 for i > 2 by (i). Choose a finite Galois extension K|k in k̄ over which A
and µm become trivial Galois modules. As a GK -module, A is isomorphic to
a finite direct sum of GK -modules of type µn , n|m. Since H j (K, µn ) is finite
for all j ≥ 0, so is H j (K, A), and the spectral sequence
E2ij = H i (G(K|k), H j (K, A)) ⇒ H n (k, A)
shows that also H n (k, A) is finite as a group with a finite filtration, whose
quotients are subquotients of the finite groups on the left-hand side.
(iv) Let tor(A) denote the torsion subgroup of A. The cohomology sequence
associated to 0 → tor(A) → A → A/tor(A) → 0 and (iii) show that we may
assume A to be ZZ-free. Let K|k be a finite Galois extension over which
A is trivial, i.e. A ∼
= ZZn as a GK -module for some n. Then H 1 (K, A) =
1
n
H (K, ZZ) = 0 and H 1 (k, A) = H 1 (G(K|k), A) is finite.
2
Using (6.3.6), we obtain the following corollary.
(7.1.9) Corollary. If k is a p-adic local field, then canonically
H 2 (k, µ) ∼
= Q/ZZ,
and, if p = char(k) > 0, then
H 2 (k, µ) =
[
p-n
1
ZZ/ZZ
n
=
M
Q` /ZZ` .
/p
`=
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Chapter VII. Cohomology of Local Fields
§2. The Local Duality Theorem
Local fields are topological fields and topological questions enter into the
game. These questions become even more important in the global theory
because of the appearance of the idèle and idèle class groups. For this reason we
premise the discussion of local and global theory with the following preparatory
Topological Remarks: Assume we are given an abelian topological group M
on which a profinite group G acts in such a way that the action is continuous
with respect to the given topology as well as with respect to the discrete
topology. Then let
A = H i (G, M )
be the i-th cohomology group of M as a discrete G-module. If i ≥ 1, we
consider these groups always as discrete topological groups. Hence the dual
H i (G, M )∨ for i ≥ 1 is always a profinite group. But for i = 0 the group
H 0 (G, M ) inherits the initial topology of M . Examples of this situation are
M = k × , where k is a local field, and the level-compact modules considered
in I §9. Suppose that
(,)
A × B −−−→ C
is a continuous bilinear pairing of topological groups. We then obtain continuous homomorphisms
α
a 7−→ fa : b 7−→ (a, b),
β
b 7−→ gb : a 7−→ (a, b),
A −→ Hom(B, C),
B −→ Hom(A, C),
inducing bijective continuous homomorphisms
A/ ker(α) −→ im(α) and B/ ker(β) −→ im(β),
where the left groups have the quotient topology and the right ones the induced
topology. We say that the pairing A × B → C is non-degenerate if both
maps α and β are injective. Furthermore, we want to recall the notation
A∗ = Hom(A, Q/ZZ) for an abelian group A. If A is a discrete abelian torsion
group, then A∗ coincides with the Pontryagin dual A∨ , but we consider A∗ as
a discrete group, while A∨ is equipped with a natural compact topology. We
always have A∨∨ = A but the equality A∗∗ = A holds if and only if A is finite.
Corollary (7.1.5) says that for every finite Galois extension K|k of a local
field k the multiplicative group K × is a class module for the Galois group
G = G(K|k) in the sense of (3.1.3). By (7.1.4), we have even a canonical
fundamental class γ ∈ H 2 (G, K × ). Therefore, by the theorem of NakayamaTate (3.1.5), we obtain the
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§2. The Local Duality Theorem
(7.2.1) Theorem. Let K|k be a finite extension of local fields with Galois group
G. Let A be a finitely generated ZZ-free G-module and A0 = Hom(A, K × ).
Then for all i ∈ ZZ the cup-product
∪
Ĥ i (G, A0 ) × Ĥ 2−i (G, A) −→ H 2 (G, K × ) =
1
ZZ/ZZ
#G
induces an isomorphism of finite abelian groups
Ĥ i (G, A0 ) ∼
= Ĥ 2−i (G, A)∗ .
The case A = ZZ and i = 3 yields the
(7.2.2) Corollary.
H 3 (G, K × ) = 0.
In the case i = 0 and A = ZZ, we have H 2 (G, ZZ)∗ ∼
= H 1 (G, Q/ZZ)∗ =
(Gab )∗∗ = Gab , and we obtain the main theorem of local class field theory, the
“local reciprocity law” (for a proof of the second statement see [212], XIV, §6):
(7.2.3) Theorem. Let k be a local field and let K|k be a finite Galois extension.
Then there is a canonical isomorphism
k × /NK|k K × ∼
= G(K|k)ab .
The norm groups NK|k K × for finite Galois extensions K|k are precisely the
open subgroups of finite index in k × .
It was this isomorphism which initiated the development of cohomology
theory in number theory. Before this development the isomorphism was obtained only in a complicated and obscure way via a detour to the theory of
global fields. It was J. TATE who put the reciprocity law on a conceptual basis
in the above cohomological setting. The law may also be proved in a direct
group theoretical way (see [160], chap. IV and V).
However, theorem (7.2.1) is still too rigid for our intended applications as it
holds only for ZZ-free Galois modules and applies only to finite Galois groups.
Our aim is to prove a duality theorem of the above type for the absolute Galois
group Gk of a local field and for Gk -modules which are finitely generated
as ZZ-modules. The essential step on this path is the explicit determination
of the dualizing module D0 = D2 (ẐZ) of the category Modt (Gk ) of discrete
Gk -modules which are torsion as abelian groups (see III §4). It is defined by
D0 = lim H 2 (K, ZZ/nZZ)∗ ,
−→
K,n
where K|k runs through the finite subextensions of k̄|k, and it is characterized
by a functorial isomorphism H 2 (K, A)∗ ∼
= HomGK (A, D0 ) for A ∈ Modt (Gk ).
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Chapter VII. Cohomology of Local Fields
(7.2.4) Theorem. The Gk -module D0 is canonically isomorphic to the Gk module µ of all roots of unity in k̄.
Proof: Let n ∈ IN, and suppose (n, p) = 1 if p = char(k) > 0. Let n I =
n
ker(D0 → D0 ). Since D0 is also the dualizing module of every open subgroup
V of G, we obtain canonically for the Gk -module µn
HomV (µn , n I) = HomV (µn , D0 ) ∼
= H 2 (V, µn )∗ ∼
= ZZ/nZZ
by (7.1.8). This shows that HomV (µn , n I) is independent of V , and we obtain
a canonical isomorphism of Gk -modules
Hom(µn , n I) ∼
= ZZ/nZZ.
Let fn : µn → n I be the element corresponding to 1 mod nZZ. The other
elements of Hom(µn , n I) are fni (ζ) = fn (ζ i ), i = 0, . . . , n − 1. fn is a Gk homomorphism. It is injective, since it has order precisely n. It is also
surjective, since otherwise we would have an x ∈ n I r im(fn ), and hence a
homomorphism µn →(x) ⊆ n I, different from the fni . If k has characteristic
zero, we obtain a Gk -isomorphism
f :µ=
[
n∈IN
∼
µn −→
[
nI
= D0 .
n∈IN
If p = char(k) > 0, this remains true, since the p-primary components of µ and
D0 are trivial, the latter because cdp Gk = 1.
2
(7.2.5) Corollary. For every prime number p we have scdp (k) = 2.
In fact, if p =/ char(k), this is true by Serre’s criterion (3.4.5), and if p =
char(k) > 0, then cdp (k) = 1 by (6.5.10), and hence scdp (k) = 2 (cf. III §3
ex.1).
As the main result of the cohomology theory of local fields we obtain the
(7.2.6) Theorem (Tate Duality). Let k be a p-adic local field . Let A be a
finite Gk -module and set A0 = Hom(A, µ). Then the cup-product
∪
H i (k, A0 ) × H 2−i (k, A) −→ H 2 (k, µ) ∼
= Q/ZZ
induces for 0 ≤ i ≤ 2 an isomorphism of finite abelian groups
∼ H 2−i (k, A)∗ .
H i (k, A0 ) −→
If k is a local field of characteristic p > 0, the same holds[true for the finite
1
ZZ/ZZ.
Gk -modules A of order prime to p, except that H 2 (k, µ) ∼
=
n
p-n
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381
§2. The Local Duality Theorem
Proof: This theorem is a special case of the abstract duality theorem (3.4.6).
We have only to show that for p =/ char(k)
Di (ZZ/pZZ) = lim
H i (K, ZZ/pZZ)∗ = 0
−→
K|k
for i = 0, 1. Since every subextension K|k of k̄|k admits a subextension K 0 |K
of a degree divisible by p, we have D0 (ZZ/pZZ) = lim ZZ/pZZ = 0. Let K|k run
−→ K
through the subextensions of k̄|k. Then from the main theorem of local class
field theory (7.2.3) we obtain a surjection (which is in fact an isomorphism)
I ×p
K × /K
Gab
K /p.
Passing to the direct limit, we obtain a surjection from lim
−→ K
K × /K ×p =
k̄ × /k̄ ×p = 1 onto lim Gab
H 1 (K, ZZ/pZZ)∗ = D1 (ZZ/pZZ), and so
K /p = lim
−→ K
−→ K
D1 (ZZ/pZZ) = 0.
2
The duality theorem (7.2.6) deals with finite Gk -modules. It is desirable to
extend it to Gk -modules A which are finitely generated as ZZ-modules. The
main reason is that it may then be applied to algebraic tori: a Gk -module A
which is finitely generated and free as a ZZ-module is the character group of
the algebraic torus T = Hom(A, Gm ) over k, and A0 = Hom(A, k̄ × ) = T (k̄) is
the Gk -module of k̄-rational points of T . This generalization is accomplished
by the following computation of the dualizing module
D2 (ZZ) = lim H 2 (U, ZZ)∗
−→
U
of Gk , i.e. the dualizing object for the category Mod(Gk ) of all Gk -modules.
Let K|k be a finite separable extension and let UK denote its group of units.
UK is compact and we consider the profinite completion of K × ,
K̂ × = lim K × /N,
←−
N
where N runs through the open subgroups of finite index. We then have an
exact commutative diagram
0JKLTMNOPQRS
UK
K×
vK
ZZ
0
0
UK
K̂ ×
v̂K
ẐZ
0,
hence K̂ × /K × ∼
= ẐZ/ZZ, which is a uniquely divisible group. The inclusion
×
×
k ,→ K is continuous and induces an injection k̂ × ,→ K̂ × . We will consider
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382
Chapter VII. Cohomology of Local Fields
the injective limit lim K̂ × , where K runs through the finite subextensions of
−→
k̄|k. If e is the ramification index of K|k, then vK|k = evk and we have a
commutative diagram of topological isomorphisms
XVWU ×
K̂ × /K
v̂K
∼
ẐZ/ZZ
e
k̂ × /k ×
v̂k
∼
ẐZ/ZZ,
noting that ẐZ/ZZ is uniquely divisible. Applying lim
−→ K
to the exact sequence
1 −→ K × −→ K̂ × −→ ẐZ/ZZ −→ 1,
we obtain an exact sequence of Gk -modules
1 −→ k̄ × −→ lim K̂ × −→ ẐZ/ZZ −→ 1,
−→
K
since lim ẐZ/ZZ ∼
= ẐZ/ZZ. Since vK (x) = vK (σx) for all σ ∈ G(K|k), the
−→ e
latter group is a trivial Gk -module and, since it is uniquely divisible, it is a
cohomologically trivial Gk -module. The exact cohomology sequence therefore
yields the
(7.2.7) Proposition.
H 0 (k, lim K̂ × ) =
−→
K
k̂ × ,
H i (k, lim K̂ × ) = H i (k, k̄ × ) for i > 0.
−→
K
(7.2.8) Theorem. The dualizing module D2 = D2 (ZZ) of Gk is canonically
isomorphic to the Gk -module lim K̂ × .
−→
K
Proof: For every finite Galois extension K|k we have the norm residue
symbol
( , K|k) : k × /NK|k K × −→ G(K|k)ab ,
which is an isomorphism, as above. By (7.2.3), the norm groups NK|k K ×
are precisely the open subgroups of finite index in k × . Hence, passing to the
projective limit, we obtain a canonical isomorphism
( , k) : k̂ × −→ Gab ∼
= H 2 (k, ZZ)∨ .
k
For a finite Galois extension K|k, we obtain the commutative diagram
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383
§2. The Local Duality Theorem
K̂_YZ[\]^ ×
H 2 (K, ZZ)∨
Gab
K
cor ∨
Ver
k̂ ×
H 2 (k, ZZ)∨
Gab
k
(cf. [160], chap.IV, (6.4) and (1.5.9)). Taking direct limits, we get an isomorphism
2
∼ D (Z
K̂ × −→
( , k̄) : lim
2 Z).
−→
K
Let A be a Gk -module which is finitely generated as a ZZ-module. By tor(A)
we denote the torsion subgroup of A, which is a Gk -module. Furthermore, we
will use the following notation for the rest of this chapter:
A0 = Hom(A, k̄ × ),
AD = Hom(A, lim
K̂ × ) = Hom(A, D2 ).
−→
K
As a generalization of Tate’s local duality theorem (7.2.6) we have the
(7.2.9) Theorem. Let A be a Gk -module which is finitely generated as a
ZZ-module and assume that #tor(A) is prime to char(k). Then, for 0 ≤ i ≤ 2,
the cup-product and the map inv
∪
inv
H i (k, AD ) × H 2−i (k, A) −→ H 2 (k, k̄ × ) −→ Q/ZZ
induce an isomorphism
∼ H 2−i (k, A)∗ .
∆i : H i (k, AD ) −→
For i = 1 these groups are finite.
Proof: We know by (7.2.5) that scd(k) = 2. We have D0 (ZZ) = 0 by the
remark following (3.4.3), and, trivially, D1 (ZZ) = lim H 1 (K, ZZ)∗ = 0. If
−→
char(k) = 0, then k̄ × is divisible and so is D2 (ZZ). If p = char(k) > 0, then
D2 (ZZ) is `-divisible for all prime numbers ` =/ char(k). Therefore we obtain
the isomorphisms by Tate’s duality theorem (3.4.3) and the subsequent remark.
The finiteness of H 1 (k, A) is part (iv) of (7.1.8).
2
The Gk -module D2 = lim
−→
K
K̂ × looks rather awkward and one may ask
whether it can be replaced by k̄ × in the theorem. This is clear for a finite
module A, since Hom(A, D2 ) = Hom(A, k̄ × ) = Hom(A, µ). Furthermore, the
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384
Chapter VII. Cohomology of Local Fields
groups AD = Hom(A, D2 ) and A0 = Hom(A, k̄ × ) inherit the natural topologies
K̂ × ∗) and of k̄ × . Therefore the groups H 0 (k, A0 ) and H 0 (k, AD )
of D2 ∼
= lim
−→
are also topological groups in a natural way.
Since H 2 (k, A) is a discrete torsion group, we may replace H 2 (k, A)∗ by the
compact topological group H 2 (k, A)∨ (which has the same underlying abstract
∼ H 2 (k, A)∨ between
group) and obtain an abstract isomorphism H 0 (k, AD ) −→
abelian topological groups.
(7.2.10) Proposition. The duality isomorphism
∼ H 2 (k, A)∨
H 0 (k, AD ) −→
is a homeomorphism. The natural injection
H 0 (k, A0 ) ,→ H 0 (k, AD )
is continuous and H 0 (k, AD ) is the profinite completion of H 0 (k, A0 ) with
respect to the open subgroups of finite index. We have isomorphisms
∼ H i (k, AD ) for i > 0
H i (k, A0 ) −→
and the groups are finite for i = 1.
Proof: The exact cohomology sequence associated to the sequence
0 −→ tor(A) −→ A −→ A/tor −→ 0,
the finiteness of the cohomology of tor(A), and the identity tor(A)0 = tor(A)D ,
allow us to assume A to be ZZ-free. Let K|k be a finite Galois extension over
which A becomes a trivial Galois module, and let G = G(K|k) be its Galois
group. We have an isomorphism of GK -modules A ∼
= ZZN for some N ∈ IN.
Now consider the commutative diagram
H 0 (K, A`abcdef 0 )G(K|k)
H 0 (K, AD )G(K|k)
(H 2 (K, A)∨ )G(K|k)
H 0 (k, A0 )
H 0 (k, AD )
H 2 (k, A)∨ .
The vertical arrow on the right is an isomorphism by (3.3.11) and the right
upper horizontal isomorphism is a homeomorphism by the definition of the
topology on D2 and because A is a trivial GK -module. The right commutative
square now gives the first statement.
By definition of K̂ × , the group H 0 (K, AD ) = Hom(A, K̂ × ) ∼
= (K̂ × )N is the
0
0
× ∼
× N
completion of H (K, A ) = Hom(A, K ) = (K ) with respect to the open
∗) The topology depends on the choice of this isomorphism. We use the natural isomorphism
constructed in (7.2.8).
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385
§2. The Local Duality Theorem
subgroups of finite index, and this remains valid for the fixed modules under
G(K|k). This shows the second statement.
We now prove the maintained isomorphisms for i > 0. From the exact
sequence
0 −→ k̄ × −→ D2 −→ ẐZ/ZZ −→ 0,
we obtain the exact sequence
0 −→ A0 −→ AD −→ Hom(A, ẐZ/ZZ) −→ 0.
Since ẐZ/ZZ is uniquely divisible, so is the Gk -module Hom(A, ẐZ/ZZ), and
hence it is cohomologically trivial. From the exact cohomology sequence it
follows that
(∗)
H i (k, A0 ) ∼
= H i (k, AD )
for i > 1. For i = 1 we need a little additional argument: for any open
subgroup H ⊆ Gk which acts trivially on A, the sequence
0 −→(A0 )H −→(AD )H −→ Hom(A, ẐZ/ZZ) −→ 0
is exact, since H 1 (H, A0 ) ∼
= H 1 (H, Hom(ZZN , k̄ × )) = H 1 (H, k̄ × )N = 0. Bei
cause Ĥ (Gk /H, Hom(A, ẐZ/ZZ)) = 0 for i ≥ 0, we obtain
H 1 (Gk /H, (A0 )H ) = H 1 (Gk /H, (AD )H ),
and from this follows (∗), by taking direct limits. Finally, the finiteness of
H 1 (k, A0 ) ∼
2
= H 1 (k, AD ) follows from (7.2.9).
The above duality theorem contains local class field theory as a special case.
Namely, taking i = 0 and A = ZZ, we get an isomorphism
∨ ∼
1
∼ H 2 (G , Z
Z)∨ = Gab ,
k̂ × = H 0 (k, D2 ) −→
k Z) = H (Gk , Q/Z
k
and from this we get the following theorem.
(7.2.11) Theorem. Let k be a local field. Then there is an exact sequence
0ijgh
k×
( ,k)
Gab
k
ẐZ/ZZ
0,
where ( , k) is the norm residue symbol of local class field theory.
The homomorphism ( , k) is by definition (cf. (3.1.6)) characterized by the
(7.2.12) Proposition. For every χ ∈ H 1 (Gk , Q/ZZ) the norm residue symbol
satisfies the formula
χ((a, k)) = inv(a ∪ δχ),
∼ H 2 (G , Z
where δ : H 1 (Gk , Q/ZZ) →
k Z).
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Chapter VII. Cohomology of Local Fields
We have seen that there are two ways to construct abelian extensions of
exponent m of the ground field k, where m is prime to char(k). First, using
Kummer theory (if µm is contained in k), we obtain an isomorphism
δ
1
k × /k ×m −→
∼ H (k, µm ).
Secondly, class field theory yields
rec
ab
k × /k ×m −→
∼ Gk /m.
We want to compare these maps. Using (1.4.6) for the exact sequence
m
0 −→ ZZ −→ ZZ −→ ZZ/mZZ −→ 0
m
and the Gk -module k̄ × (observe that the sequence 0 → µm → k̄ × → k̄ × → 0
is also exact), we get the commutative diagram
onmlk k̄ × ) ×
H 0 (G,
δ
H 2 (G, ZZ)
∪
H 2 (G, k̄ × )
−id
δ
H 1 (G, µm ) × H 1 (G, ZZ/mZZ)
From (7.2.12) it follows that
∪
H 2 (G, k̄ × ).
χ((a, k)) = inv(a ∪ δχ) = inv(−δ(a) ∪ χ)
for a ∈ H 0 (G, k̄ × ) and χ ∈ H 1 (G, ZZ/mZZ) ⊆ H 1 (G, Q/ZZ). Thus we obtain
(7.2.13) Corollary. Let m
Then the diagram
ustrqp
Gab
k /m
∈
IN be not divisible by the characteristic of k.
∪
× Hom(Gk , ZZ/mZZ)
ZZ/mZZ
rec
k × /k ×m
inv
−δ
H 1 (k, µm ) ×
H 1 (k, ZZ/mZZ)
∪
H 2 (k, µm )
is commutative. The cup-product on top is given by applying a character
χ ∈ Hom(Gk , ZZ/mZZ) to an element of Gab
k /m.
An important addendum to the local duality theorem arises from the presence
of the maximal unramified extension k̃|k. Let Γ = G(k̃|k) be its Galois group
and T = G(k̄|k̃) the inertia group. Let Õ be the valuation ring of k̃ and observe
that Õ× is a Gk -submodule of D2 . A Gk -module M is called unramified if
MT = M.
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387
§2. The Local Duality Theorem
Let A be a Gk -module which is finitely generated as a ZZ-module. We
assume that the order of the ZZ-torsion subgroup tor(A) of A is prime to the
characteristic of the residue field of k. If A is unramified, we write
Ad = Hom(A, Õ× ),
which is a submodule of AD = Hom(A, D2 ). Obviously, Ad is also unramified.
If A is finite and unramified, then
AD = A0 = Ad = Hom(A, µ) .
(7.2.14) Definition. For a Gk -module A, we set
i
Hnr
(k, A) = im(H i (Γ, AT ) −→ H i (Gk , A))
and these groups are called the unramified cohomology groups . ∗)
If A is unramified, then we have
0
Hnr
(k, A) = H 0 (k, A),
0
Hnr
(k, Ad ) ⊆ H 0 (k, A0 ) ,
and if in addition A is finite, then
2
2
Hnr
(k, A) = Hnr
(k, Ad ) = 0
as cd Γ = 1.
(7.2.15) Theorem. Let A be an unramified Gk -module which is finitely
generated as a ZZ-module and assume that #tor(A) is prime to the characteristic
i
2−i
of the residue field of k. Then the groups Hnr
(k, Ad ) and Hnr
(k, A) annihilate
each other in the pairing
∪
H i (k, AD ) × H 2−i (k, A) −→ H 2 (k, k̄ × ) ,→ Q/ZZ .
They are mutually orthogonal complements for i = 1, and for 0
is finite.
≤
i
≤
2 if A
Proof: The Γ -module Õ× is cohomologically trivial by (7.1.2)(i) and the
commutative diagram
H i (Γ,wxyzv Ad ) × H 2−i (Γ, A)
H i (k, AD ) × H 2−i (k, A)
∪
∪
H 2 (Γ, Õ× ) = 0
H 2 (k̄, D2 ) ,→ Q/ZZ
∗) The letters nr stand for “non ramifié”.
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388
Chapter VII. Cohomology of Local Fields
i
2−i
shows that the images Hnr
(k, Ad ) and Hnr
(k, A) of the upper groups are
orthogonal. If A is finite, then
0
2
Hnr
(k, Ad ) = H 0 (k, AD ) and Hnr
(k, A) = 0
recalling that cd Γ = 1,
2
0
Hnr
(k, Ad ) = 0 and Hnr
(k, A) = H 0 (k, A).
We have therefore to investigate only the case i = 1 for a Gk -module A which
is finitely generated as a ZZ-module. Using the exact sequence
0 −→ tor(A) −→ A −→ A/tor −→ 0,
we obtain a commutative diagram
{|}~
H 1 (k, tor(A))
H 1 (T, tor(A))Γ
H 1 (k, A)
H 1 (T, A)Γ ,
where the upper map is surjective, since cd Γ = 1, and the map on the righthand side is an isomorphism, since A is a trivial T -module and therefore
H 1 (T, A/tor) = Hom(T, A/tor) = 0. Thus in the commutative diagram
0€‚ƒ„…†
H 1 (Γ, A)
H 1 (T, A)Γ
H 1 (k, A)
0
ϕA
H 1 (k, AD )∗
H 1 (Γ, Ad )∗
0
the upper sequence is exact. The map on the bottom is surjective, since
inf
H 1 (Γ, Ad ) ,→ H 1 (Γ, (AD )T ) −→ H 1 (k, AD )
is injective. Since we already saw that H 1 (Γ, A) and H 1 (Γ, Ad ) annihilate each
other, we obtain the dotted map ϕA , which is necessarily surjective. We have
to show that ϕA is bijective.
In the commutative diagram
Γ
‡ˆ‰Š
H 1 (T, tor(A))
ϕtor(A)
H 1 (T, A)Γ
ϕA
H 1 (Γ, (tor(A))d )∗
H 1 (Γ, Ad )∗ ,
the lower map is injective, which one sees as follows: let Γ 0 be an open
subgroup of Γ acting trivially on A/tor. Because scd Γ = 2, we have a
surjection
cor
H 2 (Γ 0 , Hom(A/tor, Õ× )) −→ H 2 (Γ, Hom(A/tor, Õ× )) −→ 0
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389
§2. The Local Duality Theorem
and the group on the left is zero, since Õ× is cohomologically trivial. Thus
H 2 (Γ, (A/tor)d ) = 0 and we are reduced to the case of a finite module: if ϕtor(A)
is injective, then ϕA is also injective.
Since the order of tor(A) is prime to the residue characteristic p, we can
replace T by its tame part T0 . Using Kummer theory, we get a natural isomorphism
µn
T0 ∼
= lim
←−
(n,p)=1
(cf. the remarks at the beginning of §5). As an abelian group, T0 is isomorphic
Q
to `=/p ZZ` and is therefore a duality group of dimension 1 for torsion modules
Q
of order prime to p with dualizing module I ∼
= `=/p Q` /ZZ` . As a Gk -module,
I is naturally isomorphic to the prime-to-p part of the module µ of all roots of
unity. Therefore we obtain
H 1 (T0 , tor(A)) ∼
= H 0 (T0 , Hom(tor(A), µ))∗ = Hom(tor(A), µ)∗ ,
so that, using H 1 (T, tor(A)) = H 1 (T0 , tor(A)), we get
H 1 (T, tor(A))Γ ∼
= (Hom(tor(A), µ)Γ )∗ = H 1 (Γ, (tor(A))d )∗ .
This finishes the proof of the theorem.
2
If A is an unramified Gk -module which is finitely generated and free as a
ZZ-module, then we can consider Ad as a torus over Ok with character group
A. Using the technique of smooth base change, the next corollary can also be
derived (at least for the part prime to the residue characteristic) from a theorem
of S. LANG which asserts that a connected algebraic group over a finite field is
cohomologically trivial.
(7.2.16) Corollary. Let A be an unramified Gk -module which is finitely
generated and free as a ZZ-module. Then
H i (Γ, Ad ) = 0 for all i ≥ 1 .
Proof: This is clear for i ≥ 3 and we saw the assertion for i = 2 in the proof
of (7.2.15). Finally,
H 1 (Γ, Ad ) ∼
2
= (H 1 (T, A)Γ )∗ = HomΓ (T, A)∗ = 0.
For the cohomology theory of global fields, we will need also the following
duality theorem for the extension C|IR.
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Chapter VII. Cohomology of Local Fields
(7.2.17) Theorem. Let G = G(C|IR). Let A be a finitely generated G-module
and set A0 = Hom(A, C× ). Then H 2 (G, C× ) ∼
= 12 ZZ/ZZ, and the cup-product
∪
Ĥ i (G, A0 ) × Ĥ 2−i (G, A) −→ H 2 (G, C× )
induces for all i ∈ ZZ an isomorphism
∼ Ĥ 2−i (G, A)∗ .
Ĥ i (G, A0 ) −→
2
Proof: The exact sequence 1 → µ2 → C× → C× → 1 yields H 2 (G, C× ) ∼
=
H 3 (G, µ2 ) ∼
= H 1 (G, µ2 ) ∼
= 12 ZZ/ZZ. The theorem is true for the G-modules
A = ZZ/2ZZ, ZZ, ZZ(1),
(∗)
where ZZ(1) = ZZ as abelian group, and the generator σ of G acts as multiplication by −1 (A0 is then C× with the action σz = z̄ −1 ).
In fact, for A = ZZ the cohomology groups Ĥ i (G, A) and Ĥ i (G, A0 ) are
trivial for odd i and the same holds for A = ZZ(1) with even i. The cohomology
groups that occur in all other cases have order 2. If x and y are the nontrivial
elements of Ĥ i (G, A0 ) and Ĥ 2−i (G, A), then a direct elementary computation
(and then using the periodicity for the cohomology of cyclic groups) shows
that x ∪ y is the nontrivial element of H 2 (G, C× ). This proves the theorem for
the modules (∗).
Suppose we are given an exact sequence 0 → B → A → C → 0 of finitely
generated G-modules. Setting Ĥ i (M ) = Ĥ i (G, M ), we obtain a diagram
Ĥ i−1‹ŒŽ”•–—’‘“ (B 0 )
α
Ĥ 2−i+1 (B)∗
Ĥ i (C 0 )
β
Ĥ 2−i (C)∗
Ĥ i (A0 )
γ
Ĥ 2−i (A)∗
Ĥ i (B 0 )
δ
Ĥ 2−i (B)∗
Ĥ i+1 (C 0 )
ε
Ĥ 2−i−1 (C)∗
with exact rows. The partial diagrams are commutative or anti-commutative.
The five-lemma now implies that the assertion is true for A if it holds for B
and C.
If we want to prove the theorem for an arbitrary finitely generated G-module
A, we may assume that its torsion submodule is 2-primary. Since the theorem
is true for ZZ/2ZZ, which is the only finite simple 2-primary G-module, we
therefore obtain the result for finite G-modules.
Now consider the general case. First of all, we may assume the module A
to be torsion-free. Let 0 =/ a ∈ A be arbitrary. If a + σa =/ 0, it generates a
submodule isomorphic to ZZ and otherwise a = −σa generates a submodule
isomorphic to ZZ(1). Therefore the general case follows by induction on the
ZZ-rank of the module A.
2
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391
§3. The Local Euler-Poincaré Characteristic
We finish this section by fixing conventions in the archimedean case. Firstly,
we consider the extension C|IR as ramified, hence the inertia group is the full
group:
T (C|R) = G(C|R).
i
We therefore have Hnr
(K, A) = 0 for all i ≥ 0, any archimedean local field K
and any G(K|K)-module A. Furthermore, we put UK = K × and
(
UK1
=
IR×
+
C×
if K = IR,
if K = C.
Exercise 1. Let k be a local field and let k̃|k be the maximal unramified extension. Let d :
Gk → ẐZ be the surjective homomorphism coming from the canonical isomorphism G(k̃|k) ∼
=
ẐZ.
Show that the Weil group Wk of the class formation (Gk , k̄ × ) (see III §1) is the pre-image
Wk = d−1 (ZZ).
Hint: For every finite Galois extension K|k we have a commutative exact diagram
1žŸ ¡¢˜™š›œ
K×
W (K|k)
G(K|k)
1
ρ
1
g
G(K̃|K)
G(K̃|k)
G(K|k)
1,
where ρ(a) = ϕvKK (a) . Show that the image of g is the pre-image d−1 (ZZ) under d : G(K̃|k) → ẐZ,
and then pass to the limit.
Exercise 2. Let k be a local field, let k̃ be the maximal unramified extension of k and let
T = G(k̄|k̃) be the inertia subgroup of Gk . Let A be a finite Gk -module of order prime to the
residue characteristic of k (not necessarily unramified). Show that
1
1
Hnr
(k, A)⊥ = Hnr
(k, A0 )
with respect to the Tate-pairing.
§3. The Local Euler-Poincaré Characteristic
Let k be a local field and let p be its residue characteristic. If A is a
finite Gk -module of order prime to char(k) (in the case char(k) > 0), then
the cohomology groups H i (k, A) are finite groups by (7.1.8)(iii). We set
hi (k, A) = #H i (k, A) and define the Euler-Poincaré characteristic of A by
χ(k, A) =
Y
i
i
hi (k, A)(−1) =
h0 (k, A)h2 (k, A)
.
h1 (k, A)
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Chapter VII. Cohomology of Local Fields
If 0 → A → B → C → 0 is an exact sequence of finite Gk -modules of order
prime to char(k), then the alternating product of the orders of the groups in the
exact cohomology sequence
. . . −→ H i (k, A) −→ H i (k, B) −→ H i (k, C) −→ H i+1 (k, A) −→ . . .
is 1, and from this it follows that
χ(k, B) = χ(k, A)χ(k, C).
Our aim is to prove the
(7.3.1) Theorem (TATE). For every finite Gk -module A of order a prime to
char(k) we have
χ(k, A) = ||a||k ,
where || ||k is the normalized absolute value of k.∗)
The formula is simple, but the proof is surprisingly difficult. It mirrors an
explicit description of the multiplicative group K × of a finite Galois extension
K|k as a G(K|k)-module.
Let G = G(K|k) and let ` be a prime number =/ char(k). We consider
the Grothendieck group K00 (IF` [G]) of finite IF` [G]-modules (cf. [32], chap.II,
§16b):
Let F (G) be the free abelian group generated by the set of isomorphism
classes of finite IF` [G]-modules. Denoting the isomorphism class of A by
{A}, let R(G) be the subgroup in F (G) generated by all elements
{B} − {A} − {C}
arising from short exact sequences 0 → A → B → C → 0. Then K00 (IF` [G])
is defined as the quotient F (G)/R(G). We denote by [A] the class of {A}
in K00 (IF` [G]). K00 (IF` [G]) becomes a ring by linear extension of the product
[A][B] = [A ⊗IF` B].
If A is a finite IF` [G]-module, then we can view it as a Gk -module which
becomes a trivial Galois module over K, and the cohomology groups H i (K, A)
are also finite IF` [G]-modules. We define
h(K, A) =
2
X
(−1)i [H i (K, A)],
i=0
K00 (IF` [G]).
which is an element in
Theorem (7.3.1) will be a consequence of
the following theorem, which is a sharpening of it.
∗) That is, ||x|| = q −v(x) , where q is the cardinality of the residue field κ and v is the additive
k
valuation of k with value group ZZ.
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393
§3. The Local Euler-Poincaré Characteristic
(7.3.2) Theorem (SERRE). Let A be a finite IF` [G]-module. If ` is not equal to
the residue characteristic p of k, then h(K, A) = 0. If ` = p and char(k) = 0,
then
h
i
h(K, A) = − dimIFp (A)[k : Qp ] IFp [G] .
For the proof of Serre’s theorem we need the following
(7.3.3) Lemma.
(i) Let A be a finite G-module. Then
[` A] = [A` ] . ∗)
(ii) Let V be a G-module such that V` and ` V are finite. If W
submodule of finite index, then
⊆
V is a
[V` ] − [` V ] = [W` ] − [` W ] .
Proof: We prove (ii) first. Using a Jordan-Hölder series, we may assume that
V /W is a finite simple G-module. In particular, ` (V /W ) ∼
= (V /W )` . Consider
the diagram
0ª«¬­©¨£¤¥¦§
0
W
V
`
`
W
V
V /W
0
`
V /W
0.
The snake lemma gives the exact sequence
0 −→ ` W −→ ` V −→ ` (V /W ) −→ W` −→ V` −→(V /W )` −→ 0,
and hence the result. Assertion (i) follows by applying (ii) to V = A, W = 0.
2
Proof of (7.3.2): First suppose
we get
H 0 (K, µ` ) =
H 1 (K, µ` ) =
H 2 (K, µ` ) =
Therefore
that A = µ` . Using the Kummer sequence,
µ` (K),
K × /K ×` ,
H 0 (K, ZZ/`ZZ)∗ = ZZ/`ZZ.
h(K, µ` ) = [µ` (K)] − [K × /K ×` ] + [ZZ/`ZZ].
∗) For a ZZ -module V , the modules V and V are defined by the exact sequence
`
`
`
`
0 → ` V → V → V → V` → 0.
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394
Chapter VII. Cohomology of Local Fields
Let U denote the group of units of the valuation ring O of K. The exact
sequence
0 −→ U/U ` −→ K × /K ×` −→ ZZ/`ZZ −→ 0
gives us [K × /K ×` ] = [ZZ/`ZZ] + [U/U ` ], hence
(1)
h(K, µ` ) = [µ` (K)] − [U/U ` ] = [` U ] − [U` ].
Let p be the maximal ideal of O and V = 1 + p the group of principal units. V
is a finitely generated ZZp [G]-module, and we have the exact sequence
1 −→ V −→ U −→ κ× −→ 1,
with κ× the multiplicative group of the residue field of K. Since κ× is finite
we obtain from (1) and (7.3.3)
(2)
h(K, µ` ) = [` U ] − [U` ] = [` V ] − [V` ].
Now if ` =/ p, then ` V = V` = {1} since V is a pro-p-group, hence h(K, µ` ) = 0.
So let ` = p and char(k) = 0. Let W = 1 + pn be the group of n-th principal
units. It is a ZZp [G]-submodule of finite index in V , hence by lemma (7.3.3)
again, we obtain from (2)
h(K, µ` ) = [p W ] − [Wp ].
If n is sufficiently large, then the logarithm log : W → pn is an isomorphism
(see [160], chap.II, (5.5)) and, since pn is a subgroup of finite index in O, we
get
(3)
h(K, µ` ) = [p O] − [O/pO].
The extension K|k has a normal basis, K =
M
kσθ, and we may choose θ in O.
σ ∈G M
If R denotes the valuation ring of k, then M =
Rσθ is a ZZp [G]-submodule
σ ∈G
of O of finite index and p M = 0 and M/pM ∼
= R/pR[G] ∼
= IFp [G][k:Qp ] . We
now obtain from (3) via (7.3.3)
h(K, µ` ) = −[M/pM ] = −[k : Qp ][IFp [G]].
Now let A be an arbitrary finite IF` [G]-module. The cup-product on the
cochain groups C i (K, ZZ/`ZZ) ⊗ A → C i (K, A) is obviously an isomorphism,
hence we have an isomorphism of G-modules H i (K, ZZ/`ZZ) ⊗ A ∼
= H i (K, A),
and so
h(K, A) = h(K, ZZ/`ZZ) · [A].
The functor M 7→ M ∗ = Hom(M, IF` ) is exact on finite IF` [G]-modules and
thus defines an endomorphism ξ 7→ ξ ∗ of the group K00 (IF` [G]). By the duality
theorem (7.2.6), we have
h(K, ZZ/`ZZ)∗ = h(K, µ` ),
and therefore, since [IF` [G]] = [IF` [G]∗ ],
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395
§3. The Local Euler-Poincaré Characteristic
(
h(K, A) =
−[k : Qp ][IFp [G]] · [A],
0,
if p = ` ,
if p =/ ` .
Furthermore, if A0 denotes the trivial G-module with underlying group A, then
IF` [G] ⊗ A0 → IF` [G] ⊗ A, σ ⊗ a 7→ σ ⊗ σa, is an isomorphism of IF` [G]{1}
modules (this is just the dual statement to IndG A ∼
= IndG A, see I §6, p.62),
so that
[IF` [G]] · [A] = [IF` [G]] · [A0 ] = dimIF` (A)[IF` [G]].
2
This finishes the proof of theorem (7.3.2).
For the deduction of Tate’s formula
χ(A) = ||a||k ,
(∗)
we need from the representation theory of finite groups another
(7.3.4) Lemma. The group K00 (IF` [G]) ⊗ Q is generated by the images of
K00 (IF` [H]) ⊗ Q under IndH
G , where H runs through all cyclic subgroups of G
of order prime to `.
Proof: A theorem of E. Artin (see [214], 12.5 th. 26) asserts that the map
Ind ⊗ Q :
M
K00 (Q` [H]) ⊗ Q −→ K00 (Q` [G]) ⊗ Q
H ∈T
is surjective where T is the set of all cyclic subgroups of G. (This holds with
any field of characteristic zero replacing Q` .) By [214], 16.1, th. 33, there
exists a surjective homomorphism K00 (Q` [G]) ⊗ Q K00 (IF` [G]) ⊗ Q which is
natural with respect to the group G. Therefore the above map Ind ⊗ Q remains
0
surjective if we replace Q` by IF` . Let H = H (` ) × H` be a cyclic subgroup of
G where H` is the `-Sylow subgroup of H. Let M be a simple IF` [H]-module.
By (1.6.13), M H` =/ 0. Since M H` is also an H-module, we obtain M = M H` ,
(`0 )
H
∼
hence an isomorphism of IF` [H]-modules IndH
H ResH (`0 ) M = M ⊗ ZZ IF` [H` ].
(`0 )
H
Since ZZ/`ZZ is the only simple IF` [H` ]-module, the class of IndH
H ResH (`0 ) M
0
n
in K0 (IF` [H]) is equal to n[M ], where #H` = ` . Therefore the images of
0
K00 (IF` [H]) ⊗ Q and K00 (IF` [H (` ) ]) ⊗ Q under Ind ⊗ Q in K00 (IF` [G]) ⊗ Q are
the same.
2
Proof of theorem (7.3.1): Let A be a finite Gk -module of order a. We may
assume `A = 0 for some prime number ` =/ char(k). The general case follows
from this via the exact sequence 0 → ` A → A → A/` A → 0 by induction on
the order of A, since χ and || ||k are multiplicative with respect to short exact
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396
Chapter VII. Cohomology of Local Fields
sequences. Every finite Gk -module becomes a trivial Galois module over some
finite Galois extension K|k. It thus suffices to prove the formula (∗) for finite
IF` [G]-modules with G = G(K|k). The functions χ(k, A) and ϕ(A) = ||a||k are
additive, i.e. they define homomorphisms
χ, ϕ : K00 (IF` [G]) −→ Q×
+,
and we have to show χ = ϕ. Using lemma (7.3.4) and observing that Q×
+ is
torsion-free, it suffices to check this equality on elements of the form B =
IndH
G (A), where A is a finite IF` [H]-module and H is a cyclic subgroup of G
of order prime to `. Let k 0 be the fixed field of H. Shapiro’s lemma yields
χ(k, B) = χ(k 0 , A)
and for b = #B we have
0
:k]
||a||k0 = ||a||[k
= ||b||k .
k
This reduces our problem to the case where G is a cyclic group of order prime
to `. Then the Hochschild-Serre spectral sequence for the group extension
1 → GK → Gk → G → 1
and the G-module A degenerates. Thus H i (k, A) = H 0 (G, H i (K, A)). Therefore if
d : K00 (IF` [G]) → ZZ
is the homomorphism given by
d([M ]) = dimIF` (H 0 (G, M )),
then
χ(k, A) = `d(h(K,A)) .
If ` =/ p, then by (7.3.2) h(K, A) = 0, so that χ(k, A) = 1, and if ` = p, then
h(K, A) = − dim(A)[k : Qp ][IFp [G]], and since d([IFp [G]]) = dim(IFp [G]G ) =
dim(IFp ) = 1, we obtain
χ(k, A) = p−[k:Qp ] dim A = ||a||k .
2
For the fields IR and C we have a similar statement.
(7.3.5) Theorem. If k = IR or C, then for any finite Gk -module A of order a
we have
h0 (k, A)h0 (k, A0 )
= ||a||k ,
h1 (k, A)
where ||x||IR = |x| and ||x||C = |x|2 .
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397
§3. The Local Euler-Poincaré Characteristic
Proof: If k = C, then H 0 (k, A) = A and H 0 (k, A0 ) = A0 both have order a,
H 1 (k, A) = 0 and ||a||C = a2 . Let k = IR and G = G(C|IR), and let σ be the
generator of G. For x ∈ A and f ∈ A0 , we have
((1 − σ)f )(x) = f (x)/σ(f (σx)) = f (x)(f (σx)) = f ((1 + σ)x),
noting that σζ = ζ −1 for a root of unity in C. Therefore 1 − σ : A0 → A0 is
adjoint to 1 + σ : A → A, and so, in the pairing A0 × A → C× , (A0 )G and NG A
are exact annihilators. Therefore
||a||IR = #A = #(A0 )G #NG A = #H 0 (G, A0 )#(H 0 (G, A)/Ĥ 0 (G, A)).
Since A is finite, we have by [160], chap.IV, (7.3) and (1.7.1)
#Ĥ 0 (G, A) = #Ĥ −1 (G, A) = #Ĥ 1 (G, A) ,
2
and the theorem follows.
We saw in VII §2 that the Galois module µ plays an important role in the
case of local fields. Now we introduce the following general terminology.
Assume that k is any field. The subgroup µ of roots of unity contained in
the separable closure k̄ of k is a Gk -module in a natural way. There exists a
canonical isomorphism
∼
h : Aut(µ) −→
ZZ×
` ,
Y
/ char(k)
`=
given by ϕ(ζ) = ζ h(ϕ) . The right side of the equality is defined as follows: if
ζ n = 1, n ∈ IN, then ζ α := ζ a for any a ∈ ZZ with a ≡ α mod n. The action of
Gk on µ is given by a character
χcycl : Gk −→
Y
ZZ×
` .
/ char(k)
`=
(7.3.6) Definition. The character χcycl is called the cyclotomic character.
Let A be a finite Gk -module whose order is prime to char(k). For i ∈ ZZ we
denote by A(i) the Gk -module which is equal to A as an abelian group and
which is endowed with the (twisted) action
σ(a) := χcycl (σ)i · σa,
where the action on the right-hand side is the original action of Gk on A.
We call A(i) the i-th Tate twist of A. We apply the same definition to a
discrete resp. compact G-module which is a direct resp. projective limit of
finite modules of order prime to char(k).
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Chapter VII. Cohomology of Local Fields
In particular, A = A(0) and for i, j
∈
ZZ we have the rule
A(i + j) = A(i)(j) = A(j)(i).
Note that the above definition can only be applied to modules A such that the
multiplication by χcycl (σ) is a well-defined automorphism. In particular, the
module ZZ(i) does not exist (unless we consider archimedean local fields).
For (n, char(k)) = 1 and i ≥ 1 we have
ZZ/nZZ (i) ∼
= µ⊗i
n
and
ZZ/nZZ (−i) ∼
Z/nZZ).
= Hom(µ⊗i
n ,Z
For a prime number ` and i ∈ ZZ we have
ZZ` (i) = lim ZZ/`m ZZ(i)
←−
m
Q` /ZZ` (i) = lim ZZ/`m ZZ(i) .
and
−→
m
If A is finite with nA = 0, then A(i) = A ⊗ ZZ ZZ/nZZ (i) for all i ∈ ZZ.
Returning to the case of a local field, we have the
(7.3.7) Theorem. Let k be a local field with residue characteristic p and let
` =/ char(k) be a prime number. Then for all j ∈ ZZ
2
X
(
(−1)i dimIF` H i (Gk , ZZ/`ZZ (j)) =
i=0
−[k : Qp ] if ` = p ,
0
otherwise.
Proof: This follows directly from (7.3.1), since
(
||`||k =
p−[k:Qp ] if ` = p ,
1
otherwise.
2
(7.3.8) Corollary. With the notation as in (7.3.7), the following equalities hold
for all j ∈ ZZ:
2
X
(
i
(−1) rank ZZ` Hi (Gk , ZZ` (j)) =
i=0
−[k : Qp ] if ` = p ,
0
otherwise,
and
2
X
i=0
(
(−1)
i
i
rank ZZ` Hcts
(Gk , ZZ` (j))
=
−[k : Qp ] if ` = p ,
0
otherwise.
Proof: The second equality follows from the first and (2.7.12). In order to
prove the first statement, we observe that
rank ZZ` Hi (Gk , ZZ` (j)) = dimIF` Hi (Gk , ZZ` (j))/` − dimIF` ` Hi (Gk , ZZ` (j))
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399
§3. The Local Euler-Poincaré Characteristic
and
dimIF` Hi (Gk , ZZ/`ZZ(j)) = dimIF` H i (Gk , ZZ/`ZZ(−j)) .
by (2.6.9). The exact sequence
`
0 −→ ZZ` (j) −→ ZZ` (j) −→ ZZ/`ZZ(j) −→ 0
shows that
dimIF` H0 (Gk , ZZ/`ZZ(j)) = dimIF` H0 (Gk , ZZ` (j))/` ,
dimIF` Hi (Gk , ZZ/`ZZ(j)) = dimIF` Hi (Gk , ZZ` (j))/`
+ dimIF` ` Hi−1 (Gk , ZZ` (j))
for i
1. Since cd` Gk = 2, we obtain
≥
2
X
i
(−1) rank ZZ` Hi (Gk , ZZ` (j)) =
i=0
∞
X
(−1)i rank ZZ` Hi (Gk , ZZ` (j))
i=0
=
∞
X
(−1)i dimIF` H i (Gk , ZZ/`ZZ(−j))
i=0
=
2
X
(−1)i dimIF` H i (Gk , ZZ/`ZZ(−j)).
i=0
2
Now the corollary follows from (7.3.7).
(7.3.9) Corollary. With the notation as in (7.3.7), we have
dimIF` H 1 (Gk , ZZ/`ZZ) =


1 + δ + [k : Qp ] if ` = p ,
 1+δ
otherwise,
where δ = 1 or 0 according to whether the `-th roots of unity are contained in
k or not.
Proof: By duality, we have dimIF` H 2 (Gk , ZZ/`ZZ) = dimIF` H 0 (Gk , µ` ) = δ.
Since dimIF` H 0 (Gk , ZZ/`ZZ) = 1, the assertion follows from (7.3.7) with j = 0.
2
For a prime number ` =/ char(k) and i ∈ ZZ, i =/ 0, we introduce the numbers
n
o
w`i := max `n exp G(k(µ`n )|k) divides i .
Here exp(G) denotes the exponent of a finite group G. In particular, we have
w`1 = #µ`∞ (k) and w`i = w`−i .
Remark: If k(µ`∞ )|k is pro-cyclic (this can only fail if k is a dyadic number
field, i.e. a finite extension of Q2 , and ` = 2), then obviously
n
o
w`i = max `n [k(µ`n ) : k] | i .
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400
Chapter VII. Cohomology of Local Fields
(7.3.10) Proposition. Assume that k is a finite extension of degree d of Qp
and let ` be a prime number. Then
(i) H 0 (Gk , Q` /ZZ` (i)) ∼
=
(ii) H 1 (Gk , Q` /ZZ` (i)) ∼
=


Q` /ZZ`
for i = 0,
 ZZ/w i ZZ for i =
/ 0.
`


ZZ/w`1 ZZ ⊕ (Q` /ZZ` )d+1






(Q` /ZZ` )d+1




 ZZ/w 1−i ZZ ⊕ (Q /ZZ )d
`
for i = 0,
for i = 1,
for i =/ 0, 1,
for i = 0,
for i = 1,
for i =/ 0, 1,
`
`


ZZ/w`1 ZZ ⊕ (Q` /ZZ` )





Q` /ZZ`




 ZZ/w 1−i ZZ
`
(iii) H 2 (Gk , Q` /ZZ` (i)) ∼
=
` = p,
` = p,
` = p,
` =/ p,
` =/ p,
` =/ p.


H 2 (Gk , k̄ × )(`) ∼
= Q` /ZZ` for i = 1,
 0
for i =/ 1.
Remark: From the proposition above, the continuous cochain cohomology
groups with values in ZZp (i) can be easily calculated by the rule
j
Hcts
(Gk , ZZ` (i)) ∼
= H2−j (Gk , ZZ` (i − 1)) .
= H 2−j (Gk , Q` /ZZ` (1 − i))∨ ∼
Proof of (7.3.10):
If i =/ 0, then
i
H 0 (Gk , Q` /ZZ` (i)) = {ζ
∈
µ`∞ | ζ χcycl (σ) = ζ for all σ
= {ζ
∈
µ`∞ | σ i ζ = ζ for all σ
and therefore
n #H 0 (Gk , Q` /ZZ` (i)) = max `n σ i = 1 for all σ
∈
∈
∈
Gk }
G(k(µ`∞ )|k)},
o
G(k(µ`n )|k) = w`i .
This proves (i). The assertion for H 2 follows from the local duality theorem
(7.2.6):
(
Q` /ZZ` if i = 1,
2
0
∨ ∼
∼
H (Gk , Q` /ZZ` (i)) = Hcts (Gk , ZZ` (1 − i)) =
0
otherwise.
0
2
Furthermore, from (7.3.8) and the statements for H and H , we obtain

d + 1 for i = 0, 1 ` = p,



 d
for i =/ 0, 1, ` = p,
corank ZZ` H 1 (Gk , Q` /ZZ` (i)) =

1
for i = 0, 1 ` =/ p,



0
for i =/ 0, 1, ` =/ p.
It remains to calculate the cotorsion of the group H 1 (Gk , Q` /ZZ` (i)). Let
m ≥ 1. From the exact sequence 0 → ZZ/`m ZZ(i) → Q` /ZZ` (i) → Q` /ZZ` (i) → 0
follows the exact sequence
H 1 (Gk , Q` /ZZ` (i))/`m ,→ H 2 (Gk , ZZ/`m ZZ (i)) `m H 2 (Gk , Q` /ZZ` (i)) .
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§4. Galois Module Structure of the Multiplicative Group
401
Applying the projective limit, this yields for i =/ 1
lim H 1 (Gk , Q` /ZZ` (i))/`m ∼
= lim H 2 (Gk , ZZ/`m ZZ (i))
←−
m
←−
m
∼
H 0 (Gk , ZZ/`m ZZ (1 − i)))∗
= (lim
−→
m
∼
= ZZ/w`1−i ZZ ,
since H 2 (Gk , Q` /ZZ` (i)) = 0. If i = 1, then the group H 1 (Gk , Q` /ZZ` (1)) ∼
=
×
k ⊗ Q` /ZZ` is `-divisible. Therefore
∗
1
tor H (Gk , Q` /ZZ` (i))
∼
=
(
ZZ/w`1−i ZZ for i =/ 1,
0
for i = 1.
2
This finishes the proof of (7.3.10).
§4. Galois Module Structure of the
Multiplicative Group
In this section we combine some of the results of chapter V §6 with those of
the last sections in order to determine the structure of the p-adic completion
of the multiplicative group of a local field and of relation modules of certain
extensions of local fields.
(7.4.1) Theorem. Let k be a p-adic local field and let n = [k : Qp ]. Then the
absolute Galois group G = G(k̄|k) of k is generated by n + 2 elements. If
1 −→ N −→ Fn+2 −→ G −→ 1
is a presentation of G = G(k̄|k) by a free profinite group Fn+2 of rank n + 2,
then
N ab (p) ∼
= ZZp [[G ]]
as ZZp [[G ]]-modules.
If K|k is a Galois extension of p-adic local fields with Galois group G =
G(K|k), then let H = G(k̄|K). We want to determine the structure of the
ZZp [[G]]-module
m
X = H ab (p) ∼
= lim A(L) = lim L× /L×p ,
←−
L
←−
L,m
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402
Chapter VII. Cohomology of Local Fields
where L runs through all finite subextensions of K|k and the projective limit
m
is taken with respect to the norm maps. A(L) = lim L× /L×p is the p←− m
completion of the multiplicative group of the local field L, which is (via the
reciprocity map) isomorphic to G(k̄|L)ab (p). We denote the group of roots of
unity of p-power order in L by µp∞ (L).
(7.4.2) Theorem. Let k be a p-adic local field and let n = [k : Qp ].
(i) Let K|k be a Galois extension with G = G(K|k) and let
1 −→ Rn+2 −→ Fn+2 −→ G −→ 1
be a presentation of G. If X denotes the ZZp [[G]]-module G(k̄|K)ab (p),
then there exists an exact sequence
ab
0 −→ ZZp [[G]] −→ Rn+2
(p) −→ X −→ 0 .
/ K, then G is generated by n + 1 elements and there is an isomorIf µp ⊆
phism
ab
Rn+1
(p) ∼
=X.
(ii) Let σ1 , . . . , σn+2 be topological generators of G = G(k̄|k). Let ai ∈ ZZp
with σi (ζ) = ζ ai for all ζ ∈ µp∞ (k̄) and let σ̄i be the image of σi in
G = G(K|k), i = 1, . . . , n + 2. Then there exists an exact sequence
0 −→ ZZp [[G]] −→ ZZp [[G]]n+2 −→ Y −→ 0,
1
7−→ (σ̄i − ai )i ,
where Y = IG /IH IG , as in V §6. If µp∞ (K) = 1, then Y is a free
ZZp [[G]]-module of rank n + 1.
Before we prove the two theorems, which are taken from [97], we provide
two lemmas.
(7.4.3) Lemma. Let k be a local field of characteristic p and let K|k be a
finite tamely ramified Galois extension. Then UK1 /(UK1 )p is a cohomologically
trivial G(K|k)-module, and the norm map induces an isomorphism
UK1 /(UK1 )p
G(K|k)
∼ U 1 /(U 1 )p .
−→
k
k
Proof: Since the group of principal units UK1 is ZZp -torsion-free, we have the
exact sequence
p
0 −→ UK1 −→ UK1 −→ UK1 /(UK1 )p −→ 0.
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403
§4. Galois Module Structure of the Multiplicative Group
By (7.1.2)(ii), UK1 is a cohomologically trivial G(K|k)-module. It follows that
UK1 /(UK1 )p is also cohomologically trivial, and the commutative diagram
UK1 /(UK1®°¯ )p
NG(K|k)
G(K|k)
UK1 /(UK1 )p
G(K|k)
NK|k
Uk1 /(Uk1 )p
2
gives the desired result.
(7.4.4) Lemma. Let k be a local field of residue characteristic p and let K|k
be a finite Galois extension with Galois group G = G(K|k).
(i) If k is a p-adic field of degree n over Qp , then there are isomorphisms of
Qp [G]-modules
A(K) ⊗ Q ∼
= Qp [G]n ⊕ Qp ,
UK1 ⊗ Q ∼
= Qp [G]n .
(ii) If k is the field of Laurent series over a finite field and K|k is tamely
ramified, then there are isomorphisms of ZZp [G]-modules
A(K)
UK1
∼
= ZZp [G]IN ⊕ ZZp ,
∼
= ZZp [G]IN .
Proof: (i) Let UKm be the group of principal units of level m of K. For m
large enough, the p-adic logarithm induces a G-invariant isomorphism
∼ pm
log : UKm −→
⊆
OK ,
see [160], chap.II, (5.5). Tensoring by Q and noting that UKm has finite index
in UK1 , the existence of a normal basis for K|k gives us a G-isomorphism
U1 ⊗ Q ∼
= Qp [G]n .
K
Applying p-completion to the exact sequence
0 −→ UK −→ K × −→ ZZ −→ 0,
we obtain the exact sequence
0 −→ UK1 −→ A(K) −→ ZZp −→ 0.
Using the semi-simplicity of the category of finitely generated Qp [G]-modules
(2.6.12), we obtain
A(K) ⊗ Q ∼
= U 1 ⊗ Q ⊕ Qp ,
K
hence the result.
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404
Chapter VII. Cohomology of Local Fields
(ii) Let k = IFq ((t)), where q = pf0 , and let K|k be a finite tamely ramified
Galois extension. By the properties of tamely ramified extensions, one knows
that there
exists an unramified extension K 0 of K which is of the form K 0 =
√
e
k(ζ, t), where ζ is a primitive e-th root of unity and e = q f − 1, f ≥ 1. Using
(7.4.3), we may assume that K = K 0 . Then the Galois group G = G(K|k) has
the following structure:
G = hσ, τ | στ σ −1 = τ q , σ f = 1, τ e = 1i,
where
√
√
e
σ(ζ) = ζ q and σ(√
t) = e√t ,
τ (ζ) = ζ and τ ( e t) = ζ e t .
If H = hτ i, then G/H = G(k(ζ)|k) ∼
= G(λ|κ), where λ ∼
= IFqf is the residue
∼
field of K and κ = IFq is the residue field of k. The existence of a normal basis
σ̄ i (α),
i = 0, . . . , f − 1,
of λ gives us an isomorphism λ ∼
= κ[G/H] of IFq [G/H]-modules.
≤
≤
For 0 r e − 1, let λ(r) = εr λ ⊆ λ[H] be the 1-dimensional eigenspace
corresponding to the idempotent
εr =
X
1 e−1
ζ −rj τ j ,
e j=0
i.e. τ (β) = ζ r β for β ∈ λ(r). Obviously, λ(r) is an IFq [H]-module. However,
it is even an IFq [G]-module, as σεr = εr σ in IFq [G], i.e. σ(β) = β q for β ∈ λ(r).
Therefore the map
e−1
M
λ(r) −→ κ[G],
r=0
defined by σ i (α) 7→ σ i εr for σ i (α) ∈ λ(r), i = 0, . . . , f − 1, r = 0, . . . , e − 1,
is an isomorphism of IFq [G]-modules. In particular, the modules λ(r) are
IFq [G]-projective.
Let i1 < i2 < · · · < ie(p−1) be natural numbers such that 1 ≤ iν < ep and
p - iν , and let ie(p−1)+1 = ep. For each 0 ≤ r ≤ e − 1, there are exactly p − 1
numbers iν such that iν ≡ r mod e. Therefore we obtain an isomorphism of
IFq [G]-modules
e(p−1)
M
∼ κ[G]p−1 .
λ(iν ) −→
ν=1
For each j
≥
0, we have an IFq [G]-isomorphism
UKepj+iν (UK1 )p /UKepj+iν+1 (UK1 )p
∼ λ(i ),
−→
ν
√
e
[1 + α ( t)epj+iν ] 7→ α .
Since the modules λ(iν ) are IFq [G]-projective, it follows that
UKepj (UK1 )p /UKep(j+1) (UK1 )p ∼
= κ[G](p−1) ,
and so we get IFp [G]-isomorphisms
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§4. Galois Module Structure of the Multiplicative Group
(∗)
405
UK1 /UKepj (UK1 )p ∼
= IFp [G]f0 (p−1)j
for all j ≥ 1. Passing to the projective limit, we get UK1 /(UK1 )p ∼
= IFp [G]IN .
IN
Since ZZp [G] is the free compact ZZp [G]-module (on countably many generators), the diagram
φ
±²³´ IN
ZZp [G]
UK1
IFp [G]IN
UK1 /(UK1 )p
can be commutatively completed by a continuous homomorphism φ, which
is surjective by the topological Nakayama lemma (5.2.18)(i) (for ZZp ). Using
the fact that UK1 is ZZp -torsion-free, a second application of the topological
Nakayama lemma implies that ker(φ) = 0, hence φ is an isomorphism.
In order to show the second isomorphism, we consider the exact sequence
0 → UK1 → A(K) → ZZp −→ 0. Using (5.2.14) and (2.7.6), we have
1
Ext1ZZp [G] (ZZp , UK1 ) ∼
(G, ZZp [G]IN ) = 0,
= Hcts
and so an isomorphism A(K) ∼
= UK1 ⊕ ZZp , which is the desired result.
2
Proof of (7.4.1): First let K|k be a finite tamely ramified Galois extension.
Then G = G(K|k) is generated by two elements, σ and τ say, acting on µp∞ (K)
by ζ σ = ζ a and ζ τ = ζ b , a, b ∈ ZZp . We obtain an exact sequence
ϕ
ZZp [G]2 −→ ZZp [G] −→ µp∞ (K)∨ −→ 0,
where ker(ϕ) = (σ − a, τ − b), which induces the exact sequence
(∗)
0 −→ ZZp [G] −→ ZZp [G]2 −→ M0 −→ 0,
with M0 ' D(µp∞ (K)∨ ) since (µp∞ (K)∨ )+ = 0. Setting Y = IG /IH IG ,
where H = G(k̄|K), we get from (5.6.9) and (7.2.4) the homotopy equivalence
M0 ' Y,
and for X = G(k̄|K)ab (p) we have the exact sequence
0 −→ X −→ Y −→ IG −→ 0
by (5.6.5). By Maschke’s theorem, finitely generated Qp [G]-modules are
projective, so that
Y ⊗Q∼
= X ⊗ Q ⊕ IG ⊗ Q ,
and given (7.4.4)(i) and the exact sequence (∗), it follows that
Y ⊗Q∼
= Qp [G]n ⊕ Qp ⊕ IG ⊗ Q ∼
= Qp [G]n+1 ∼
= M0 ⊗ Q ⊕ Qp [G]n .
From (5.6.11), we get the isomorphism
(∗∗)
Y ∼
= M0 ⊕ ZZp [G]n .
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406
Chapter VII. Cohomology of Local Fields
If
1 −→ Rn+2 −→ Fn+2 −→ G −→ 1
is a presentation of the finite group G by a free profinite group Fn+2 , then
ab
(p) → ZZp [G]n+2 → IG → 0.
we obtain from (5.6.6) an exact sequence 0 → Rn+2
Since ZZp [G]n+2 is projective, we get, using the isomorphism (∗∗) and the exact
sequence (∗), an commutative exact diagram
0µ¶·¸¹º»¼½¾¿
ab
(p)
Rn+2
ZZp [G]n+2
IG
0
IG
0,
α
X
0
Y
where the kernel of α is isomorphic to ZZp [G]. Thus we obtain a G-invariant
ab
surjection β : Rn+2
(p) X, whose kernel is isomorphic to ZZp [G], and an
isomorphism
β∗
ab
2
(ZZ/#G ZZ)(p) ∼
(p)) −→
= H 2 (G, Rn+2
∼ H (G, X) .
Now the exact diagram
1ÉÊÀÁÂÃÄÅÆÇÈ
ab
(p)
Rn+2
Fn+2 /[Rn+2 , Rn+2 ]R(p)
G
1
G /[H , H ]H (p)
G
1,
β
1
X
where R(p) = ker(Rn+2 Rn+2 (p)) and H (p) = ker(H H (p)), can
be completed to a commutative diagram, since the corresponding 2-cocycles
of the group extensions are mapped by β∗ onto each other (after possibly
multiplying with a unit in ZZp ); use scd Fn+2 = scd G = 2 and (3.6.4)(iii) and
I §5 ex.4.
Let ktr be the maximal tamely ramified extension of k. Passing to the
projective limit over all finite Galois extensions K|k inside ktr , we obtain by
the usual compactness argument a surjection
Fn+2 G /[G(k̄|ktr ), G(k̄|ktr )] .
Thus the profinite group G /[G(k̄|ktr ), G(k̄|ktr )] is generated by n+2 elements.
Since G(k̄|ktr ) is a pro-p-group, the Frattini argument (3.9.1) implies that G
itself is generated by n + 2 elements.
In order to prove the second assertion, we first observe that N ab (p) is ZZp [[G ]]ab
projective by (5.6.7), so that NH
(p) is ZZp [G]-projective for all open normal
subgroups H of G , where G = G /H . From the exact sequence
(∗)
ab
ab
0 −→ NH
(p) −→ Rn+2
(p) −→ X −→ 0
for X = H ab (p) (recalling that scd G = 2), we obtain
Rab (p) ⊗ Q ∼
= X ⊗ Q ⊕ N ab (p) ⊗ Q ,
n+2
H
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§4. Galois Module Structure of the Multiplicative Group
407
and hence, using again (7.4.4)(i),
ab
Qp ⊕ Qp [G]n+1 ∼
(p) ⊗ Q
= Qp ⊕ Qp [G]n ⊕ NH
so that
ab
NH
(p) ⊗ Q ∼
= Qp [G] .
ab
It follows that NH
(p) ∼
= ZZp [G] using (5.6.15). We complete the proof of
(7.4.1) by passing to the limit over all finite quotients G of G ,
2
Proof of (7.4.2): Let G = G /H for some closed normal subgroup H of
G (not necessarily of finite index). Again we obtain the exact sequence (∗)
ab
used in the proof of (7.4.1) and, since NH
(p) ∼
= ZZp [[G]], we obtain the first
statement of (i).
/ K, then the ZZp [[G]]-module Y is free of rank n + 1, which can
If µp ⊆
be shown in the same manner as in the proof of (7.4.1). Thus we get an
ab
∼ X. As above, this implies that G can be generated
(p) −→
isomorphism Rn+1
by n + 1 elements.
Now we prove assertion (ii). Since ai
∈
ZZ×
p , we get the exact sequence
ZZp [[G ]]n+2 −→
ZZp [[G ]]
−→ ZZp −→ 0 ,
−1
ei
7−→ ai (σ̄i − 1)
where {ei | i = 1, . . . , n + 2} is a basis of ZZp [[G ]]n+2 , see (5.6.6). Tensoring by
ZZp (−1), using the isomorphism
∼
ZZp [[G ]]
ZZp [[G ]] ⊗ ZZp ZZp (−1) −→
g⊗1
7−→ χcycl (g) · g ,
and taking G(k̄|K)-coinvariants, we obtain the exact sequence
ZZp [[G]]n+2 −→ ZZp [[G]] −→ µp∞ (K)∨ −→ 0 .
ei
7−→ (σ̄i−1 − ai )
Since (µp∞ (K)∨ )U is finite for every open normal subgroup U of G, we have
(µp∞ (K)∨ )+ = 0. Thus the last sequence implies an exact sequence
0 −→ ZZp [[G]] −→ ZZp [[G]]n+2 −→ M0 −→ 0
1
7−→ (σ̄i − ai )i ,
with M0 ' D(µp∞ (K)∨ ). As in the proof of (7.4.1), we obtain isomorphisms
YU ∼
= (M0 )U ⊕ ZZp [G/U ]n
for any open normal subgroup U of G. By (5.6.11), this implies the existence
of an isomorphism Y ∼
2
= M0 ⊕ ZZp [[G]]n , showing the assertion.
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408
Chapter VII. Cohomology of Local Fields
For a local field k the absolute Galois group, the inertia group and the
ramification group is denoted by Gk , Tk and Vk , respectively. Furthermore,
Gk = Tk /Vk is the Galois group of the maximal tamely ramified extension ktr
of k. The next theorem is taken from [94].
(7.4.5) Theorem. Let k be a local field with residue characteristic p.
(i) If k is a p-adic field with n = [k : Qp ], then there is an exact sequence of
ZZp [[Gk ]]-modules
0 −→ ZZp [[Gk ]] −→ ZZp [[Gk ]]n+1 −→ Vkab −→ 0 .
(ii) If k is a local field of characteristic p, then there is an isomorphism of
ZZp [[Gk ]]-modules
Vkab ∼
= ZZp [[Gk ]]IN ,
i.e. Vkab is a free compact ZZp [[Gk ]]-module of countably infinite rank.
Proof: (i) By (7.4.2), we have an exact sequence of ZZp [[Gk ]]-modules
0 −→ ZZp [[Gk ]] −→ ZZp [[Gk ]]n+2 −→ Y −→ 0 ,
where Y = IGk /IGktr IGk , and with X = G(k̄|ktr )ab = Vkab we have the exact
sequence
0 −→ X −→ Y −→ IGk −→ 0 .
By (5.6.12), the augmentation ideal IGk is a free ZZp [[Gk ]]-module of rank 1,
hence
Y ∼
= X ⊕ ZZp [[Gk ]] .
This gives the desired result.
(ii) Let K|k be a finite tamely ramified Galois extension. Then, by (7.4.3),
the norm map
NG(K|k) : UK1 /(UK1 )p Uk1 /(Uk1 )p
is surjective, and furthermore we have NG(K|k) UKeK pj = Ukek pj ; see [212] chap.V.
Therefore
NG(K|k) : UK1 /UKeK pj (UK1 )p Uk1 /Ukek pj (Uk1 )p
is well-defined and surjective. In the proof of (7.4.4)(ii) we have shown the
assertion (∗): For j ≥ 1 there is an isomorphism
∼ IF [G(K|k)]f0 (p−1)j
UK1 /UKeK pj (UK1 )p −→
p
of IFp [G(K|k)]-modules. Passing to the projective limit over all finite tamely
ramified Galois extension K|k, we obtain
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§5. Explicit Determination of Local Galois Groups
409
UK1 /(UK1 )p
Vkab /p ∼
= lim
←−
K|k
∼
lim UK1 /UKeK pj (UK1 )p
= lim
←− ←−
K|k
j
K|k
j
∼
lim IFp [G(K|k)]f0 (p−1)j
= lim
←− ←−
∼
IFp [[Gk ]]f0 (p−1)j ∼
= lim
= IFp [[Gk ]]IN
←−
j
Since Vkab is ZZp -torsion-free, we may apply the topological Nakayama lemma
in a similar way as in the proof of (7.4.4)(ii), to obtain the desired isomorphism
Vkab ∼
2
= ZZp [[Gk ]]IN .
§5. Explicit Determination of Local Galois Groups
The absolute Galois group Gk of a local field is prosolvable. It is the
projective limit of the groups G(K|k) of the finite Galois extensions K|k which
contain the inertia group T (K|k) and the ramification group V (K|k) as normal
subgroups. V (K|k) is a p-group, and T (K|k)/V (K|k) and G(K|k)/T (K|k)
are cyclic, hence G(K|k) is a solvable group. Furthermore, the inertia group
Tk = lim T (K|k) and the ramification group Vk = lim V (K|k) are normal
←− K
←− K
subgroups of Gk . From (7.1.8)(i) and (7.4.5) we obtain:
(7.5.1) Proposition. The ramification group Vk of the absolute Galois group
Gk of a local field k with residue characteristic p is a free pro-p-group of
countably infinite rank.
The group Gk has several interesting quotients. The simplest is the quotient
by the inertia group Tk . This is the Galois group Γ = G(k̃|k) of the maximal
unramified extension k̃|k. It is canonically isomorphic to the absolute Galois
group Gκ of the finite residue field κ, hence to ẐZ, and has the Frobenius
automorphism σk as a canonical topological generator.
We next consider the quotient Gk of Gk by the ramification group Vk . This
is the Galois group
Gk = G(ktr |k)
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410
Chapter VII. Cohomology of Local Fields
of the maximal tamely ramified extension ktr |k. Gk is a group extension of
Gk̃ = G(ktr |k̃) by Γ , i.e. we have an exact sequence
1 −→ Gk̃ −→ Gk −→ Γ −→ 1,
which splits after choosing of a pre-image σ of σk ∈ Γ . In other words, Gk is
the semi-direct product of Gk̃ , which is an abelian group, and Γ . We obtain a
complete description of Gk by determining explicitly Gk̃ as a Γ -module. This
is easily achieved:
0
Let p be the residue characteristic of k and let µ(p ) be the group of roots
0
of unity in k̄ of order prime to p. µ(p ) is a Γ -module, which is canonically
isomorphic to κ̄× , via the reduction map. The value group
[of the normalized
1
ZZ, hence
valuation of k is ZZ, and that of its extension to ktr is ∆ =
n
p-n
∆/ZZ =
M
(p0 )
Q` /ZZ` =: (Q/ZZ)
.
/p
`=
By [160], chap.II,(9.15), we have canonically
Gk̃ ∼
= Hom(∆/ZZ, κ̄× ). ∗)
The Frobenius automorphism σk ∈ Γ acts on both groups by x 7→ xq (q = #κ),
since this is its action on κ̄× . Thus Gk̃ is canonically isomorphic to the additive
Y
(p0 )
(p0 )
Γ -module ẐZ (1), which is isomorphic to ẐZ =
ZZ` as an abelian group,
/p
`=
and on which σk acts as multiplication by q. We have thus obtained the
(7.5.2) Proposition. If Gk denotes the Galois group of the maximal tamely
ramified extension ktr of a local field k, then we have a split group extension
1 −→ ẐZ
(p0 )
(1) −→ Gk −→ Γ −→ 1.
We may reformulate this result as follows (cf. [87]).
(7.5.3) Theorem (IWASAWA). The Galois group Gk of the maximal tamely
ramified extension of a local field k is isomorphic to the profinite group G
generated by two elements σ, τ with the only relation
στ σ −1 = τ q .
∗) This isomorphism comes from Kummer theory via the following observation: let K|k̃
be a finite extension of degree e with (e, p) = 1 and let Π, π be uniformizers of K, k̃. Then
Πe = π · u, where u ∈ K is a unit. Note that u1/e ∈ K and (Π/u1/e )e = π. Hence
K = k̃(π 1/e ).
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411
§5. Explicit Determination of Local Galois Groups
Proof: Let F be the free profinite group generated by two elements σ and
τ . Let N be the normal closed subgroup of F generated by στ σ −1 τ −q , and
set G = F/N . G is the profinite group generated by the images σ̄, τ̄ of σ, τ
with the defining relation σ̄ τ̄ σ̄ −1 = τ̄ q . The homomorphism F → Γ , given by
σ 7→ σk , τ 7→ 1, induces a surjection G → Γ . The kernel is the closed normal
subgroup Z topologically generated by τ̄ . In fact, Z is normal in G because
σ̄ τ̄ σ̄ −1 = τ̄ q , and G/Z is generated by the image of σ̄, i.e. is procyclic with a
surjection G/Z → Γ which must be an isomorphism.
Writing Z additively, the action of σk on Z becomes multiplication by q.
Since it is an automorphism of Z, the p-Sylow subgroup of Z must be trivial.
Y
(p0 )
In other words, Z is a quotient of ẐZ =
ZZ` .
/p
`=
Now consider the Galois group Gk = G(ktr |k). Let τ 0 be a topological
(p0 )
generator of Gk̃ = G(ktr |k̃) ∼
= ẐZ (1) and σ 0 a pre-image of σk ∈ Γ . Then
σ 0 τ 0 σ 0−1 = τ 0q , and the surjective homomorphism F → Gk , given by σ 7→ σ 0 ,
τ 7→ τ 0 , factors through G. We obtain an exact commutative diagram
1ÐËÌÍÎÏÑÒÓÔÕ
G
Z
α
1
ẐZ
(p0 )
Γ
1
Γ
1,
β
(1)
Gk
(p0 )
where α is surjective. But α is necessarily an isomorphism, since ẐZ (1) is
Y
(p0 )
(p0 )
the procyclic group ẐZ = `=/p ZZ` , and Z is a quotient of ẐZ . Therefore β
is an isomorphism, and the theorem is proved.
2
We want to take a closer look at the group Gk . For this we introduce the
following notation.
If G is a profinite group and g an element of G, then we define the α-power g α
of g for α ∈ ẐZ as follows: Consider the homomorphism
ϕ : ẐZ hgi ⊆ G,
which is given by 1 7→ g. Then we define g α = ϕ(α). Observe that for α ∈ ZZ
we obtain the usual powers of g.
Y
ZZr given by
For a prime number ` consider the projectors π` , ∆` ∈ ẐZ ∼
=
r
(
(π` )r =
1 for r = `
0 for r =/ ` ,
(
(∆` )r =
0 for r = `
1 for r =/ ` .
Then
π` ẐZ = ZZ`
and ∆` ẐZ = ẐZ
(`0 )
,
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Chapter VII. Cohomology of Local Fields
(`0 )
where ZZ` and ẐZ are embedded as direct factors into ẐZ. Further, ∆` + π` = 1,
∆` π` = 0 and for every α ∈ ẐZ:
π`α = π`
and ∆α` = ∆` .
If G is a profinite abelian group, which (written additively) is a ZZr -module,
then raising an element to the π` -power is the zero map if r =/ ` and is the
identity if r = `.
For a prime number ` =/ p and a prime number r|(` − 1) we define an (` − 1)-th
root of unity of r-power order
e(`, r) ∈ µ`−1 (r) ⊆ ZZ`
by
p≡
Y
e(`, r) mod `
r|`−1
(e(`, r) depends on p although we do not indicate this in the notation). We
make the convention that e(`, r) = 1 if r - (` − 1) and we put
e(`) =
Y
e(`, r) ∈ ZZ` ,
r|`−1
i.e. π` p = e(`) · u, where u ∈ ZZ` is a principal unit.
(7.5.4) Lemma. (i) Let r be a prime number. Then for every n ∈ IN, there
exists a prime number ` =/ p such that the r-power root of unity e(`, r) is at
least of order rn .
(ii) For q = pf , 1 ≤ f ∈ IN, the homomorphism
0
ψ : ẐZ −→ Aut(ZZ(p ) )
α 7−→ (x 7→ q α x)
is injective.
√
Proof: (i) Consider the fields k0 = Q(µrn ) and k = k0 ( r p). The set of prime
numbers ` which are completely decomposed in k is
{` | ` ≡ 1 mod rn , p
`−1
r
≡ 1 mod `} ,
since a prime number ` splits completely in k if and only if ` - p r, ` ≡ 1
√
mod rn and r p ∈ ZZ` . But
√
√
r p ∈ Z
Z` ⇔ r e(`) ∈ ZZ`
`−1
⇔ e(`) r = 1
`−1
⇔ p r ≡ 1 mod ` .
Therefore the density of the set
S = {` | ` ≡ 1 mod rn , p
`−1
r
6≡ 1 mod `}
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§5. Explicit Determination of Local Galois Groups
413
is [k0 : Q]−1 − [k : Q]−1 = 1/rn and, in particular, S is not empty. Thus for
` ∈ S, the root of unity e(`, r) has order rm with m ≥ n. This proves (i).
In order to prove (ii), assume that α is an element in the kernel of ψ, i.e. for
all ` =/ p the equality q α π` = π` holds. Then, in particular,
e(`, r)f α = (e(`)f )α = 1 ∈ ZZ`
Y
r|`−1
and hence also e(`, r)f α = 1 for all ` =/ p, r|(` − 1).
For an arbitrary prime number r and an arbitrary n ∈ IN, we can use (i) in order
to find a prime number ` =/ p such that
e(`, r)r
n−1
=/ 1.
Hence f α ∈ rn ẐZ and since r and n were arbitrary, we conclude that f α = 0.
Finally, since 1 ≤ f ∈ IN and since ẐZ is torsion-free (as an abelian group), we
obtain α = 0.
An alternative possibility to see assertion (ii) is the following: the homomorphism ψ (after replacing q by q −1 ) describes the action of G(κ̄|κ) on the
dual Hom(κ̄× , Q/ZZ) of the multiplicative group κ̄× . Obviously, this action is
faithful.
2
For a prime number r (possibly equal to p), we set
σ r = σ πr ,
τr = τ πr
where σ and τ are generators of the group Gk = hσ, τ | τ σ = τ q i, q = pf .
(7.5.5) Proposition. With the notation as above, we have for arbitrary prime
numbers r, and ` =/ p





−f
τ`qe(`) −1 if r = ` ,
f
[σr , τ` ] =  τ`e(`,r) −1 if r|` − 1 ,


 1
if r =/ ` , r - ` − 1 .
Proof: For r =/ ` we have
π` q πr = π` pπr f = (π` p)πr f = e(`, r)f ,
since all other components in the decomposition of π` p are annihilated (i.e.
sent to 1) when raised to the πr -th power. Similarly,
π` q π` = π` pπ` f = (π` p)π` f = π` qe(`)−f .
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Chapter VII. Cohomology of Local Fields
It follows that
[σr , τ` ] = τ`q
πr −1
= τ`π` (q
πr −1)





−f
τ`qe(`) −1 if r = ` ,
f
= τ`e(`,r) −1 if r|` − 1 ,



 1
if r =/ ` , r - ` − 1 .
2
This proves the proposition.
(7.5.6) Corollary. For every prime number r there exists a prime number
` =/ p such that σr and τ` do not commute. In particular, σp does not commute
with τ .
Proof: By (7.5.5), we have the equality
[σr , τ` ] = τ`e(`,r)
f −1
for every prime number r with r|` − 1, and from (7.5.4)(i) it follows that there
are prime numbers ` such that the order of e(`, r) is bigger than the r-part of
the fixed number f .
2
(7.5.7) Corollary.
(i) The ramification group Vk of the absolute Galois group Gk of a local
field k is the maximal normal pro-p subgroup of Gk .
(ii) The subgroup Tk /Vk = hτ i of Gk = Gk /Vk is the unique maximal abelian
normal subgroup of Gk (which exists in Gk !).
Proof: (i) Since Vk is a normal pro-p subgroup of Gk and hVk , σp i is a pro-p
Sylow subgroup of Gk , the result follows from the previous corollary.
(ii) Every abelian normal subgroup of Gk is contained in hτ i by (7.5.5) and
(7.5.4)(i). Thus the result follows.
2
We next study the maximal pro-p-factor group Gk (p) of Gk , where p is the
residue characteristic of the local field k. This is the Galois group G(k(p)|k)
of the maximal p-extension k(p)|k, i.e. of the composite of all finite Galois
extensions of p-power degree.
First, we have to compare the cohomology for a Gk (p)-module A with
respect to Gk (p) and Gk . This will be done in the next proposition in a slightly
more general form. Recall the notion G(c) for the maximal pro-c-quotient of a
profinite group G with respect to a full class c of finite groups.
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§5. Explicit Determination of Local Galois Groups
415
(7.5.8) Proposition. Let c be a full class of finite groups, ` a prime number
such that ZZ/`ZZ ∈ c and A a Gk (c)-module. Then for every i ≥ 0 the inflation
map
H i (Gk (c), A)(`) −→ H i (Gk , A)(`)
is an isomorphism.
Proof: The degree of k(c)|k is infinitely divisible by `, since the maximal
unramified `-extension of k is contained in k(c) and has Galois group ZZ` .
Using (7.1.8)(i) and (6.1.3), we obtain cd` (k(c)) ≤ 1. Let H = G(k̄|k(c)). Then
H i (H, A)(`) = 0 for i ≥ 1. Indeed, since cohomology commutes with direct
limits and since A is a trivial H-module, one reduces to the cases A = ZZ, ZZ/`ZZ
and, using the exact sequence 0 → ZZ → Q → Q/ZZ → 0, we finally reduce to
the case A = ZZ/`ZZ. By cd` H ≤ 1, the assertion is obvious for i ≥ 2.
Since ZZ/`ZZ ∈ c, the group H has no nontrivial homomorphism to an `-group,
showing the case i = 1.
Now the spectral sequence
H i (Gk (c), H j (H, A))(`) ⇒ H i+j (Gk , A)(`)
gives isomorphisms
H i (Gk (c), A)(`) ∼
= H i (Gk , A)(`) for all i ≥ 0 .
2
We can now explicitly determine the structure of the pro-`-group Gk (`). A
consequence of (7.5.3) is the
(7.5.9) Proposition. Let k be a local field with residue characteristic p and let
` be a prime number different to p.
/ k, then Gk (`) ∼
If µ` ⊆
= ZZ` ; if µ` ⊆ k, then Gk (`) is a Poincaré group of
dimension 2 (i.e. a Demuškin group) of rank 2 with dualizing module µ(`), the
group of all `-power roots of unity in k(`). Furthermore,
Gk (`) ab ∼
= ZZ/`s ZZ ⊕ ZZ` ,
where `s = #µ(k)(`).
From (6.1.7), we obtain the
(7.5.10) Theorem. Let k be a local field with char(k) = p. Then Gk (p) is a
free pro-p-group of countably infinite rank.
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Chapter VII. Cohomology of Local Fields
(7.5.11) Theorem. Let k be a p-adic local field.
(i) If µp is not contained in k, then Gk (p) is a free pro-p-group of rank N + 1
with N = [k : Qp ].
(ii) If µp ⊆ k, then Gk (p) is a Poincaré group of dimension 2 (i.e. a Demuškin
group) of rank N + 2 . The dualizing module of Gk (p) is the group µ(p)
of all p-power roots of unity in k(p).
Proof: By the (7.5.8) and by the duality theorem (7.2.6), we obtain
(1) H 1 (Gk (p), ZZ/pZZ) = H 1 (Gk , ZZ/pZZ) ∼
= H 1 (Gk , µp )∗ ∼
= (k × /k ×p )∗ ,
(2) H 2 (Gk (p), ZZ/pZZ) = H 2 (Gk , ZZ/pZZ) ∼
= H 0 (Gk , µp )∗ .
From [160], chap.II, (5.7)(i), we get k × /k ×p ∼
= (ZZ/pZZ)r with r = N + 1 if
/ k, or N + 2 if µp ⊆ k. Therefore, by (3.9.1), the pro-p-group Gk (p) has
µp ⊆
rank
rk(Gk (p)) = N + 1 or N + 2,
/ k or µp ⊆ k.
according to whether µp ⊆
⊆
If µp / k, then from (2) it follows that H 2 (Gk (p), ZZ/pZZ) = 0, and Gk (p) is
a free pro-p-group by (3.9.5). This proves (i).
Assume µp ⊆ k. Then (2) implies that H 2 (Gk (p), ZZ/pZZ) ∼
= ZZ/pZZ and
i
H (Gk (p), ZZ/pZZ) = 0 for i > 2 by (7.5.8) and (7.1.8), so that cdp Gk (p) = 2.
By the duality theorem (7.2.6), the cup-product yields a non-degenerate pairing
H 1 (Gk (p), ZZ/pZZ) × H 1 (Gk (p), ZZ/pZZ) → H 2 (Gk (p), ZZ/pZZ) ∼
= ZZ/pZZ.
Since µp ⊆ k, the p-part µ(p) of the dualizing module of µ of Gk (see (7.2.4))
is a Gk (p)-module. Therefore (7.5.8) shows that µ(p) is the dualizing module
of Gk (p) for the category of p-torsion Gk -modules. From (3.7.2) it follows
now that Gk (p) is a Poincaré group.
2
When µp ⊆ k we get an explicit description of Gk (p) by applying the results
of (3.9.11) and (3.9.19).
(7.5.12) Theorem (DEMUŠKIN). Let k be a p-adic local field of degree N =
[k : Qp ] and let ps be the largest p-power such that µps ⊆ k. If ps > 2, then
Gk (p) is the pro-p-group defined by N + 2 generators x1 , . . . , xN +2 subject to
the one relation
s
xp1 (x1 , x2 )(x3 , x4 ) · · · (xN +1 , xN +2 ) = 1.
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§5. Explicit Determination of Local Galois Groups
Proof: By class field theory the abelianized group Gk (p)ab is isomorphic
to the pro-p-completion k̂ × of the multiplicative group k × which, by [160],
× ∼
chap.II, (5.7), is isomorphic to ZZ ⊕ ZZ/(pf − 1)ZZ ⊕ ZZ/ps ZZ ⊕ ZZN
p , i.e. k̂ =
+1
. Therefore ps is the number which we have denoted by q in
ZZ/ps ZZ ⊕ ZZN
p
(3.9.11). Moreover, we have
H 1 (Gk (p), ZZ/pZZ) ∼
= Hom(Gk (p), ZZ/pZZ) ∼
= (ZZ/pZZ)N +2 ,
2
so that rk(Gk (p)) = N + 2. The theorem follows now from (3.9.11).
The structure of Gk (2) when ps = 2 has been determined by J.-P. SERRE if
N is odd and in general by J. LABUTE (cf. (3.9.19) and [209], [117]). Gk (2)
is again generated by N + 2 generators x1 , . . . , xN +2 with one defining relation
ρ, but the shape of this relation depends on further conditions:
If N is odd, then
ρ = x21 x42 (x2 , x3 )(x4 , x5 ) · · · (xN +1 , xN +2 ).
In particular, if k = Q2 , then Gk (2) is generated by three elements with the
defining relation x2 y 4 (y, z) = 1.
If N is even, then we have to take the 2-part of the cyclotomic character
χ : Gk → ZZ×
2 into account, which is obtained from the action of Gk on the
∞
group µ2 of all roots of unity of 2-power order. We have End(µ2∞ ) = ZZ2
and the action of Gk gives the homomorphism χ : Gk → Aut(µ2∞ ) = ZZ×
2 . The
structure of Gk (2) depends now on the image of χ:
f
If im(χ) is the closed subgroup of ZZ×
2 generated by −1 + 2 (f
≥
2), then
f
ρ = x2+2
(x1 , x2 )(x3 , x4 ) · · · (xN +1 , xN +2 ).
1
If im(χ) is generated by −1 and 1 + 2f (f
≥
2), then
f
ρ = x22 (x1 , x2 )x23 (x3 , x4 ) · · · (xN +1 , xN +2 ).
The cohomological methods leading to these results were extended to give
an explicit description of the entire absolute Galois group Gk of a local field. In
the case that k is the field of Laurent series over a finite field, this was done by
H. KOCH [108]. If k is a p-adic local field, the structure of Gk was determined
by U. JANNSEN and K. WINGBERG [99] for p =/ 2; and by V. DIEKERT [39] for
p = 2 under the condition that k(µ4 )|k is unramified. The structure of GQ2 is
not known.
The result for local fields of positive characteristic is the following.
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Chapter VII. Cohomology of Local Fields
(7.5.13) Theorem (KOCH). Let k be a local field of characteristic p with residue
field κ of cardinality q. Then Gk is isomorphic to the semi-direct product
Gk ∼
=
∗F
Gk
ω
o Gk ,
where Gk = G(ktr |k) is the Galois group of the maximal tamely ramified
extension of k, and Gk Fω is the free pro-p-Gk operator group of countably
infinite rank (cf. (4.3.8)).
In other words, the group Gk has generators σ, τ and xi , i ∈ IN, subject to
the following defining conditions resp. relations:
∗
A) The closed normal subgroup topologically generated by xi , i
pro-p-group.∗)
∈
IN, is a
B) The elements σ, τ satisfy the relation στ σ −1 = τ q .
Proof: By (7.5.1) and (7.4.5)(ii), the ramification group Vk is a free pro-pgroup, and we have an isomorphism of compact ZZp [[Gk ]]-modules
Vkab ∼
= ZZp [[Gk ]]IN ,
i.e. Vkab is a free ZZp [[Gk ]]-module of countably infinite rank. Since the order of
G(ktr |k̃) is prime to p, we have cdp Gk = cdp G(k̃|k) = 1. Therefore the group
Gk is p-projective by (3.5.6), and the exact sequence
1 −→ Vk −→ Gk −→ Gk −→ 1
splits. Fixing a splitting Gk ∼
= Vk o Gk , the ramification group Vk becomes
a pro-p-Gk operator group. Therefore we find a surjective homomorphism of
pro-p-Gk operator groups
ψ:
∗F
Gk
ω
Vk ,
∗
where Gk Fω is the free pro-p-Gk operator group of countably infinite rank and
ψ is defined by mapping the generators of Fω to lifts of the ZZp [[Gk ]]-generators
of Vkab . Since Vk is a free pro-p-group and the map
ψ̄ : (
∗F
Gk
ω
∼ V ab
)ab = ZZp [[Gk ]]IN −→
k
induced by ψ is an isomorphism, ψ is bijective by (1.6.15). Therefore we
obtain an isomorphism
(ψ, id) :
∗F
Gk
ω
∼ V o G ∼ G .
o Gk −→
k
k =
k
∗) This topological condition could be replaced by an infinite set of algebraic relations.
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419
§5. Explicit Determination of Local Galois Groups
Now let k be a p-adic field. We describe the structure of Gk in the case
p =/ 2. It depends on the degree N = [k : Qp ], the cardinality q = #κ of the
residue field κ, the order ps of the group µps of all p-power roots of unity in the
maximal tamely ramified extension ktr |k and on two further numbers g, h ∈ ZZp
which are defined as follows. By (7.5.3), the Galois group Gk = G(ktr |k) is
generated by two elements σ, τ with defining relation στ σ −1 = τ q . The actions
of σ and τ on µps are given by two numbers g, h ∈ ZZp such that
ζ σ = ζ g , ζ τ = ζ h for ζ
∈
µtr .
In the following we denote the commutator xyx−1 y −1 by [x, y] in contrast
to the commutator (x, y) = x−1 y −1 xy used in III §8. If one had taken there
the commutator [x, y] instead of (x, y), the shape of the Demuškin relation
would not change because it is the same modulo F 3 (the third filtration step of
the descending p-central series), and then the iteration process works as well.
Which commutator to use is just a matter of personal taste.
(7.5.14) Theorem (JANNSEN-WINGBERG). The group Gk is isomorphic to the
profinite group generated by N + 3 generators σ, τ, x0 , . . . , xN , subject to the
following defining conditions resp. relations.
A) The closed normal subgroup, topologically generated by x0 , . . . , xN is a
pro-p-group.
B) The elements σ, τ satisfy the “tame” relation
στ σ −1 = τ q .
C) In addition, the generators satisfy one further relation:
(i) for even N
s
xσ0 = hx0 , τ ig xp1 [x1 , x2 ][x3 , x4 ] · · · [xN −1 , xN ],
(ii) for odd N
s
xσ0 = hx0 , τ ig xp1 [x1 , y1 ][x2 , x3 ][x4 , x5 ] · · · [xN −1 , xN ],
where
p−1
hx0 , τ i = (xh0
p−2
τ xh0
π
τ · · · xh0 τ ) p−1
(π = πp being the element of ẐZ with π ẐZ = ZZp ), and where y1 is a certain
element in the subgroup generated by x1 , σ, τ , described below.
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Chapter VII. Cohomology of Local Fields
The definition of the element y1 is a little bit subtle. Let α : Gk →(ZZ/ps ZZ)×
be the character describing the action of Gk = G(ktr |k) on µps , and let β :
⊆ Gk
Gk → ZZ×
p be a lift of α (not necessarily homomorphic). For ρ ∈ hσ, τ i
∈
and x Gk set
π
p−2
{x, ρ} := (xβ(1) ρ2 xβ(ρ) ρ2 · · · xβ(ρ ) ρ2 ) p−1 .
Writing τ2 = τ π2 and σ2 = σ π2 , where π2 is the element of ẐZ with π2 ẐZ = ZZ2 ,
y1 is given by
p+1
τ p+1
b
a
2
y1 = x12 {x1 , τ2p+1 }σ2 τ2 {{x1 , τ2p+1 }, σ2 τ2a }σ2 τ2 +τ2 .
Here a, b ∈ ZZ are chosen in such a way that
2
b
× 2
/ (IFp ) .
−α(στ a ) mod p ∈ (IF×
p ) and − α(στ ) mod p ∈
For the proof we refer to [99] and [39]. It is based on a theory of H. KOCH
[111] which axiomizes the fact that for every finite, tamely ramified extension
K|k the group GK (p) is a Demuškin group. We would like to mention the
following special cases.
For p > 2, the group GQp has four generators σ, τ, x0 , x1 satisfying the
relations
τσ = τp ,
τ p+1
p−1
2
xσ0 = hx0 , τ i xp1 [x1 , x12 {x1 , τ2p+1 }σ2 τ2
p−1
p+1
p+1
2 +τ 2
2
{{x1 , τ2p+1 }, σ2 τ2 2 }σ2 τ2
],
and for the group GQp (ζp ) there are generators σ, τ, x0 , . . . , xp−1 satisfying
τσ = τp ,
xσ0 = (x0 τ )π xp1 [x1 , x2 ] · · · [xp−2 , xp−1 ] .
Remarks: 1. Although we know by (7.4.1) that Gk can be generated by N + 2
elements, it is more convenient to use N +3 generators in order to obtain “nice”
relations.
In the case that µp ⊆ k, there is also a satisfactory description with N + 2
generators, cf. [99], §1.4(d).
2. Generators and relations for Gk were also assigned by A. V. JAKOVLEV in
[92]. However, some mistakes required a comprehensive correction. This has
been sketched only for the case of even N , and produced three relations, one
of them being a somewhat complicated limit (cf. [93]).
We finish this section by showing that the absolute Galois group Gk of a
p-adic local field k, p =/ 2, possesses nontrivial outer automorphisms. This is
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§5. Explicit Determination of Local Galois Groups
421
easy to see if N = [k : Qp ] > 1. Let σ, τ, x0 , x1 , . . . , xN be the generators of
Gk described in (7.5.14), satisfying the tame relation στ σ −1 = τ q and the wild
relation
s
xσ0 = hx0 , τ ig xp1 [x1 , −] · · · [xN −1 , xN ] .
If we define ψ : Gk −→ Gk by
ψ(y) = y for y = σ, τ, x0 , . . . , xN −1 and ψ(xN ) = xN · xN −1 ,
then ψ is an automorphism of Gk . Indeed, since
[xN −1 , xN ] = [xN −1 , xN xN −1 ] ,
the generators σ, τ, x0 , . . . , xN −1 , xN · xN −1 satisfy both relations if N > 1.
Now suppose that ψ is an inner automorphism, i.e. there is an element ρ ∈ Gk
such that ψ(z) = z ρ for all z ∈ Gk . Recall that Vki denotes the i-th term
of the p-central series of the ramification group Vk with Vk1 = Vk . Since
xρN −1 = ψ(xN −1 ) = xN −1 and since xN −1 Vk2 generates a free IFp [[Gk ]]-module in
Vk /Vk2 , by [99], §2, we obtain that ρ ∈ Vk . It follows that xN Vk2 = xN · xN −1 Vk2
which is a contradiction. Thus, if N = [k : Qp ] > 1, we have constructed a
nontrivial outer automorphism of Gk .
The case k = Qp is more difficult. We use the following result from [250].
(7.5.15) Theorem. Let
∼ G /V 3
ψ3 : Gk /Vk3 −→
k
k
be an automorphism of Gk /Vk3 which induces the identity on the factor group
Gk = Gk /Vk . Then there exists an automorphism ψ of Gk which coincides
with ψ3 modulo Vk2 .
Remark: This result was proven in [250], Satz 2, but was stated there in an
incorrect manner. The result above is exactly what was needed for all results
in the two papers [99] and [250], except for the statement in [99], §5.1. There
an automorphism of GQp was defined and claimed to be a nontrivial outer
automorphism. But the argument used the incorrect formulation of [250],
Satz 2 and therefore the constructed automorphism might be inner.
In order to proceed, we need some more notation. The homomorphism
α : Gk →(ZZ/ps ZZ)× induces an involution ∗ on IFp [[Gk ]] which is defined by
ρ∗ = α(ρ)ρ−1 for ρ ∈ Gk . We define the element E of IFp [[Gk ]] by
X
1 e−1
E = lim
τ 2i α(τ )−i ,
K e
i=0
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422
Chapter VII. Cohomology of Local Fields
where K runs through all finite tamely ramified Galois extensions of k and e
denotes the ramification index of K|k. Then E ∗ = E and E is an idempotent
which is central in IFp [[Gk ]], because
σE = lim
K
X
1 e−1
τ 2iq α(τ )−i σ = E q σ = Eσ.
e i=0
From now on, let k = Qp . Then q = p, α(σ) = 1 and α(τ ) is a primitive
(p − 1)-th root of unity. For
ε = 1 − 2E
we have ε∗ = ε and ε2 = 1. In particular, ε is a central unit in IFp [[GQp ]]. We
write G, G and V for GQp , GQp and VQp , respectively.
Since cdp G = 1, the group G is p-projective by (3.5.6), and the exact
sequence
1 −→ V −→ G −→ G −→ 1
splits. Fixing a splitting G ∼
= V o G, the ramification group V becomes
a pro-p-G operator group. Therefore we find a surjective homomorphism of
pro-p-G operator groups
f:
∗F
G
2
V,
∗
where G F2 is the free pro-p-G operator group of rank 2 with basis {x0 , x1 }
and f is defined by mapping the generators x0 and x1 to lifts of the ZZp [[G]]generators of V ab . We obtain a surjection
F := F2 o G V o G ∼
= G.
∗
G
Let ψ : F → F be defined by
ψ(y) = y for y = σ, τ, x0
and ψ(x1 ) = xε1
(where the “sum” ε is chosen in some ordering). Then ψ is an automorphism
of F since it is an automorphism modulo F 2 (ε is a unit in IFp [[G]]).
We will show that for every finite tamely ramified Galois extension K|k the
automorphism ψ induces an automorphism
∼ G (p)/G (p)3 .
ψK : GK (p)/GK (p)3 −→
K
K
In order to prove this, let κK and λK be elements of IFp [[G]] defined as
κK =
fX
−1
σ i α(σ)−i ,
i=0
λK
X
1 e−1
=
τ i α(τ )−i ,
e i=0
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423
§5. Explicit Determination of Local Galois Groups
where e and f denote the ramification index and the residue degree of K
respectively. Let FK be the pre-image of GK in F. By [99], §2 (observe that
the index of the p-central series differs there from our notation by −1), the
relation rK in GK (p) satisfies
p
κK λK
rK ≡ (x−σ
mod FK (p)3 .
0 hx0 , τ ix1 [x1 , y1 ])
Let δ
∈
IFp [[G]] be such that y1 = xδ1 mod F 2 . Then we get
κK λK pεκK λK
κK λK
ψ(rK ) ≡ (x−σ
x1
[xε1 , xεδ
0 hx0 , τ i)
1 ]
∗
κK λK pκK λK
]κK λK
≡ (x−σ
x1
[x1 , xεδε
0 hx0 , τ i)
1
κK λK pκK λK
≡ (x−σ
x1
[x1 , xδ1 ]κK λK
0 hx0 , τ i)
≡ rK mod FK (p)3 ,
since ελK = λK and ε is central in IFp [[G]], so that εδε∗ = δεε∗ = δε2 = δ. This
gives us the automorphism ψK .
Now, in the limit over all K, ψ induces an automorphism of V /V 3 which is
G-invariant because ψ|G = id. Thus we obtain an automorphism of G/V 3 ∼
=
3
V /V o G, which is also denoted by ψ. By (7.5.15), it extends to an automorphism ϕ of G which coincides with ψ modulo V 2 .
Suppose that ϕ is an inner automorphism, i.e. there exists an element ρ ∈ G
such that ϕ(z) = z ρ for all z ∈ G. Then xρ1 ≡ ϕ(x1 ) ≡ ψ(x1 ) ≡ xε1 mod V 2 .
Since x1 generates a free IFp [[G]]-module in V /V 2 , by [99], §2, we obtain ε = ρ
mod V ∈ G, which is a contradiction. Thus we have shown that GQp possesses
nontrivial outer automorphisms.
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Chapter VIII
Cohomology of Global Fields
§1. Cohomology of the Idèle Class Group
Having established the cohomology theory for local fields, we now begin its
development for global fields, i.e. algebraic number fields and function fields
in one variable over a finite field. The cohomology theory treats both types of
fields equally.
The role that the multiplicative group of fields played in the local theory is
now taken over for a global field k by the idèle class group. Let k be a global
field, k̄ a separable closure of k and Gk = G(k̄|k) its absolute Galois group.
The idèle group Ik of k is defined as the restricted product
Ik =
Y
kp× ,
p
where p runs through all primes of k including the archimedean ones if k is
a number field, kp is the completion of k at p and the restricted product is
taken with respect to the unit groups Up in kp× . The multiplicative group k ×
of k injects diagonally into Ik and we define the idèle class group of k as the
quotient
Ck := Ik /k × .
We refer the reader to [160], chap.VI, §1 for basic properties of the idèle and
the idèle class group.
If K is a finite separable extension of k, then Ik naturally injects into IK and
we call the direct limit
I = lim
IK
−→
K|k
the idèle group of k̄. I is a discrete Gk -module and one easily observes that
IK = I GK
for every finite separable extension K|k. The quotient
C := I/k̄ ×
is the idèle class group of k̄. We have
C = lim
CK
−→
K|k
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426
Chapter VIII. Cohomology of Global Fields
and a straightforward application of Hilbert’s Satz 90 implies that for every
finite separable extension K|k,
CK = C GK .
For every (possibly infinite) separable extension K of k we put IK = I GK ,
CK = C GK , and if K|k is Galois, we set
H i (K|k) = H i (G(K|k), CK ),
and we also write Ĥ i (K|k) for Ĥ i (G(K|k), CK ).
The basis of our results in this chapter is the following theorem, called the
class field axiom. It is an immediate consequence of the so-called first and
second fundamental inequalities for global fields. For a proof of these results,
we refer the reader to [6], chap.5,6, or, for the number field case, to [160],
chap.VI, (4.4).
(8.1.1) Theorem (Class Field Axiom). For a finite cyclic extension K|k we
have


 [K : k] for i = 0,
i
#Ĥ (K|k) =

 1
for i = 1.
Since H 1 (K|k) ∼
= Ĥ −1 (K|k) for K|k cyclic, the class field axiom is a statement
about the kernel and cokernel of the norm map
NK|k
Ö
CK /IG(K|k)
CK
Ck
(here IG(K|k) is the augmentation ideal in ZZ[G(K|k)]), and can therefore be
considered as a noncohomological assertion. Starting with this input, we will
calculate the cohomology groups of CK for arbitrary Galois extensions.
Let K|k be a finite separable extension. Setting
IK (p) =
Y
×
KP
,
P|p
we obtain a decomposition
IK =
Y
IK (p),
p
where p runs through all primes of k and the restricted product is taken with
respect to the subgroups
UK (p) :=
Y
UK,P ,
P|p
×
where UK,P denotes the group of units in KP
.
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427
§1. Cohomology of the Idèle Class Group
Passing to various extension fields, we frequently have to choose compatible
prolongations of primes. Therefore we make the following fixed choice. For
every prime p of k, we choose a k-embedding ip : k̄ ,→ k̄p , in particular a
prime p of k̄ above p, see [160], chap.II, (8.1). We denote the subextensions of
k̄|k by K|k and for each such extension the prime of K lying under p by the
pointed letter P. . Thus every separable extension K|k comes equipped with a
distinguished prime P. above p. By abuse of notation we write
Kp := ip (K)kp
for the extension KP. of the local field kp which corresponds to the prime P.
of K (and which is the completion of K at P. if K|k is finite). We adopt the
×
same convention for the unit groups and write UK,p for group of units in KP
..
Furthermore, we write Gp (K|k) (or simply Gp if K is clear from the context)
for the decomposition group of P. in G(K|k), i.e. Gp (K|k) = G(Kp |kp ).
Now let K|k be a finite Galois extension and let G = G(K|k) be its Galois
group. The idèle group IK is then a G-module and
IK =
Y
IK (p)
p
is a decomposition into G-modules. Then
IK (p) =
Y
×
KσP
. =
σ ∈G/Gp
Y
G
p
×
×
σKP
. = IndG (Kp ),
σ ∈G/Gp
G
and similarly UK (p) = IndGp (UP. ). By Shapiro’s lemma, we obtain isomorphisms
Ĥ i (G, IK (p)) ∼
= Ĥ i (Gp , Kp× )
for all i ∈ ZZ. Furthermore, if p is unramified in K, we have for all i ∈ ZZ
Ĥ i (G, UK (p)) ∼
= Ĥ i (Gp , UP. ) = 0
by (7.1.2)(i). From this follows the
(8.1.2) Proposition. For a finite Galois extension K|k, we have
M
Ĥ i (G, IK ) ∼
Ĥ i (Gp , K × )
=
p
p
for all i ∈ ZZ, where p runs through all primes of k.
Proof: Let S run through the finite sets of primes of k containing the primes
p ramified in K and the infinite primes if k is a number field. Setting
S
IK
=
Y
p∈S
IK (p) ×
Y
UK (p),
/S
p∈
we have IK = lim
I S and
−→ K
S
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428
Chapter VIII. Cohomology of Global Fields
S
)
Ĥ i (G, IK
Ĥ i (G, IK ) = lim
−→
S
= lim Ĥ i (G,
−→
S
= lim
−→
IK (p)) × Ĥ i (G,
p∈S
Y
Y
Ĥ (G, IK (p)) ×
UK (p))
i
Ĥ (G, UK (p))
/S
p∈
Ĥ i (Gp , Kp× ) =
M
Ĥ i (Gp , Kp× ).
2
p
p∈S
S
Y
/S
p∈
Y
i
p∈S
S
∼
= lim
−→
Y
By Hilbert’s Satz 90 and by (7.2.2), we obtain the
H 1 (G, IK ) = H 3 (G, IK ) = 0.
(8.1.3) Corollary.
By proposition (8.1.2), we may associate to every cohomology class c ∈
Ĥ i (G, IK ) its local components cp ∈ Ĥ i (Gp , Kp× ): cp is the image of c under
the composite of the maps
πp
resp
Ĥ i (G, IK ) −→ Ĥ i (Gp , IK ) −→ Ĥ i (Gp , Kp× ),
×
×
where πp is induced by the projection IK → KP
. = Kp . Since these maps
commute with inf , res, cor, we obtain in a straightforward way the
(8.1.4) Proposition. Let L
Then for i ≥ 1
(i)
inf
K|k
L|k (c)p
= inf
⊇
K
⊇
k be finite Galois subextensions of k̄|k.
Kp |kp
Lp |kp (cp )
for c ∈ H i (G(K|k), IK ),
k
for c ∈ H i (G(L|k), IL ),
(ii) res kK (c)P = res KpP (cp )
(iii) cor K
k (c)p =
X
K
.
σ∗−1 corkpσP (cσP. )
for c ∈ H i (G(L|K), IL ),
σ ∈G/Gp
×
×
i
where σ∗ is the map H i (G(KP. |kp ), KP
. ) −→ H (G(KσP. |kp ), KσP. ) induced
by σ : KP. → σKP. = KσP. . For the last two formulae it suffices to require
only that L|k is Galois.
Although the idèle group I = lim
−→ K|k
Q
p
IK of k̄ is not the restricted product
k̄p× , we obtain the following direct decomposition for its cohomology by
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§1. Cohomology of the Idèle Class Group
applying lim
−→ K|k
with respect to the inflation maps to the formula (8.1.2):
H i (k, I) ∼
=
M
H i (Gp (k̄|k), (k̄)×
p)
for all i ≥ 1.
p
Now the question naturally occurs of whether the canonical surjection
G(k̄p |kp ) Gp (k̄|k) is an isomorphism, or equivalently, whether (k̄)p = (kp ).
(8.1.5) Proposition. Let p be a prime of the global field k. Then
(k̄)p = (kp ).
For the proof we need
(8.1.6) Krasner’s Lemma. Let κ be a complete field with respect to a nonarchimedean valuation and let Ω be an algebraic closure of κ. Let α ∈ Ω be
separable over κ and let α = α1 , . . . , αn be the conjugates of α over κ. Suppose
that for β ∈ Ω we have
|α − β| < |α − αi |
for i = 2, . . . , n,
where | | denotes the unique extension of the valuation to Ω. Then κ(α) ⊆ κ(β).
Proof: Consider the extension κ(α, β)|κ(β) and let K|κ(β) be its Galois
closure. Let σ ∈ G(K|κ(β)). Then σ(β − α) = β − σ(α). Since |σ(x)| = |x|
for all x (by the uniqueness of the extension of the absolute value), we have
|β − σ(α)| = |β − α| < |αi − α|
for i = 2, . . . , n. Therefore
|α − σ(α)| < max{|α − β|, |β − σ(α)|} < |α − αi |
for i = 2, . . . , n. It follows that σ(α) = α, and so α ∈ κ(β).
2
Proof of (8.1.5): We have the natural inclusion (k̄)p ⊆ (kp ). The proposition
is trivial if p is archimedean, so assume that p is finite. Let α be in (kp ) and
let f ∈ kp [X] be its minimal polynomial. Since k is dense in kp , we can
choose a polynomial g ∈ k[X] near to f . Then |g(α)| = |g(α) − f (α)| is small.
Q
Writing g(X) = (X − βj ) with βj ∈ k̄ ⊆ (k̄)p , we see that |α − β| is small
for some root β of g(X). In particular, we can choose g(X) and then β such
that |β − α| < |αi − α| for all conjugates αi ∈ (kp ) of α, αi =/ α. By Krasner’s
lemma, we obtain α ∈ kp (β) = (k(β))p ⊆ (k̄)p .
2
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430
Chapter VIII. Cohomology of Global Fields
Passing to the limit over all finite extensions K of k inside k̄, (8.1.2) and
(8.1.5) immediately imply
(8.1.7) Proposition.
H i (k, I) ∼
=
M
H i (kp , k̄p× ) for all i ≥ 1.
p
Since H 1 (k, I) = 0, the inflation maps
H 2 (G(K|k), IK ) → H 2 (k, I)
are injective. We identify the first group with its image. Then the equality
H 2 (k, I) = lim H 2 (G(K|k), IK ) becomes
−→
K|k
H 2 (k, I) =
[
H 2 (G(K|k), IK ).
K|k
In the local case we have seen this with the multiplicative groups in place
of I. Like the unramified extensions in the local case, a decisive role is played
here by the cyclic extensions K|k because of their periodic cohomological
behaviour. For this reason the following analogue (8.1.9) of the local situation
is of crucial importance. But first we observe the
(8.1.8) Proposition. Let L|k be a Galois extension and let p be a prime number.
Suppose in the number field case that L is totally imaginary if p = 2. If p∞
divides the local degrees [Lp : kp ] for all finite primes p of k, then
H 2 (L, I)(p) = 0
and
H 2 (G(L|k), IL )(p) ∼
= H 2 (k, I)(p).
Proof: Let K|k run through the finite subextensions of L|k. Then H 1 (L, I) =
lim
H 1 (K, I) = 0, hence the sequence
−→
K,res
0 −→ H 2 (G(L|k), IL ) −→ H 2 (k, I) −→ H 2 (L, I)
is exact. Therefore it suffices to prove H 2 (L, I)(p) = 0. Using (8.1.7) and
passing to the direct limit, we obtain
H 2 (L, I)(p) =
M
G (L|k)
p
IndG(L|k)
H 2 (Lp , L̄×
p )(p).
p
For finite primes we have H 2 (Lp , L̄×
p )(p) = 0 by (7.1.6), and for the archimedean
primes the group H 2 (Lp , L̄×
)(p)
is
trivially zero for p =/ 2 and by assumption
p
also for p = 2.
2
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431
§1. Cohomology of the Idèle Class Group
Let k(µ) be the extension of k obtained by adjoining all roots of unity in
k̄ to k. This extension has an abelian Galois group. In the function field
case, k(µ) is the extension obtained by passing to the algebraic closure of the
constant field and G(k(µ)|k) ∼
= ẐZ. We use the notation k̃ = k(µ) in this case.
If k is a number field, let T be the torsion subgroup of G(k(µ)|k). Then the
field k̃ := k(µ)T is an extension of k with Galois group ẐZ which we call the
cyclotomic ẐZ-extension of k ∗) . The decomposition group of a finite prime
Gp (k̃|k) ⊆ G(k̃|k) is open and, in particular, isomorphic to ẐZ. Since ẐZ is
torsion-free, archimedean primes are unramified in k̃|k.
(8.1.9) Proposition.
[
H 2 (k, I) =
H 2 (G(K|k), IK ) .
K|k
cyclic
Moreover, it suffices to take the union over all cyclic extensions of k inside k̃
if k is a function field or a totally imaginary number field. If k is a number
field having a real place, it suffices to take the union over all cyclic extensions
of k inside k̃1 , where k1 |k is an arbitrarily chosen totally imaginary quadratic
extension of k.
Proof: Let x ∈ H 2 (k, I). Using (8.1.7), we decompose x into an archimedean
and a nonarchimedean part: x = xa + xn . By (8.1.8), we find a finite extension K of k inside k̃ such that res kK xn ∈ H 2 (K, I) vanishes. This finishes the
proof if k is a function field or a totally imaginary number field. If k has a
real place, let k1 be an arbitrarily chosen totally imaginary quadratic extension
of k. Let K1 |k be the uniquely defined cyclic extension of k inside k̃ of degree
2 · [K : k]; in particular, K1 is a quadratic extension of K. The extensions
k1 |k and K1 |k are linearly disjoint because the first is totally ramified and
the second is unramified at all real places. The extension k1 K1 |K has Galois
group ZZ/2ZZ ⊕ ZZ/2ZZ and thus contains a unique subextension L|K of degree 2
distinct from K1 and k1 K. We have the following diagram of fields.
×ØÙÚÛÜÝÞß
k1 K1
k1 K
k1
L
K1
K
k
∗) The cyclotomic ẐZ-extension is the composite of the cyclotomic ZZ -extensions for all
p
prime numbers p. We will recall the definition of the cyclotomic ZZp -extension in XI §1.
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432
Chapter VIII. Cohomology of Global Fields
Since L is totally imaginary and cyclic over k, we conclude that res kL x ∈
H 2 (L, I) is zero.
2
Again let K|k be a finite Galois extension with Galois group G and decomposition groups Gp = G(Kp |kp ). For every prime p, we have by VII §1 the
invariant map
1
∼
invKp |kp : H 2 (Gp , Kp× ) −→
ZZ/ZZ.
[Kp :kp ]
From the decomposition
H 2 (G, IK ) ∼
=
M
H 2 (Gp , Kp× ),
p
we obtain a canonical homomorphism
1
ZZ/ZZ,
invK|k : H 2 (G, IK ) −→ [K:k]
given by
invK|k (c) =
X
invKp |kp (cp ).
p
This invariant map is compatible with inf , res, cor as the following proposition
shows.
(8.1.10) Proposition. If L ⊇ K ⊇ k are finite separable extensions with L|k
Galois, then we have the commutative diagrams
àåäãâá
H 2 (G(L|K),
IL )
res
invL|K
cor
H 2 (G(L|k), IL )
1
ZZ/ZZ
[L:K]
[K:k] incl
invL|k
1
ZZ/ZZ
[L:k]
.
Moreover, if K|k is Galois, then invL|k is an extension of invK|k .
Proof: The proposition is an immediate consequence of (8.1.3) and of the
analogous properties of the local inv’s as invariant maps of a class formation
(see (3.1.8)). If
c ∈ H 2 (G(K|k), IK ) ⊆ H 2 (G(L|k), IL ),
then
invL|k (c) =
X
p
invLp |kp (cp ) =
X
invKp |kp (cp ) = invK|k (c).
p
Letting P run through the primes of K and choosing always a prime P0 |P of
L, we obtain for c ∈ H 2 (G(L|k), IL )
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§1. Cohomology of the Idèle Class Group
invL|K (res c) =
X
=
X
invLP0 |KP (reskK (c)P ) =
X
P
k
invLP0 |KP (resKpP (cp ))
P
[KP : kp ] invLP0 |kp (cp )
P
=
XX
p
=
[KP : kp ] invLP0 |kp (cp )
P|p
X
[K : k] invLP0 |kp (cp )
p
= [K : k] invL|k (c).
Finally, for c ∈ H 2 (G(L|K), IL ) we obtain
invL|k (cor c) =
X
invLP0 |kp (corK
k (c)p ) =
XX
p
=
K
invLP0 |kp (corkpP (cP ))
p P|p
XX
invLP0 |KP (cP )
= invL|K (c).
2
p P|p
By the compatibility of inv with the inflation, we also obtain invariant maps
for infinite Galois subextensions K|k of k̄|k,
1
invK|k : H 2 (G(K|k), IK ) −→ [K:k]
ZZ/ZZ,
by passing to the direct limit over the finite Galois subextensions Kα |k of K|k
and setting
[
1
1
ZZ/ZZ =
ZZ/ZZ.
[K:k]
[Kα :k]
α
In particular, we have an invariant map
invk : H 2 (k, I) −→ Q/ZZ
for the idèle group I of the separable closure k̄.
For every finite Galois extension K|k with Galois group G, local class field
theory provides us with the norm residue symbol
( , K|k) : Ik −→ Gab
given by
(α, K|k) =
Y
(αp , Kp |kp ).
p
The product on the right is finite, i.e. well-defined, because (αp , Kp |kp ) = 1 for
all primes p such that αp ∈ Up and p is unramified in K|k. This norm residue
symbol is linked with the invariant map invK|k as follows. For every character
χ ∈ H 1 (G, Q/ZZ), the exact sequence
0 −→ ZZ −→ Q −→ Q/ZZ −→ 0
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434
Chapter VIII. Cohomology of Global Fields
yields an element δχ ∈ H 2 (G, ZZ). Moreover, we have for every idèle α ∈ Ik =
H 0 (G, IK ), the cup-product α ∪ δχ ∈ H 2 (G(K|k), IK ).
(8.1.11) Proposition. For every χ ∈ H 1 (G, Q/ZZ) and α ∈ Ik , we have
χ((α, K|k)) = invK|k (α ∪ δχ).
Proof: The result follows from its local analogue (7.2.12). If χp is the
restriction of χ to Gp = G(Kp |kp ), then αp ∪ δχp ∈ H 2 (Gp , Kp× ) is obviously
the local component of α ∪ δχ, so that
invK|k (α ∪ δχ) =
X
invKp |kp (αp ∪ δχp )
p
=
X
χp ((αp , Kp |kp )) = χ((α, K|k)).
2
p
The next proposition is a first step towards theorem (8.1.22) below, which
claims that the pair (Gk , C) is a class formation.
(8.1.12) Proposition. For every finite Galois extension K|k
H 1 (G(K|k), CK ) = 0.
Proof: If K|k is cyclic, this is part of the class field axiom (8.1.1). If
[K : k] = pn is a prime power, then there exists a cyclic subextension L|k of
K|k of degree p, and we can proceed by induction on [K : k] using the exact
sequence
0 −→ H 1 (G(L|k), CL ) −→ H 1 (G(K|k), CK ) −→ H 1 (G(K|L), CK ).
If K|k is arbitrary, we consider the fixed fields Σp of the p-Sylow subgroups
of G(K|k). Then, by (1.6.10), the restriction map
res : H 1 (G(K|k), CK ) ,→
Y
H 1 (G(K|Σp ), CK )
p
is injective and we are done.
(8.1.13) Corollary.
2
H 1 (k, C) = 0.
By (6.3.4), the group H 2 (k, k̄ × ) is canonically isomorphic to the Brauer
group Br(k) of central simple algebras over the global field k.
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435
§1. Cohomology of the Idèle Class Group
(8.1.14) Proposition. Let k be a global field. Then
(i)
Br(k) =
[
Br(K|k).
K|k
cyclic
(ii) Let K|k be an infinite Galois extension such that p∞ divides the local
degrees [Kp : kp ] for all finite primes p of K and assume in the number
field case that K is totally imaginary if p = 2. Then
Br(K)(p) = 0.
Remark: We could have sharpened assertion (i) in a similar manner to (8.1.9).
Proof: For every finite Galois extension K|k, (8.1.13) and H 1 (K, I) = 0 =
H 1 (K, k̄ × ) give the exact commutative diagram
0æçèéêëìíîïðñ
H 2 (K, k̄ × )
H 2 (K, I)
0
H 2 (k, k̄ × )
H 2 (k, I)
0
H 2 (G(K|k), K × )
H 2 (G(K|k), IK )
0
0.
Considering this diagram for cyclic extensions K|k, we obtain assertion (i)
from (8.1.9). Since Br(K)(p) injects into H 2 (K, I)(p), the second statement
follows from (8.1.8).
2
(8.1.15) Proposition. Let K|k be a cyclic extension with Galois group G.
Then the sequence
0 −→ H 2 (G, K ×ò ) −→ H 2 (G, IK )
invK|k
1
[K:k]
ZZ/ZZ −→ 0
is exact.
1
Proof: We start by showing that invK|k : H 2 (G, IK ) → [K:k]
ZZ/ZZ is surjective. We first assume that [K : k] is a prime power, say pn . Let K 0 be the
unique extension of degree p inside K|k. If all primes of k would split in K 0 ,
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436
Chapter VIII. Cohomology of Global Fields
then the norm map NK 0 |k : IK 0 → Ik would be surjective, and so would be the
norm map NK 0 |k : CK 0 → Ck . Since this contradicts #Ĥ 0 (G(K 0 |k), CK 0 ) = p,
we find a prime p of k which is inert in K 0 , and hence also in K. Therefore
G = Gp = G(Kp |kp ), and we know that
invKp |kp : H 2 (Gp , Kp× ) −→
1
[Kp :kp ]
ZZ/ZZ
is bijective. We conclude that
1
ZZ/ZZ
invK|k : H 2 (G, IK ) → [K:k]
is surjective if [K : k] is a prime power. The general case now easily follows
from (8.1.10).
Since G is cyclic, H 3 (G, K × ) = H 1 (G, K × ) = 0. Moreover, H 1 (G, CK ) = 0,
and so the exact sequence
0 → K × → IK → CK → 0
yields the exact sequence
0 −→ H 2 (G, K × ) −→ H 2 (G, IK ) −→ H 2 (G, CK ) −→ 0;
in particular, the map H 2 (G, K × ) −→ H 2 (G, IK ) is injective and, by (8.1.1),
its cokernel has order
#H 2 (G, CK ) = #Ĥ 0 (G, CK ) = [K : k].
It remains to show that inv is trivial on the image of H 2 (G, K × ). We prove
this without the assumption of K|k being cyclic. So let α ∈ Br(k) be arbitrary.
If k is a function field, then, by (8.1.14) and the following remark, we may
assume that α ∈ Br(K|k), where K = k(ζn ) for some n prime to char(k). If
k is number field, let K be a finite extension of k which is Galois over Q and
such that α ∈ Br(K|k). Then
invk (α) = invK|k (α) = invK|Q (corkQ (α)).
Hence we may assume that k = Q and, by the same argument as in the function
field case, α ∈ Br(K|Q) for a cyclic subextension K of Q(ζn )|Q for some n.
In both cases, let G = G(K|k) and let χ be a generator of H 1 (G, Q/ZZ). Then
δχ is a generator of H 2 (G, ZZ), and the cup-product
δχ∪ : Ĥ 0 (G, K × ) −→ H 2 (G, K × )
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437
§1. Cohomology of the Idèle Class Group
is the periodicity isomorphism (see (1.7.1)). Therefore every element of
H 2 (G, K × ) is of the form ā ∪ δχ with a ∈ k × . By (8.1.11), we have
invK|k (ā ∪ δχ) = χ((a, K|k)).
It therefore remains to show that (a, K|k) = 1 for a ∈ k × . Hence it suffices to
show that (a, k(ζn )|k)ζn = ζn , where k is a function field and (n, char(k)) = 1
or k = Q and n arbitrary. Let k be a function field. Then, for any place p of k,
we have
vp (a)
.
(a, kp )ζn = ζn#k(p)
The ‘product formula’ yields
Y
#k(p)vp (a) = 1,
p
hence
Q
(a, k)ζn = ζn
p
#k(p)vp (a)
= ζn .
The argument in the case k = Q is similar, but the computation of the local
norm residue symbols at the primes p | n is much more involved, see, e.g.,
[160], chap.VI, (5.3) and chap.V, (2.4). This proves the proposition.
2
Using (8.1.14)(i) and (8.1.9), we obtain from (8.1.15) the following
(8.1.16) Corollary. We have an exact sequence
0 −→ H 2 (k, k̄ ×ó ) −→ H 2 (k, I)
invk
Q/ZZ −→ 0 .
By (8.1.7), we have
H 2 (k, I) ∼
=
M
H 2 (kp , k̄p× ),
p
and the summands are the Brauer groups Br(kp ) of the local fields kp . Hence
corollary (8.1.16) gives us the famous Hasse principle for central simple
algebras:
(8.1.17) Theorem. We have an exact sequence
0÷ôõö
Br(k)
M
Br(kp )
invk
Q/ZZ
0,
p
where invk is the sum of the local invariant maps invkp : Br(kp ) → Q/ZZ.
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438
Chapter VIII. Cohomology of Global Fields
(8.1.18) Corollary. Let p be a prime number and let K|k be an infinite separable extension of the global field k which we assume to be totally imaginary
if k is a number field and p = 2. If p∞ divides the local degrees [Kp : kp ] for
all nonarchimedean primes p of k, then
cdp GK ≤ 1.
Proof: If p = char(K), this is always true by (6.5.10). Otherwise we have
to show that Br(L)(p) = 0 for every finite separable extension L|K (see
(6.5.11)). Passing to inductive limits, (8.1.17) implies that Br(L)(p) injects
Q
into p Br(Lp )(p). Furthermore, Br(Lp )(p) ∼
= H 2 (Lp , µp∞ ) = 0 by (7.1.8)(i)
or by assumption if p = 2 and p|∞.
2
We now carry over the results on the cohomology of the idèles to the cohomology of the idèle class group. Recall the notation
H i (K|k) = H i (G(K|k), CK ).
(8.1.19) Lemma. For every finite Galois extension K|k, we have
#H 2 (K|k) | [K : k].
Proof: If K|k is cyclic, the assertion follows from the class field axiom
(8.1.1). If K|k is a p-extension and L|k a cyclic subextension of degree p, then
the exact sequence (observe that H 1 (K|L) = 0)
0 −→ H 2 (L|k) −→ H 2 (K|k) −→ H 2 (K|L)
shows, using induction on [K : k],
#H 2 (K|k) | #H 2 (K|L) · #H 2 (L|k) | [K : L] · [L : k] = [K : k].
In the general case, let Σp be a p-Sylow field of K|k. Since the restriction map
res : H 2 (K|k) ,→
M
H 2 (K|Σp )
p
is injective, we obtain
Y
Y
#H 2 (K|k) |
#H 2 (K|Σp ) |
[K : Σp ] = [K : k].
p
If N
⊇
K
⊇
2
p
k are two finite Galois extensions in k̄|k, then the sequence
inf
res
0 −→ H 2 (K|k) −→ H 2 (N |k) −→ H 2 (N |K)
is exact by (1.6.7), since H 1 (N |K) = 0. As for the idèles, we identify H 2 (K|k)
with its image in H 2 (k, C) so that
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439
§1. Cohomology of the Idèle Class Group
H 2 (k, C) =
[
H 2 (K|k),
K|k
where K|k varies over the finite Galois subextensions of k̄|k.
We would like to deduce from the idèlic invariant maps
1
invK|k : H 2 (G(K|k), IK ) → [K:k]
ZZ/ZZ
invariant maps for the groups H 2 (K|k). This is not possible in a direct way,
since the map H 2 (G(K|k), IK ) → H 2 (G(K|k), CK ) is in general not surjective.
But it becomes possible if we first pass to the direct limit.
As before let k̃|k be the cyclotomic ẐZ-extension in the number field case and
the ẐZ-extension obtained by passing to the algebraic closure of the constant
field in the function field case. Since scd ẐZ = 2, we have H 3 (k̃|k, k̃ × ) = 0, and
therefore the exact sequence
0 −→ k̃ × −→ Ik̃ −→ Ck̃ −→ 0
induces the exact cohomology sequence
0 −→ H 2 (k̃|k, k̃ × ) −→ H 2 (k̃|k, Ik̃ ) −→ H 2 (k̃|k, Ck̃ ) −→ 0 .
Passing to the limit over all finite subextensions of k̃|k, (8.1.15) induces the
exact sequence
0 −→ H 2 (k̃|k, k̃ × ) −→ H 2 (k̃|k, Ik̃ ) −→ Q/ZZ −→ 0 .
Therefore we obtain an isomorphism
∼ Q/Z
Z.
invk̃|k : H 2 (k̃|k, Ck̃ ) −→
For a finite separable extension K|k we have a commutative diagram by
(8.1.10)
invK̃|K
øùüûú C )
H 2 (K̃|K,
Q/ZZ
K̃
res
H 2 (K̃|k, CK̃ )
[K:k]
inf
H 2 (k̃|k, Ck̃ )
invk̃|k
Q/ZZ .
(8.1.20) Proposition. The sequence
0 −→ H 2 (k, k̄ × ) −→ H 2 (k, I) −→ H 2 (k, C) −→ 0
is exact.
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440
Chapter VIII. Cohomology of Global Fields
Proof: Let K|k be an arbitrary finite Galois extension of degree n = [K : k]
and let kn be the unique extension of k in k̃ of degree n. We claim that
H 2 (K|k) = H 2 (kn |k) ,
where we identify H 2 (K|k) and H 2 (kn |k) with their images in H 2 (k, C) under
the inflation maps. In fact, (8.1.19) and the class field axiom (8.1.1) imply that
#H 2 (K|k) [K : k] = [kn : k] = #H 2 (kn |k),
and it therefore suffices to show the inclusion "⊇". But this follows from the
exact commutative diagram
0ýþÿ
H 2 (K|k)
H 2 (k, C)
res
H 2 (K, C)
H 2 (k̃|k, Ck̃ )
res
H 2 (K̃|K, CK̃ )
invk̃|k
invK̃|K
Q/ZZ
[K:k]
Q/ZZ,
in which invk̃|k and invK̃|K are isomorphisms as shown above. Since H 2 (kn |k) ⊆
H 2 (k̃|k) has order n = [K : k], it is mapped by the middle arrow res, and thus
by the upper arrow res , to zero, so that
H 2 (kn |k) ⊆ H 2 (K|k),
which shows the claim. Hence we obtain
H 2 (k, C) =
[
H 2 (K|k) =
[
H 2 (kn |k) = H 2 (k̃|k, Ck̃ ).
n
K
In particular, this implies
H 2 (k, C) =
[
H 2 (K|k) .
K|k
cyclic
The same assertion holds if we replace C by I or k̄ × , from (8.1.9) or (8.1.14)(i),
respectively. Now for K|k cyclic, we have the exact sequence
0 −→ H 2 (K|k, K × ) −→ H 2 (K|k, IK ) −→ H 2 (K|k, CK ) −→ 0,
because H 1 (K|k, CK ) = 0 and H 3 (K|k, K × ) ∼
= H 1 (K|k, K × ) = 0. From this
and from the above observation now follows the statement of the proposition.
2
(8.1.21) Corollary. We have a canonical isomorphism
∼
invk : H 2 (k, C) −→
Q/ZZ.
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441
§1. Cohomology of the Idèle Class Group
From this corollary we obtain for every finite Galois extension K|k a canonical invariant map
1
∼
ZZ/ZZ
invK|k : H 2 (K|k) −→
[K:k]
using the commutative diagram
0
H 2 (K|k)
H 2 (k, C)
invK|k
0
res
H 2 (K, C)
invk
1
ZZ/ZZ
[K:k]
Q/ZZ
0
invK
[K:k]
Q/ZZ
0.
The results above and the compatibility of inv with inf and res , which follows
from (8.1.10), therefore imply the
(8.1.22) Theorem. The G-module C is a formation module with respect to
the invariant maps invK|k (see (3.1.8)).
We now obtain the main theorem of global class field theory from (3.1.6):
the “global reciprocity law”.
(8.1.23) Theorem. Let K|k be a finite Galois extension of global fields with
Galois group G(K|k). Then there is a canonical isomorphism
Ck /NK|k CK ∼
= G(K|k)ab .
By the results of III §1 (see the remark 5 on p.157), we obtain a reciprocity
homomorphism
rec : Ck −→ Gab
k ,
which has a dense image, and whose kernel is the group of universal norms,
i.e. the intersection
\
NK|k CK = NGk C .
K
As before we also call rec the norm residue symbol and we write it in the
form rec(α) = (α, k).
Essential is the following theorem, which is called the existence theorem for
global class field theory. For a proof, see [6], chap. 8, th. 1. For number fields,
see also [123], chap.XI §2 th. 1 or [160], chap.VI th. 6.1.
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442
Chapter VIII. Cohomology of Global Fields
(8.1.24) Theorem. The norm groups NK|k CK , where K|k is a finite separable
extension, are exactly the open subgroups of finite index in Ck .
The idèle group Ik = p kp× is a locally compact topological group, k × is a
discrete, closed subgroup (see [160], chap.VI, (1.5)), and so Ck is also a locally
compact topological group. We have a canonical homomorphism, called the
“absolute value”,
Q
| | : Ik −→ IR×
+ , |α| =
Y
|αp |p ,
p
which is trivial on k × , and hence induces a homomorphism
| | : Ck −→ IR×
+.
The kernel Ck0 = {x ∈ Ck | |x| = 1} is a compact group (cf. [22], chap.II, §16,
Theorem, or [160], chap.VI, (1.6) for the number field case).
If k is a number field, then the homomorphism | | is surjective: Since IR×
+
has no nontrivial finite quotient, Ck and Ck0 have the same image under the
0
reciprocity map, which is all of Gab
k since Ck is compact and (Ck , k) is dense
in Gab
k . This yields the
(8.1.25) Proposition. In the number field case we have an exact sequence of
topological groups
0
NGk C
Ck
( ,k)
Gab
k
0.
If k is a function field, with constant field κ of cardinality q, we have the
following situation. The group Ik , and hence Ck , is totally disconnected by
(1.1.9)(iii); in particular, Ck0 is an abelian profinite group. For α ∈ Ck we have
|α| = q − deg(α) , where
deg : Ck → ZZ,
deg(α mod k × ) =
X
vp (αp ) · [κ(p) : κ] ,
p
∼
∼ Z be the canonical
is the degree map. Let π : Gab
k G(k κ̄|k) = Gκ = Ẑ
projection. Then, for all α ∈ Ck , we have π((α, k)) = Frobdeg(α) , where
Frob denotes the Frobenius automorphism x 7→ xq on κ̄. Therefore deg is
surjective∗) and we obtain an exact sequence of topological groups
deg
0 −→ Ck0 −→ Ck −→ ZZ −→ 0.
Since Ck0 is profinite the intersection of all open subgroups of finite index in
Ck is zero. The degree map extends to a map from C k := lim
Ck /NK|k CK
←−
K|k
∗) This can also be seen directly: one can construct an injection coker(deg) ,→ Br(κ) = 0.
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443
§2. The Connected Component of Ck
onto ẐZ. We therefore obtain a commutative exact diagram
0
Ck0
Ck
deg
0
Ck0
Ck
deg
ZZ
0
ẐZ
0,
which yields, using (8.1.23), the
(8.1.26) Proposition. In the function field case we have an exact sequence of
topological groups
0"#!
( ,k)
Ck
Gab
k
ẐZ/ZZ
0.
§2. The Connected Component of Ck
We now study the topological properties of the idèle class group Ck and,
in particular, the connected component Dk of 1 in Ck . The results are due to
J. TATE (see [6], chap.9) and are quoted in [160], chap.VI, §1, ex.1–10.
We denote the connected component (of 1) of the idèle class group Ck by
Dk . If k is a function field, then Ik , and hence Ck , is totally disconnected, so
that Dk = 1.
Therefore we assume for the rest of this section that k is a number field.
We introduce the following notation:
S∞ = S∞ (k) the set of archimedean primes of k ,
SIR = SIR (k) the set of real primes of k ,
SC = SC (k) the set of complex primes of k ,
r1 = r1 (k) the number of real primes of k ,
r2 = r2 (k) the number of complex primes of k ,
r
= r(k)
= r1 (k) + r2 (k) .
The continuous surjection
| | : Ck −→ IR×
+
has a continuous section s : IR×
+ → Ck : for any infinite prime p, the restriction
×
of | | to the subgroup IR+ of kp× ⊆ Ck is an isomorphism, and its inverse gives
a section of the homomorphism | |. Since IR×
+ is connected, the image of s is
contained in Dk and we obtain compatible isomorphisms of topological groups
Ck ∼
= D0 × IR× ,
= C 0 × IR× , Dk ∼
k
+
k
+
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Chapter VIII. Cohomology of Global Fields
where Dk0 = Ck0 ∩ Dk is the connected component of Ck0 . The group Dk has
the following characterizations.
(8.2.1) Theorem.
(i) Dk is the closure of the image of Ũ =
Y
Up1 under the projection
p|∞
×
1
Ik → Ck , where Up1 = IR×
+ if p is real and Up = C if p is complex.
(ii) Dk is the intersection of all open subgroups of Ck of finite index.
(iii) Dk is the group of universal norms Nk̄|k C =
\
NK|k CK and Dk0 is the
K|k
0
group of the universal norms Nk̄|k C =
\
0
.
NK|k CK
K|k
(iv) For any finite extension K|k the norm maps NK|k : DK → Dk and
0
NK|k : DK
→ Dk0 are surjective.
(v) Dk is the group of all divisible elements of Ck , i.e. elements x which are
n-th powers, x = y n , for every n ∈ IN.
(vi) D is divisible, i.e. it is the maximal divisible subgroup in C .∗)
k
k
Remark: All open subgroups in Ck have finite index. We will prove this fact
in the context of restricted ramification in (8.3.14).
Proof: (i) Since Ũ is the connected component of 1 in I, the assertion follows
from (1.1.9)(iii).
(ii) Since Ck /Dk is totally disconnected, the intersection of all its open subgroups Ūi is trivial by (1.1.9)(i). But Ck /Dk = Ck0 /Dk0 is compact, i.e. the Ūi
are of finite index. The pre-images Ui in Ck are open subgroups of finite index
and Dk is their intersection. On the other hand, any open subgroup contains
Dk . This proves (ii).
(iii) By the existence theorem (8.1.24), the open subgroups of finite index in Ck
are just the norm groups NK|k CK of the finite Galois extensions K|k, so that
(ii) implies that Dk = Nk̄|k C. As IR×
+ is cohomologically trivial, we have an
0
0 ∼
isomorphism Ck /NK|k CK = Ck /NK|k CK for every finite Galois extension
K|k. This implies
Nk̄|k C 0 = Ck0 ∩ Nk̄|k C = Ck0 ∩ Dk = Dk0 .
∗) If A is an abelian group, then the subgroup of divisible elements contains the maximal
divisible subgroup of A, but in general is not divisible itself.
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§2. The Connected Component of Ck
(iv) By assertion (iii), Dk0 is the group of universal norms of the level-compact
0
module C 0 , hence NK|k DK
= Dk0 for every finite Galois extension K|k. The
analogous assertion for D follows from the commutative diagram
0-$%&'()*+,.
0
DK
NK|k
0
IR×
+
DK
NK|k
NK|k
Dk0
IR×
+
Dk
0
0
×
[K:k]
noting that the norm map NK|k : IR×
and is
+ → IR+ is just the map x 7→ x
therefore surjective.
(v) Ck /Dk = Ck0 /Dk0 is a compact, totally disconnected Hausdorff group, i.e. a
profinite group. If x ∈ Ck is divisible, then its image in Ck /Dk is contained in
every open subgroup, i.e. the image is 1 and hence x ∈ Dk . Hence (v) follows
from (vi).
0
(vi) Since IR×
+ is divisible, it suffices to show that Dk is divisible. Let ` be a
prime number. If the finite extension K of k contains the `-th roots of unity,
then for sufficiently large S containing all primes above ` and all archimedean
primes,
the group (CK )` · ŪKS is the norm group of the Kummer extension
q
Q
Q
K( ` OS× )|K. Here ŪKS = UKS K × /K × and UKS = p∈/S Up × p∈S 1 (see [123],
chap.XI §2 th. 1 or the proof of [160], chap.VI th. 6.1). Hence, for any such S,
DK
⊆
(CK )` Ū S .
As the absolute value | | is trivial on Ū S , we obtain
0
DK
⊆
\
0 ` S
0 `
(CK
) Ū = (CK
).
S
Using (iv) this implies
0
Dk0 = NK|k DK
(∗)
⊆
0 `
(NK|k CK
).
For each a ∈ Dk0 , let a1/` denote the set of all elements of Ck0 whose `-th power
0
) ∩ (a1/` ) are nonempty.
is a. From (∗) we see that the sets XK = (NK|k CK
`
0
is closed,
The kernel of the map Ck0 → Ck0 , x 7→ x` , is compact and NK|k CK
T
and so XK is compact. Therefore the intersection K|k XK is not empty. By
(iii), an element of this intersection is an element of Dk0 whose `-th power is a.
This proves (vi).
2
By (8.2.1), we obtain from (8.1.25) the
(8.2.2) Corollary. We have an exact sequence of topological groups
02/01
Dk
Ck
( ,k)
Gab
k
0.
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446
Chapter VIII. Cohomology of Global Fields
In order to give an explicit description of the connected component Dk ∼
=
Z
Z
/
Z
Z
,
where
× IR×
,
we
have
first
to
consider
the
group
+
Dk0
ZZ = IR ×
Y
ZZp = IR × ẐZ
p
is the group of integral elements in the adèle ring AQ of Q.
(8.2.3) Proposition. The topological group ZZ/ZZ is compact, connected and
uniquely divisible. It is called the solenoid.∗)
Proof: The canonical projection ZZ = ẐZ × IR ẐZ induces an exact sequence
0 −→ IR −→ ZZ/ZZ −→ ẐZ/ZZ −→ 0 .
This shows that ZZ/ZZ is uniquely divisible, since IR and ẐZ/ZZ have this property.
Furthermore, IR is dense in ZZ/ZZ: let λ = (z, x) mod ZZ be an arbitrary element
in (ẐZ × IR)/ZZ, let m ∈ ZZ be given (describing a neighbourhood of ZZ) and
let a ∈ ZZ such that a ≡ z mod m. Then λ = (z − a, x − a) mod ZZ and
(z − a, x − a) is contained in the set (mẐZ, 0) + (0, x − a) ⊆ mẐZ × IR, which is
mapped into a neighbourhood of x−a ∈ IR ⊆ ZZ/ZZ. Now, since IR is connected
and the closure of a connected set is connected, we conclude that the solenoid
is connected. (Another possible method to show the connectedness of ZZ/ZZ
is to use the exact sequence at the beginning of the proof and to show the
connectedness of ẐZ/ZZ .)
Since ZZ is closed in ZZ, the quotient ZZ/ZZ is a Hausdorff topological group.
Finally, the compact set ẐZ×[0, 1] ⊆ ZZ is mapped under the canonical projection
ZZ ZZ/ZZ onto ZZ/ZZ. Hence the solenoid is compact.
2
We consider the group
Uk = Ū × Ũ ,
where
Ū =
Y
Up
and
Ũ =
p-∞
Y
Up1 ,
p|∞
and we write every idèle α ∈ Uk as a product α = ᾱα̃ with ᾱ ∈ Ū , α̃ ∈ Ũ .
The abelian profinite group Ū is a (multiplicative) ẐZ-module in a natural
way: The exponentiation
IN × Ū → Ū , (n, α) 7→ αn ,
extends continuously to an exponentiation
ẐZ × Ū −→ Ū , λ 7−→ αλ .
∗) For another description of the solenoid, see ex.1.
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447
§2. The Connected Component of Ck
(8.2.4) Lemma. If ε1 , . . . , εr−1 are ZZ-independent units of k, r = r1 + r2 , then
Q
the corresponding idèles ε̄1 , . . . , ε̄r−1 ∈ Ū = p-∞ Up are ẐZ-independent.
Proof: By Dirichlet’s unit theorem, the group generated by ε1 , . . . , εr−1 (with
ordinary integers as exponents) has finite index d in Ok× . Consider a relation
z
r−1
ε̄z11 · · · ε̄r−1
= 1 , zi
(∗)
∈
ẐZ.
We have to show z1 = · · · = zr−1 = 0. Let n ∈ IN. For each zi we find an
νr−1
integer νi ∈ ZZ such that νi ≡ zi mod 2dn. Then ε = εν11 · · · εr−1
is an element
of k and we may write
ν
ν
r−1 ν1
r−1
ε = ε̄1ν1 · · · ε̄r−1
ε̃1 · · · ε̃r−1
.
Dividing the right-hand side by the left-hand side of (∗), we obtain
ν
r−1
ε = ε̄ν11 −z1 · · · ε̄r−1
−zr−1 ν1
ε̃1
ν
r−1
· · · ε̃r−1
.
At each finite prime each idèle ε̃νi i has component 1; the remaining factors have
ẐZ-exponents divisible by 2dn. This means that ε is a 2dn-th power in kp× for
each finite prime p. But from this it follows that ε is a dn-th power in k × , i.e.
ε = η dn , η ∈ k × , see (9.1.11)(ii). This η must be a unit and consequently η d is
µr−1
contained in the subgroup generated by the ε1 , . . . , εr−1 , i.e. η d = εµ1 1 · · · εr−1
,
µi ∈ ZZ. We now get
ν
µ
n
r−1
r−1
ε = εν11 · · · εr−1
= εµ1 1 n · · · εr−1
.
Since the εi are ZZ-independent, we must have νi = µi n. Because νi ≡ zi
mod 2dn, this shows that each zi is divisible by n, i.e. zi ∈ nẐZ. But n was
arbitrary, hence zi = 0.
2
We shall now determine the structure of Dk0 . It contains a torus Tk which is
defined as follows: for each complex prime p we have the embedding kp× ,→ Ck
and we obtain an embedding
Y
kp× ,→ Ck .
p∈SC
Each factor kp× = C× contains the unit circle Sp1 = {z
put
Y
Tk =
Sp1 .
∈
kp× | |z| = 1} and we
p∈SC
It is a compact connected subgroup of Dk0 , also described by the topological
isomorphism
exp :
Y
IR/ZZ −→ Tk ,
θ = (θp ) 7−→ (e2πiθp )p∈SC .
p∈SC
We may regard Tk as a subgroup of the idèle group Ũ
⊆
Uk .
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448
Chapter VIII. Cohomology of Global Fields
For p ∈ S∞ , we have an exponentiation IR × Up → Up , (x, αp ) 7→ αpx =
e
, where log is the natural logarithm of IR×
+ if p is real, and is any fixed
×
branch of the logarithm of C if p is complex. For fixed α ∈ Uk , the map
IR → Ũ , x 7→ α̃x , is continuous, and so is the map
x log αp
Z̄Z = ẐZ × IR −→ Uk ,
λ = (z, x) 7−→ ᾱz α̃x .
Let O+ (k) be the subgroup of Ok× of totally positive units of k, i.e. ε ∈ O+ (k)
if and only if εp > 0 for every real prime p. Since O+ (k) has finite index in Ok× ,
Dirichlet’s unit theorem implies that it is a finitely generated group of rank
rk(O+ (k)) = #S∞ (k) − 1 = r − 1,
and we choose a fixed r − 1-tuple
ε = (ε1 , . . . , εr−1 )
of ZZ-independent units. In addition to Tk , we consider the subgroup
λ
r−1
I(ε) = {ελ1 1 · · · εr−1
| λ1 , . . . , λr−1
∈
ZZ}
of Uk together with the continuous surjective homomorphism
expε : ZZr−1 −→ I(ε),
λ
r−1
(λ1 , . . . , λr−1 ) 7−→ ελ1 1 · · · εr−1
.
Note that Tk is a subgroup of Uk0 = {x ∈ Uk | |x| = 1} and the same holds
for I(ε), since εzi i ∈ Up for every finite prime p, so that |ελi i | = |ε̃xi i | = |ε̃i |xi =
|εi |xi = 1 (where λi = (zi , xi ) ∈ ẐZ × IR = ZZ). We get a continuous homomorphism
exp : (IR/ZZ)r2 × ZZr−1 −→ Uk0
(∗)
with image Tk I(ε).
(8.2.5) Theorem (TATE). The homomorphism (∗) induces a topological isomorphism
∼ D0 .
exp : (IR/ZZ)r2 × (ZZ/ZZ)r−1 −→
k
For the connected component Dk of Ck , this yields a topological isomorphism
Dk ∼
= (IR/ZZ)r2 × (ZZ/ZZ)r−1 × IR .
Proof: The second statement follows from the first since Dk ∼
= Dk0 × IR×
+ . In
+
0
order to prove the first, observe that O (k) is a subgroup of Uk , and we obtain
a homomorphism
ZZr−1 −→ Uk0 /O+ (k) .
Since ελi i ∈ O+ (k) when λi = (n, n) ∈ ZZ ⊆ ZZ, this homomorphism factors
through (ZZ/ZZ)r−1 . We obtain a continuous homomorphism
expε : (ZZ/ZZ)r−1 −→ Ck0 ,
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449
§2. The Connected Component of Ck
whose image is contained in the connected component Dk0 since ZZ/ZZ is connected by (8.2.3).
λ
r−1
I. Injectivity of exp: Let α ∈ Tk and β = expε (λ) = ελ1 1 · · · εr−1
, where λ =
(λ1 , . . . , λr−1 ) ∈ ZZr−1 , λi = (zi , xi ). For the injectivity of exp we have to show:
suppose that αβ ∈ Uk0 is a principal idèle,
αβ = a ∈ k × ,
then λ ∈ ZZr−1 . Looking only at the components at the finite primes, we have
(1)
z
r−1
ε̄z11 · · · ε̄r−1
= a.
The group generated by ε1 , . . . , εr−1 has finite index in O+ (k) so that ad =
µr−1
εµ1 1 · · · εr−1
with µi ∈ ZZ. Raising (1) into the d-th power, we obtain
dz
r−1
ε̄1dz1 −µ1 · · · ε̄r−1
−µr−1
=1
and by lemma (8.2.4), dzi − µi = 0 in ẐZ. From this it follows that the zi are
contained in ZZ. Now
zr−1
ε = εz11 · · · εr−1
is an element in k which coincides with a in kp for p - ∞, hence ε = a. Looking
at the components at the infinite primes, we obtain from αβ = ε
x
z
r−1
r−1
α̃ε̃x1 1 · · · ε̃r−1
= εz11 · · · εr−1
.
Taking absolute values | |p for every p ∈ S∞ , we get
|ε1 |px1 −z1 · · · |εr−1 |xp r−1 −zr−1 = 1.
Taking the logarithms of these equations, we conclude that xi −zi = 0, since the
vectors (log |εi |p )p∈S∞ , i = 1, . . . , r − 1, are linearly independent by Dirichlet’s
unit theorem. This proves λi = (zi , xi ) ∈ ZZ, i.e. λ ∈ ZZr−1 , and thus the
injectivity, as desired.
II. Surjectivity of exp: Let D0 ⊆ Dk0 be the image of exp and let a ∈ Dk0 . For
the proof that a ∈ D0 we use the fact that a is divisible by (8.2.1)(v). So we
may write a = b2hm where h is the class number of k and m a highly divisible
integer. The ideal class group Clk is the quotient Ik /Vk k × with
Vk =
Y
p-∞
Up ×
Y
kp× ,
p|∞
so the class bh can be represented by an idèle in Vk , hence b2h by an idèle
β ∈ Uk . Since a ∈ Dk0 , we have β ∈ Uk0 . The element a is therefore represented
by the idèle β m = β̄ m β̃ m . Choosing a suitable highly divisible m, the idèle β̄ m
is contained in a neighbourhood of 1 which is as small as we like. We shall
prove that the idèle class c of β̃ m belongs to D0 . From this it follows that a is in
the closure of D0 . But D0 is compact, hence closed and consequently a ∈ D0 .
Set α̃ = β̃ m ∈ Uk0 . Let pν , ν = 1, . . . , r, range over the archimedean
primes. From the independence of the units ε1 , . . . , εr−1 it follows that the
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450
Chapter VIII. Cohomology of Global Fields
vectors ai = (log |εi |p1 , . . . , log |εi |pr−1 ), i = 1, . . . , r − 1, in IRr−1 are linearly
independent. Setting a = (log |α̃|p1 , . . . , log |α̃|pr−1 ), we find x1 , . . . , xr−1 ∈ IR
such that
x1 a1 + · · · + xr−1 ar−1 = a,
i.e.
|α̃|pν = |ε̃1 |xpν1 · · · |ε̃r−1 |xpνr−1
for ν = 1, . . . , r − 1.
Since |α̃| = 1, we also have
|α̃|pr = |ε̃1 |xpr1 · · · |ε̃r−1 |xprr−1 .
Let λi = (0, xi ) ∈ ZZ. Since α̃ is totally positive, we have
x
r−1
α̃p = (ε̃x1 1 · · · ε̃r−1
)p
for every real prime p. For complex p, both sides differ by an element of
value 1. Hence we can write
x
λ
r−1
r−1
α̃ = ε̃x1 1 · · · ε̃r−1
γ̃ = ελ1 1 · · · εr−1
γ̃
with γ̃ ∈ Tk . This proves that the class c of α̃ is contained in the image D0 of
exp as contended.
2
It is not unimportant to know also the cohomology of the connected component DK of a finite Galois extension K|k. It is given by the
(8.2.6) Corollary. Let K|k be a finite Galois extension with Galois group
G = G(K|k) and let m be the number of real primes of k that become complex
in K. Then
(
(ZZ/2ZZ)m
if i is even,
i
∼
Ĥ (G, DK ) =
0
if i is odd,
and H 0 (G, DK ) = Dk .
×
as a G-submodule of the
Proof: We may consider the product P∈SC KP
idèle group IK as well as of DK ⊆ CK . In this product we have the canonical
G-submodule
Y
1
TK =
SP
,
Q
P∈SC
1
SP
×
KP
×
= C . By the above theorem and by (8.2.3),
where
is the unit circle in
the quotient
0
DK
/TK ∼
= (ZZ/ZZ)r(K)−1
is uniquely divisible, hence cohomologically trivial. Therefore the exact cohomology sequence yields
0
Ĥ i (G, DK ) = Ĥ i (G, DK
) = Ĥ i (G, TK ).
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451
§2. The Connected Component of Ck
Let M be the set of primes p of k lying under the complex primes of K. For
every p ∈ M we choose a fixed prime P0 |p of K and consider the decomposition group GP0 . We regard the group IR/ZZ as a GP0 -module by letting
σ ∈ GP0 act by σθ = θ if σ = 1, and σθ = −θ if σ =/ 1. Then
×
IR/ZZ −→ KP
, θ 7−→ e2πiθ ,
0
1
is an injective GP0 -homomorphism with image SP
, and we obtain an injective
0
G-homomorphism
GP
GP
×
IndG 0 (IR/ZZ) −→ IndG 0 (KP
)=
0
Y
×
KP
= IK (p)
P|p
and hence an injective G-homomorphism
M
GP
IndG 0 (IR/ZZ) −→
p∈M
Y
IK (p)
p∈M
with image TK . Shapiro’s lemma now yields
Ĥ i (G, TK ) =
M
GP
Ĥ i (G, IndG 0 (IR/ZZ)) ∼
=
p∈M
=
M
M
Ĥ i (GP0 , IR/ZZ)
p∈M
Ĥ i (GP0 , IR/ZZ) ,
p∈N
where N is the set of real primes of k becoming complex in K. Let p ∈ N;
then GP0 is generated by an element σ of order 2. If i is even, then
Ĥ i (GP0 , IR/ZZ) ∼
= Ĥ 0 (GP0 , IR/ZZ) = (IR/ZZ)GP0 /NGP0 (IR/ZZ).
The fixed module consists of the elements θ ∈ IR/ZZ with σθ = −θ = θ, i.e. is
1
ZZ/ZZ, and the norm group consists of the elements NGP0 θ = θ+σθ = θ−θ = 0.
2
This proves Ĥ i (GP0 , IR/ZZ) = ZZ/2ZZ.
If i is odd, then
Ĥ i (GP0 , IR/ZZ) ∼
= Ĥ −1 (GP0 , IR/ZZ) = NGP (IR/ZZ)/(σ − 1)(IR/ZZ).
0
For θ ∈ IR/ZZ, we have (σ − 1)θ = σθ − θ = −2θ, so that (σ − 1)(IR/ZZ) =
2(IR/ZZ) = IR/ZZ, and thus Ĥ i (GP0 , IR/ZZ) = 0.
Finally, since TKG = Tk and NK|k DK ⊆ Dk , we obtain from Ĥ 0 (G, DK ) =
0
G
) = Ĥ 0 (G, TK ) the assertion DK
= TK NK|k DK ⊆ Dk , which gives
Ĥ 0 (G, DK
0
H (G, DK ) = Dk .
2
Exercise 1. Show that the solenoid ZZ/ZZ is the Pontryagin dual Homcts (Q, IR/ZZ) of Q. It may
n
also be identified with the projective limit lim S 1 over the maps S 1 → S 1 , z 7→ z n (n ∈ IN),
of the unit circle S 1 = {z
∈
C | |z| = 1}.
←− n
Exercise 2. The Pontryagin dual Hom(Dk , IR/ZZ) of the connected component Dk of Ck is
topologically isomorphic to ZZr2 × Qr−1 × IR, where r2 is the number of complex primes and
r is the number of all infinite primes.
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452
Chapter VIII. Cohomology of Global Fields
§3. Restricted Ramification
In the foreground of the Galois cohomological considerations in the previous
chapters has been the absolute Galois group Gk of the fields. In the case of a
global field k, cohomology theory gives much more subtle and deeper lying
arithmetic laws if we study not just the absolute Galois groups, but also Galois
groups “with restricted ramification”. This theory is of great importance, so
that we pay particular attention to its development.
Let k be a global field, which we regard as a fixed ground field. Let S be a
nonempty set of primes of k containing the set S∞ of infinite (i.e. archimedean)
primes if k is a number field. In some of the following formulas for a global
field k the expressions S∞ , SIR and SC will occur. If k is a function field, then
these terms should be redundant.
In place of the separable closure k̄|k, we now consider the maximal subextension kS |k which is unramified outside S. We denote its Galois group by
GS = G(kS |k).
If K|k is any finite subextension of kS |k, we set
GS (K) = G(kS |K).
We also let the symbol S stand for the set of primes of K which lie above the
primes in S. Thus P ∈ S means that P is a prime of K lying above a prime
p ∈ S of k. The ring of S-integers of K is defined by
OK,S = {a ∈ K | vP (a) ≥ 0
for all P ∈/ S}.
×
Its group of units OK,S
and its ideal class group ClS (K) play a particularly
important role. The last group is the quotient of the usual ideal class group
ClK of K by the subgroup generated by the classes of all prime ideals in S. It
is finite (in the function field case note that S is nonempty) and is called the
S-ideal class group. The ring of integers of kS is denoted by
OS =
[
OK,S ,
K|k
where the union is taken over all finite extensions K of k inside kS . Finally,
we set
×
IN(S) = {n ∈ IN | n ∈ Ok,S
}.
If k is a number field, then these are the natural numbers such that vp (n) = 0 for
all p ∈/ S. If k is a function field, then they are the numbers prime to char(k).
The following proposition is a special case of (6.1.10).
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453
§3. Restricted Ramification
(8.3.1) Proposition. Let K|k be a Galois extension with K
i
H (K|k, OK,S ) = 0 for all
⊆
kS . Then
i > 0.
(8.3.2) Corollary. If p = char(k) > 0 and L|k is a p-closed extension inside kS ,
then

for i = 1,
 Ok,S /℘Ok,S
i
H (L|k, ZZ/pZZ) =

0
for i > 1,
p
where ℘(x) = x − x.
Proof: Let K|k be a finite subextension of L|k and let a ∈ OK,S . If a is not of
the form αp − α for some α ∈ OK,S , consider the cyclic p-extension K(α)|K
given by f (α) = 0, f = X p − X − a. We have
(
−1 if p ≡ 1 mod 4,
p(p−1)/2
0
d(α) = disc(f ) = (−1)
Res(f, f ) =
+ 1 else.
Obviously, α lies in OK(α),S , i.e. α is an integer in K(α)P for P|p and p ∈/ S.
For these primes the discriminant of the field extension K(α)P |Kp divides
d(α) = ±1, whence the extension is unramified. Thus K(α)|K is unramified
outside S. Since L|k is a p-closed extension inside kS , we conclude that
℘ : OL,S → OL,S is surjective and obtain the exact sequence
℘
0 −→ ZZ/pZZ −→ OL,S −→ OL,S −→ 0 .
Taking cohomology, the result follows from (8.3.1).
2
Since constant field extensions are unramified, p∞ divides the order of
G(L|k). Applying the last corollary to the fixed field of L with respect to a
p-Sylow group of G(L|k), (3.3.6) implies the
(8.3.3) Corollary. If p = char(k) > 0 and if L|k is a p-closed extension inside
kS , then
cdp G(L|k) = 1.
For the group of units of OS , we have the exact Kummer sequence:
(8.3.4) Proposition. If n ∈ IN(S), then µn ⊆ kS and the group of units OS× of
n
OS is n-divisible, i.e. the map OS× −→ OS× , ε 7−→ εn , is surjective. In other
words, we have an exact sequence of GS -modules
n
0 −→ µn −→ OS× −→ OS× −→ 0.
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Chapter VIII. Cohomology of Global Fields
Proof: If p ∈/ S, then vp (n) = 0, hence kp (µn )|kp is unramified. Therefore
k(µn )|k is unramified outside S, i.e. µn ⊆ kS . Let ε ∈ OS× and let K =√
k(µn , ε).
n
For every finite prime P ∈/ S, ε and n are√units in KP , hence KP ( ε) | KP
is unramified. Therefore the extension K( n ε)|K is unramified outside S, i.e.
√
n
ε ∈ OS× .
2
In the sequel we denote the finite subextensions of kS |k by K|k. As in §1
we select a fixed embedding ip : kS ,→ k̄p for every prime p of k, in particular
a prime p̄ of kS , and denote by P. the prime of K lying under p̄. We set
kS,p = ip (kS )kp and
Kp = KP. = ip (K)kp .
We consider the S-idèle group
IK,S :=
Y
×
KP
.
P∈S
×
It contains the group of S-units OK,S
as a discrete subgroup and we set
×
CK,S = IK,S /OK,S
.
In spite of the analogy with the formation of the idèle class group CK = IK /K × ,
it is not this group which takes the role of CK in the “S-theory”. The reason
is the failure of Galois descent, i.e. if K|k is Galois, then Ck,S is not always
G(K|k)
the fixed module CK,S . It will become clear in a moment that we have to
consider the group
CS (K) = IK /K × UK,S
instead of CK,S , where UK,S is the compact subgroup
UK,S =
Y
{1} ×
P∈S
Y
UP
/S
P∈
of the full idèle group IK . Since K × ∩ UK,S = 1, we may regard UK,S as a
subgroup of CK = IK /K × and may also write
CS (K) = CK /UK,S .
This group is called the S-idèle class group. Note that if K|k is Galois,
then UK,S is a cohomologically trivial G(K|k)-module. Namely, by the same
argument as for (8.1.2), we have
Ĥ i (G(K|k), UK,S ) =
M
Ĥ i (G(KP. |kp ), UP. ),
/S
p∈
and since KP. |kp is unramified for p ∈/ S, Ĥ i (G(KP. |kp ), UP. ) = 1 by
(7.1.2)(i). The difference of the groups CK,S and CS (K) is given by the
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455
§3. Restricted Ramification
(8.3.5) Proposition. CK,S is an open subgroup of CS (K) and there is an exact
sequence
π
0 −→ CK,S −→ CS (K) −→ ClS (K) −→ 0.
In particular, CK,S = CS (K) if S omits only finitely many primes.
Proof: Consider the canonical injection IK,S → IK , α 7→ α̃, and the induced
homomorphism
j : IK,S −→ IK /K × UK,S = CS (K).
The image of j is closed. If α ∈ IK,S is an idèle such that j(α) = 1, then α̃ = aũ
with a ∈ K × , ũ ∈ UK,S . This means that
αP = α̃P = a
for P ∈ S
×
∈ OK,S .
and
1 = ãP = auP
for P ∈/ S,
×
OK,S
hence α = a and a
Therefore ker(j) =
and j induces an injection
CK,S ,→ CS (K) with closed image. Its cokernel is
IK /IK,S UK,S K × = (
M
×
(KP
/UP ))/im(K × ) = (
/S
P∈
M
ZZ)/im(K × ),
/S
P∈
which can be identified with the finite ideal class group ClS (K). In particular,
CK,S has finite index in CS (K) and is thus open.
If S omits only finitely many prime ideals, then OK,S is a Dedekind ring with
only finitely many prime ideals and is hence a principal ideal domain by [16],
chap.VII, §2, prop.1. Therefore in this case ClS (K) = 0 and CK,S = CS (K).
2
The proposition shows that the groups CK,S and CS (K) are in general
different. But they become equal in the limit over all K ⊆ kS . We set
IS = lim
IK,S ,
−→
K|k
CS = lim CK,S ,
−→
K|k
US = lim
UK,S ,
−→
K|k
C(kS ) = lim CK ,
−→
K|k
where K|k runs through all finite subextensions of kS |k. These are GS modules and US is a cohomologically trivial GS -module, since UK,S is a
cohomologically trivial G(K|k)-module if K|k is Galois. Taking the fixed
module of CS under GS (K) = G(kS |K), we do not recover CK,S but CS (K).
(8.3.6) Proposition. The GS -module CS has the following properties:
(i) CS = lim
CS (K) = C(kS )/US ,
−→
K|k
(ii)
CSGS (K)
= CS (K).
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456
Chapter VIII. Cohomology of Global Fields
Proof: (i) We have lim
−→ K|k
ClS (K) = 0, since every ideal in OK,S becomes
a principal ideal in a suitable finite unramified Galois extension L|K; if K
is a number field, we take the maximal subfield of the Hilbert class field of
K which is completely split at the primes in S (the principal ideal theorem,
see [160], chap.VI, (7.5)). The same argument holds in the function field case
since ClS (K) is finite for S =/ ∅. Therefore (i) follows from (8.3.5).
(ii) The exact sequence 0 → US → C(kS ) → CS → 0 yields the exact cohomology sequence
0 −→ UK,S −→ CK −→ CSGS (K) −→ H 1 (GS (K), US ) = 0,
2
hence CSGS (K) = CK /UK,S = CS (K).
If K|k is Galois, then we have the exact sequence of G(K|k)-modules
0 −→ UK,S −→ CK −→ CS (K) −→ 0.
Since UK,S is cohomologically trivial, we obtain the
(8.3.7) Proposition. For every finite Galois subextension K|k of kS |k and
every i ∈ ZZ,
Ĥ i (G(K|k), CS (K)) ∼
= Ĥ i (G(K|k), CK ).
We now compute the cohomology of the three GS -modules in the exact
sequence
0 −→ OS× −→ IS −→ CS −→ 0.
(8.3.8) Proposition (Cohomology of IS ).
(i) H 0 (GS , IS ) = Ik,S .
(ii) H 1 (GS , IS ) = 0.
(iii) H 2 (GS , IS )(p) =
M
Br(kp )(p)
for every prime number p ∈ IN(S).
p∈S
(iv) H 3 (GS , IS ) = 0.
Proof: (i) is a consequence of (8.1.2). For i
just as in §1
H i (G(K|k), IK,S ) =
M
≥
1 and K|k Galois, we obtain
×
H i (G(KP |kp ), KP
)
p∈S
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457
§3. Restricted Ramification
and furthermore
H i (GS , IS ) = lim
H i (G(K|k), IK,S ).
−→
K|k
From this (ii) follows because of Hilbert’s Satz 90. The assertion (iv) is a
× ∼
consequence of H 3 (G(KP |kp ), KP
) = H 1 (G(KP |kp ), ZZ) = 0 (see (7.1.7)).
For (iii) we have the invariant map of local class field theory
(∗)
×
∼
invp : H 2 (G(KP |kp ), KP
) −→
1
[KP :kp ]
ZZ/ZZ
for nonarchimedean primes p. If k is a function field, then we have constant
extensions of every degree, i.e. lim
of (∗) gives Q/ZZ. If k is a number field
−→ K|k
and p ∈ IN(S), then k(µp∞ )|k is an extension inside kS |k of degree divisible by
p∞ , hence
×
lim H 2 (G(KP |kp ), KP
)(p) = Qp /ZZp
−→
K|k
for nonarchimedean primes p. For p ∈ S∞ the group G(KP |kp ) is of order
√ 1 or
2 depending on whether KP = kp or KP =/ kp . If 2 ∈ IN(S), then K = k( −1)
×
is contained in kS , and so H 2 (G(KP |kp ), KP
) = 12 ZZ/ZZ if p is real. This proves
the proposition.
2
(8.3.9) Proposition (Cohomology of CS ). The pair (GS , CS ) is a class formation, and we have
(i) H 0 (GS , CS ) = CS (k),
(ii) H 1 (GS , CS ) = 0,
(iii) H 2 (GS , CS ) ∼
=
1
#GS
ZZ/ZZ,
(iv) H 3 (GS , CS ) = 0.
Proof: (i) is the assertion (ii) of (8.3.6). The other equalities follow from
H i (G(K|k), CS (K)) = H i (G(K|k), CK ) , i ≥ 1,
(see (8.3.7)) and from the fact that the Gk -module C = lim
−→ K|k
CK , where K|k
runs through all finite Galois subextensions of the separable closure k̄|k, is a
formation module by (8.1.22).
2
As a corollary of the proposition above and (3.1.6), we obtain the global
reciprocity law for restricted ramification.
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458
Chapter VIII. Cohomology of Global Fields
(8.3.10) Theorem Let K|k be a finite Galois extension inside kS with Galois
group G(K|k). Then there is a canonical isomorphism
CS (k)/NK|k CS (K) ∼
= G(K|k)ab .
(8.3.11) Proposition (Cohomology of OS× ).
×
(i) H 0 (GS , OS× ) = Ok,S
.
(ii) H 1 (GS , O× ) ∼
= ClS (k).
S
(iii) For every prime number p ∈ IN(S)
H 2 (GS , OS× )(p) = ker
M
H 2 (kp , µp∞ ) −→ H 2 (GS , CS )(p)
p∈S
∼
= ker
M
Σ
Qp /ZZp −→ Qp /ZZp
or
p∈S\S∞
∼
= ker
M
Q2 /ZZ2 ⊕
M
Σ
1
ZZ/ZZ −→ Q2 /ZZ2
2
p∈SIR
p∈S\S∞
according to whether p =/ 2 or p = 2 (if k is a function field, then the
expressions S∞ and SIR are redundant).
(iv) H 3 (GS , OS× )(p) = 0
for every prime number p ∈ IN(S).
Proof: (i) is trivial and the other statements follow from the exact cohomology
sequence
δ
−→ H i−1 (GS , CS ) −→ H i (GS , OS× ) −→ H i (GS , IS ) −→ H i (GS , CS ) −→ .
For i = 1 we obtain by (8.3.8) and (8.3.9) the exact sequence
δ
Ik,S −→ CS (k) −→ H 1 (GS , OS× ) −→ 0.
The cokernel of the left arrow is ClS (k) by (8.3.5), proving (ii).
If k is a function field or p =/ 2, then (iii) follows from the commutative diagram
0783456
H 2 (GS , OS× )(p)
H 2 (GS , IS )(p)
L
H 2 (GS , CS )(p)
invp
M
Qp /ZZp
Σ
Qp /ZZp ,
p∈S\S∞
where the lower horizontal map is the summation over the components and
the zero on the left in the upper row is a consequence of H 1 (GS , CS ) = 0. If
p = 2 and k is a number field, then the same argument gives the result using
L
L
H 2 (GS , IS )(2) ∼
= p∈S\S∞ Q2 /ZZ2 ⊕ p∈SIR 12 ZZ/ZZ.
Finally, (iv) follows from the surjectivity of the upper right-hand arrow and
from the equality H 3 (GS , IS ) = 0.
2
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459
§3. Restricted Ramification
(8.3.12) Corollary. Let k be a global field and let p ∈ S be a prime number.
(i) If k is a function field or p =/ 2, then H 2 (GS , OS× ) is p-divisible.
(ii) If k is a number field and p = 2, then the canonical map
∼
H 2 (GS , OS× )/2 −→
M
H 2 (kp , k̄p× )
p∈SIR
is an isomorphism. In particular, H (GS , OS× ) is 2-divisible if and only
if k is totally imaginary.
2
(iii) If k is a number field and p = 2, then the restriction map
M
∼
H 3 (GS , ZZ/2ZZ) −→
H 3 (kp , ZZ/2ZZ)
p∈SIR
is an isomorphism.
Proof: The first two assertions follow immediately from the fact that the
group H 2 (GS , OS× )(p) is isomorphic to the kernel of the summation over the
local components.
Using (8.3.11)(iv), it follows from the exact sequence
2
0 −→ µ2 −→ OS× −→ OS× −→ 0
that H 2 (GS , OS× )/2 ∼
= H 3 (kp , ZZ/2ZZ)
= H 3 (GS , ZZ/2ZZ). Since H 2 (kp , k̄p× ) ∼
for p ∈ SIR , we obtain the last assertion from (ii).
2
The groups CS (k) are locally compact topological groups exactly like Ck .
Concerning its connected component we have the
(8.3.13) Theorem. Assume that k is a number field. The connected component
DS (k) of CS (k) = Ck /Uk,S is given by
DS (k) = Dk Uk,S /Uk,S .
It is divisible, and there is an exact sequence of topological groups
09:;<
DS (k)
CS (k)
( ,kS |k)
Gab
S
0.
In particular, DS (k) is the group of universal norms NGS CS .
Proof: Consider the continuous projection
(∗)
Ck −→ CS (k) = Ck /Uk,S .
For general topological reasons, the connected component DS (k) of CS (k) is
the closure of the image Dk Uk,S /Uk,S of the connected component Dk of Ck .
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460
Chapter VIII. Cohomology of Global Fields
But this image is already closed, since Uk,S is compact and hence (∗) is a proper
map. This proves the first assertion. The divisibility follows from that of Dk .
For the second statement we use the exact sequence (8.2.2):
0@=>?
Dk
Ck
Gab
0.
For each nonarchimedean prime p the norm residue symbol ( , k) maps the
subgroup kp× ⊆ Ck onto the decomposition group of Gab
k with respect to p and
the group of units Up ⊆ kp× onto the inertia group (see [160], chap.VI, (5.6) and
chap.V, (6.2)). Therefore the compact group Uk,S is mapped onto the subgroup
ab
H of Gab
k = G(k |k) which is generated by the inertia groups for the primes
not in S. The fixed field of H is therefore the maximal subextension kSab |k
of k ab |k which is unramified outside S. This gives us an exact commutative
diagram
ABCDEFGH
Uk,S
G(k ab |kSab )
0
( ,k)
0
Dk
Ck
G(k ab |k)
0
of topological groups. The snake lemma yields an exact sequence of topological groups
0 −→ Dk Uk,S /Uk,S −→ Ck /Uk,S −→ G(kSab |k) −→ 0,
and since G(kSab |k) = G(kS |k)ab = Gab
S , the theorem is proved.
2
(8.3.14) Corollary. Let k be a number field. For a subgroup N of CS (k) the
following conditions are equivalent:
(i) N is the norm group NK|k CS (K) of a finite Galois extension K|k inside
kS |k.
(ii) N is an open subgroup of CS (k).
Each open subgroup N of CS (k) contains DS (k) and has finite index.
Proof: By (8.3.9), the pair (GS , CS ) is a class formation. The isomorphism
(1)
∼ Gab
CS (k)/DS (k) −→
S
is the projective limit of the isomorphisms of finite groups
(2)
CS (k)/NK|k CS (K) ∼
= G(K|k)ab ,
where K|k runs through the finite Galois subextensions of kS |k. Therefore the
norm groups are precisely the open subgroups of finite index in CS (k) which
contain DS (k). It remains to show that each open subgroup in CS (K) has finite
index and contains DS (k). As DS (k) is divisible, it suffices to show that each
open subgroup in CS (k) has finite index.
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461
§3. Restricted Ramification
As CS (k) is a topological quotient of Ck , we may assume that S is the set
of all primes. Let W ⊆ Ck be an open subgroup. The pre-image of W in Ik is
open and therefore contains a subgroup of the form
V =
Y
Vp ,
p
where Vp ⊆ Up is an open subgroup for all p, Vp = Up for almost all p and
Vp ⊆ Up1 for archimedean p. For nonarchimedean p, an open subgroup in
Up has finite index and for archimedean p the group Up1 has no proper open
subgroup. Hence V has finite index in the group
Uk =
Y
Up ×
p-∞
Y
Up1
⊆
Ik ,
p|∞
and it suffices to show that the image of Uk in Ck has finite index. We have
M
M
Ik /Uk ∼
ZZ ⊕
ZZ/2ZZ,
=
/S∞
p∈
p∈SIR
which gives an isomorphism
Ik /k × Uk ∼
= Cl0 (k)
where Cl0 (k) is the ideal class group of k in the narrow sense, which is known
to be finite. This concludes the proof.
2
For a function field k we have a continuous injection
CI k
( ,k)
Gab
k .
It is not surjective, since in Ck we have the group Ck0 = {x ∈ Ck | |x| = 1}, and
the quotient Ck /Ck0 ∼
= ZZ is not profinite. With the same arguments as above
in this situation, we obtain the
(8.3.15) Theorem. If k is a function field, then we have an exact sequence of
topological groups
0LMJK
CS (k)
( ,kS |k)
Gab
S
ẐZ/ZZ
0.
(8.3.16) Corollary. Let k be a function field. For a subgroup N of CS (k) the
following conditions are equivalent:
(i) N is the norm group NK|k CS (K) of a finite Galois subextension K|k of
kS |k.
(ii) N is an open subgroup of finite index.
(iii) N is an open subgroup which is not contained in the group
CS0 (k) = {x ∈ CS (k) | |x| = 1}.
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Chapter VIII. Cohomology of Global Fields
As before let k be a global field, S a nonempty set of primes of k containing
the set S∞ of infinite primes if k is a number field, and let kS |k be the maximal
extension unramified outside S. Now we study the cohomological dimension
of GS . In the function field case we have the following result (for the case
S = ∅, which is not considered here, we refer to X §1).
(8.3.17) Theorem. If k is a function field, then for all prime numbers p
scdp GS = 2.
Proof: Since the constant field extensions are contained in kS , every prime
number p divides the order of GS , hence cdp GS > 0 and therefore scdp GS ≥ 2,
because the strict cohomological dimension is never 1. It therefore suffices to
prove scd GS ≤ 2. By (3.6.4), this is equivalent to the existence of a levelcompact formation module for GS with trivial universal norms. We construct
such a formation module from the GS -module CS = C(kS )/US , which is a
formation module by (8.3.9).
Let L|K be a finite normal intermediate extension in kS |k. The norm group
N = NL|K CL is an open subgroup of finite index in CK . The property of being
×
unramified at a prime p is equivalent to Up ⊆ NLP |Kp L×
P = N ∩ Kp by class
field theory (see [160], chap.VI, (5.8)). This means that the norm groups of
finite Galois subextensions L|K of kS |K are open subgroups N of finite index
in CK containing UK,S = UK,S K × /K × . By the existence theorem of class field
theory, they are precisely all such groups (8.3.16), and hence UK,S K × /K × is
their intersection. Therefore CS (K) = CK /UK,S becomes a dense subgroup
of the compact group
C S (K) = lim
CK /N.
←−
N
The canonical surjective map
deg : CS (K) → ZZ,
deg(α) =
X
vp (αp ) deg(p) ,
p
extends to C S (K) and yields an exact commutative diagram
0QRSTNOPUVWX
CS0 (K)
CS (K)
deg
0
CS0 (K)
C S (K)
deg
ZZ
0
ẐZ
0,
which shows that the quotient C S (K)/CS (K) is isomorphic to ẐZ/ZZ and is
consequently uniquely divisible. Because of this fact, for a normal intermediate extension L|K with group G, the G-module C S (L)/CS (L) has trivial
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463
§3. Restricted Ramification
cohomology, so that
H i (G, C S (L)) = H i (G, CS (L))
for i ≥ 1,
and from CS (L)G = CS (K) it follows that C S (L)G = C S (K). This shows that
C̄S (K) is a formation module since CS is.∗) It
the G-module C S = lim
is level-compact
because
\
−→ K|k
because C̄SGS (K)
= C S (K), and it has trivial universal norms
NL|K CL = UK,S K × /K × ,
i.e.
K ⊆L⊆kS
\
NL|K C S (L) = 0.
K ⊆L⊆kS
C S is the desired formation module and the theorem is proved.
2
Now we assume that k is a number field. In this case the existence of
the infinite primes causes severe difficulties for the determination of the strict
cohomological dimension scdp GS . However, the cohomological dimension
cdp GS is more easy to handle.
For a prime number p we put
Sp = Sp (k) = {p a prime of k dividing p} .
(8.3.18) Proposition. Let Sp ∪ S∞
totally imaginary if p = 2. Then
⊆
S and assume that the number field k is
cdp GS
≤
2.
Proof: kS contains the group µp of p-th roots of unity since Q(µp )|Q, and
hence also k(µp )|k, is unramified outside p. The group OS× of S-units of kS
contains µp and is p-divisible by (8.3.4), i.e. the sequence
p
0 −→ µp −→ OS× −→ OS× −→ 0
(∗)
is exact. Let K|k be any finite subextension of kS |k. Proposition (8.3.11)
shows that H 3 (GS (K), OS× )(p) = 0, and by (8.3.12) the group H 2 (GS (K), OS× )
is p-divisible. Therefore the exact cohomology sequence associated to (∗)
yields H 3 (GS (K), µp ) = 0. Now let Gp be a p-Sylow subgroup of GS . Let
Σp be its fixed field and let K|k run through the finite subextensions of Σp |k.
Noting that µp ⊆ Σp , we get
H 3 (Gp , ZZ/pZZ) ∼
H 3 (GS (K), µp )(−1) = 0,
= H 3 (Gp , µp )(−1) = lim
−→
K
hence cdp GS =
cdp Gp ≤
2.
∗) If S = ∅, then C is no longer a formation module.
S
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Chapter VIII. Cohomology of Global Fields
(8.3.19) Proposition. Let S2 ∪ S∞ ⊆ S and let K be an (possibly infinite)
extension of the number field k inside kS . Then cd2 GS (K) = ∞ if and only if
K has a real place, and cd2 GS (K) ≤ 2 otherwise.
√
Proof: As S2 ∪ S∞ ⊆ S the field kS contains K( −1). Therefore, if K has a
real place, then GS (K) contains its decomposition group which is isomorphic
to ZZ/2ZZ. Hence cd2 GS (K) = ∞ by (3.3.5).
If cd2 GS (K) = ∞, then H n ((GS (K))2 , ZZ/2ZZ) =/ 0 for all n and every 2Sylow subgroup (GS (K))2 of GS (K) by (3.3.6) and (3.3.2)(iii). We therefore
find a finite extension K 0 |K inside kS such that H 3 (GS (K 0 ), ZZ/2ZZ) =/ 0. From
(8.3.12)(iii) we obtain
0 =/ H 3 (GS (K 0 ), ZZ/2ZZ) ∼
= lim
−→
L
M
H 3 (Lp , ZZ/2ZZ) ,
p∈SIR (L)
where L runs through the finite extensions of k in K 0 . We conclude that K 0 ,
and hence also K, has at least one real place.
Finally, √
assume that cd2 GS (K) is finite.
Then cd2 GS (K) is equal to
√
But K( −1) is the union of totally imaginary
cd2 GS (K( −1)) by (3.3.5)(ii). √
number fields, hence cd2 GS (K( −1)) ≤ 2 by (8.3.18).
2
Remark: The assumptions of (8.3.18) may be weakened to Sp ⊆ S and, if
p = 2, S contains no real primes. The statement of (8.3.19) may be sharpened
in the following way: Assume that S2 ⊆ S; then cd2 (GS (K)) = ∞ if and only
if S contains a real prime, and cd2 (GS (K)) ≤ 2 otherwise, see (10.6.1).
Together with (8.3.11), we obtain from (8.3.18) the following finiteness
theorem.
(8.3.20) Theorem. Let A be a GS -module which is finitely generated as a
ZZ-module and whose torsion submodule tor(A) is of an order which is a unit
in Ok,S . Then the following hold:
(i) If S and A are finite, then the cohomology groups H n (GS , A) are finite
for all n ≥ 0.
(ii) If S is finite or A is torsion-free, then H 1 (GS , A) is finite.
Proof: (i) Let K|k be a finite Galois subextension of kS |k over which A
becomes a trivial Galois module and which contains the n-th roots of unity,
where n = #A, and which is totally imaginary in the number field case. Then
A is isomorphic over K to a direct sum of modules µpν , p|n. If the assertion
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§3. Restricted Ramification
holds for A = µpν and base field K, the general case follows using the spectral
sequence
H i (G(K|k), H j (GS (K), A)) ⇒ H i+j (GS , A).
Therefore let A = µpν . By induction, using the exact sequence 0 → µp → µpν
→ µpν−1 → 0 and its associated exact cohomology sequence, we may assume
A = µp with p ∈ IN(S). Therefore, as in the proof of (8.3.18), we have the
exact sequence
p
0 −→ µp −→ OS× −→ OS× −→ 0,
and the associated exact cohomology sequence
p
H i−1 (GS , OS× ) −→ H i−1 (GS , OS× ) −→ H i (GS , µp ) −→ . . .
p
. . . −→ H i (GS , OS× ) −→ H i (GS , OS× ).
×
× p
For i = 1 we obtain the finiteness of H 1 (GS , µp ) since Ok,S
/(Ok,S
) and
× ∼
1
H (GS , OS ) = ClS (k) are finite. For i = 2 we use the finiteness of the set
p
S and see by (8.3.11)(iii) that the kernel of H 2 (GS , OS× ) → H 2 (GS , OS× ) is
finite, and thus also H 2 (GS , µp ), again using H 1 (GS , OS× ) ∼
= ClS (k). Since
n
≥
H (GS , µp ) = 0 by (8.3.18) for n 3, part (i) of the theorem is proved.
(ii) Let A be torsion-free and let K|k be a finite Galois subextension of kS |k
over which A becomes a trivial Galois module, i.e. A = ZZr for some r ≥ 0.
Then the exact sequence
0 −→ H 1 (G(K|k), A) −→ H 1 (GS , A) −→ H 1 (GS (K), ZZr ) = 0
gives the desired result. If S is finite and is A arbitrary, then the exact sequence
H 1 (GS , tor(A)) −→ H 1 (GS , A) −→ H 1 (GS , A/tor(A))
2
gives the result using (i) and the result for torsion-free modules.
Let N run through the open subgroups of finite index in Ik,S and put
I¯k,S = lim Ik,S /N,
←−
N
i.e. I¯k,S is the profinite completion of Ik,S . We can collect many of the results
obtained in the form of two commutative exact diagrams.
(8.3.21) Corollary.
(i) Let k be a function field. Then we have the commutative exact diagram
0fdYZ[\]^_`abce
×
Ok,S
dense
0
×
Ok,S
Ik,S
dense
I¯k,S
CS (k)
ClS (k)
0
ClS (k)
0.
dense
GS (k)ab
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Chapter VIII. Cohomology of Global Fields
(ii) Let k be a number field. Then we have the commutative exact diagram
vwxughijklmnopqrst
M
NGp k̄p×
DS (k)
Ik,S
CS (k)
ClS (k)
0
GS (k)ab
ClS (k)
0.
p∈S∞
0
×
Ok,S
dense
0
×
Ok,S
dense
I¯k,S
×
×
Here Ok,S
denotes the closure of the image of Ok,S
in I¯k,S . The left-hand
vertical arrow in (ii) is injective if S contains at least one nonarchimedean
prime.
∼
Remark: If S is finite, local class field theory induces an isomorphism I¯k,S →
Q
Q
ab
ab
∼
¯
ˆ p∈S Gk , where
p∈S Gkp . For infinite S, we obtain an isomorphism Ik,S →
p
the latter group is the free corestricted proabelian product of the groups Gab
kp
with respect to their inertia subgroups; see [154].
Exercise 1. Show the isomorphism H 2 (GS , OS× ) ∼
= Br(kS |k).
Exercise 2. Let k be a number field. Let K|k be a finite Galois subextension of kS |k with
Galois group G, and let DS (K) = DK ∩ UK,S . If k is totally imaginary, then
(i) H 0 (G, DS (K)) = DS (k),
(ii) H 0 (G, DS (K)) = DS (k) ⇐⇒ H 1 (G, DS (K)) = 0,
(iii) Ĥ i (G, DS (K)) ∼
= Ĥ i (G, DS (K)) for all i ∈ ZZ.
Hint: DK is a cohomologically trivial G-module.
×
Exercise 3. Show that the group Ck,S = Ik,S /Ok,S
contains the connected component Dk
of Ck .
§4. The Global Duality Theorem
In this section we consider the profinite group
GS = G(kS |k)
of the maximal extension kS |k which is unramified outside the given nonempty
set of primes S (where S∞ is contained in S if k is a number field). The abstract
duality theorem (3.1.9) applies at once to any Galois extension K|k inside kS
in view of (8.3.9).
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§4. The Global Duality Theorem
(8.4.1) Theorem. Let K|k be a Galois extension which is unramified outside S,
let G be its Galois group and let A be a finitely generated and ZZ-free G-module.
Then for all i ∈ ZZ the cup-product
∪
inv
Ĥ i (G, Hom(A, CS (K))) × Ĥ 2−i (G, A) −→ H 2 (G, CS (K)) −→ Q/ZZ
yields a topological isomorphism
Ĥ i (G, Hom(A, CS (K))) ∼
= Ĥ 2−i (G, A)∨ .
If K|k is finite, these are isomorphisms of finite abelian groups.
Remark: Recall that if K|k is infinite, the topologies on the occurring cohomology groups are defined as the direct resp. projective limit topologies of
the groups on the finite levels, which are finite and endowed with the discrete
topology. The topology of the S-idèle class group does not play any role here.
The theorem (8.4.1), concerning only ZZ-free G-modules, is too rigid for
the applications that we have in mind. In the local case we have proved a
cohomological duality for finite Galois modules A and A0 = Hom(A, k̄ × ). We
would like to prove an analogous theorem for the global Galois group GS ,
where the idèle class group CS offers itself as an analogue of the multiplicative
group k̄ × . However, for several reasons CS cannot take over the role of k̄ × :
we cannot apply the duality theorem (3.4.3) of Tate, since on the one hand it is
not known whether scd GS ≤ 2 if S is finite and k is a number field, and on the
other hand for large sets S (e.g. S has density equal to 1) the module CS is not
divisible. The duality theorem (3.4.6) seems to be more convenient, since one
only needs the cohomological dimension of GS , but again there is a problem
with the divisibility of the dualizing module of GS . We will see in X §9 that
in fact CS is p-divisible if p ∈ S and S is finite. But this will be a consequence
of the results of this and the following sections.
We have still the abstract duality theorem (3.1.11)(i), which does not require
the divisibility of CS . Instead it requires a level-compact formation module
and divisible universal norm groups. CS satisfies the last condition, not the
first one. But we may pass to a modified formation module satisfying both
conditions without changing the cohomology. This was the idea of G. POITOU
and K. UCHIDA. In what follows the reader should recall the topological remarks
before (3.1.11).
As before, K|k denotes the finite subextensions of kS |k. We consider the
GS -module CS = C(kS )/US with the fixed modules
CS (K) = CSGS (K) = CK /UK,S .
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Chapter VIII. Cohomology of Global Fields
For the following discussion we have to distinguish between the case of number
fields and of function fields. First let k be a number field. UK,S is contained in
the compact group
||
0
CK
= ker(CK −→ IR×
+ ),
and we replace CS (K) by
0
CS0 (K) = CK
/UK,S .
The quotient CS (K)/CS0 (K) ∼
= IR×
+ is uniquely divisible. We set
CS0 = lim
CS0 (K) = C 0 (kS )/US
−→
K|k
where K|k runs through the finite subextensions of kS |k.
(8.4.2) Proposition. Let k be a number field. Then the following is true.
(i) The GS -module CS0 is a level-compact formation module with divisible
universal norm group NkS |k CS0 .
(ii) If A is a GS -module which is finitely generated as a ZZ-module, then for all
i ∈ ZZ the natural map CS0 → CS induces an isomorphism
∼ Ĥ i (G , Hom(A, C )).
Ĥ i (GS , Hom(A, CS0 )) −→
S
S
Proof: The GS -module US is cohomologically trivial as mentioned on several previous occasions. Therefore, applying H 0 (G(kS |K), −) to the exact
sequence
0 −→ US −→ C 0 (kS ) −→ CS0 −→ 0,
we obtain the exact sequence
0
0 −→ UK,S −→ CK
−→(CS0 )G(kS |K) −→ 0,
and so
(CS0 )G(kS |K) = CS0 (K).
Thus CS0 is a level-compact GS -module. By (8.3.13), NkS |K CS = DS (K) is
||
divisible, hence also DS0 (K) = ker(DS (K) −→ IR×
+ ). Since |NK 0 |K (x)| = 1 if
0
0
and only if |x| = 1 for x ∈ CS (K ), where K |K is a finite subextension of
kS |K, we get NkS |K CS0 = DS0 (K), and hence NkS |K CS0 is divisible.
Now let A be a GS -module which is finitely generated as a ZZ-module and
let K|k be a finite Galois extension inside kS over which A becomes trivial.
Consider the exact sequence of G(K|k)-modules
0 −→ CS0 (K) −→ CS (K) −→ Q(K) −→ 0
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§4. The Global Duality Theorem
in which Q(K) = CS (K)/CS0 (K) ∼
= IR is uniquely divisible. It follows that
Hom(A, Q(K)) is also uniquely divisible and the sequence
0 −→ Hom(A, CS0 (K)) −→ Hom(A, CS (K)) −→ Hom(A, Q(K)) −→ 0
is exact. We therefore obtain natural isomorphisms
∼ Ĥ i (G(K|k), Hom(A, C (K)))
Ĥ i (G(K|k), Hom(A, CS0 (K))) −→
S
2
for all i ∈ ZZ.
For function fields there is the following similar result. For each finite
subextension K|k of kS |k consider the compact group
CK /N,
C S (K) = lim
←−
N
where N runs through the open subgroups of finite index in CK which contain
UK,S . As before let
C S = lim
C S (K).
−→
K|k
(8.4.3) Proposition. Let k be a function field. Then the following is true.
(i) The GS -module C S is a level-compact formation module with trivial universal norm group NkS |k C S = 0.
(ii) If A is a GS -module which is finitely generated as a ZZ-module, then for all
i ∈ ZZ the natural homomorphism CS → C S induces an isomorphism
∼ Ĥ i (G , Hom(A, C )).
Ĥ i (GS , Hom(A, CS )) −→
S
S
Proof: The first assertion was already shown in the proof of (8.3.17) and the
second follows in the same way as in (8.4.2) using the exact sequence
0 −→ CS (K) −→ C S (K) −→ ẐZ/ZZ −→ 0,
where K|k is a finite Galois extension inside kS trivializing A. Since ẐZ/ZZ is
uniquely divisible, we obtain isomorphisms
∼ Ĥ i (G(K|k), Hom(A, C (K)))
Ĥ i (G(K|k), Hom(A, CS (K))) −→
S
for all i ∈ ZZ.
2
Applying the duality theorem (3.1.11)(i) to an arbitrary global field, we
obtain the
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Chapter VIII. Cohomology of Global Fields
(8.4.4) Theorem. Let A be a discrete GS -module which is finitely generated
as a ZZ-module. If k is a number field, then the pairing
∪
Ĥ i (GS , Hom(A, CS0 )) × Ĥ 2−i (GS , A) −→ H 2 (GS , CS0 ) ∼
=
1
ZZ/ZZ
#GS
induces topological isomorphisms
Ĥ i (GS , Hom(A, CS0 )) ∼
= Ĥ 2−i (GS , A)∨
for all i ≤ 0. This assertion holds for all i ∈ ZZ if A is assumed to be ZZ-free.
If k is a function field, then the statements above hold with CS0 replaced by C S .
Remark: In order to apply also part (ii) of (3.1.11) to (GS , CS0 ) (resp. (GS , C S ))
one has to investigate the question of finding those primes p for which the group
CS is p-divisible. We will do this in chapter X, see (10.11.8).
We formulated the theorem with CS0 (resp. C S ) instead of CS to obtain the
additional information that the duality isomorphisms are topological. If one
wants to define a topology on the cohomology groups of CS (A) directly, one
runs into the problem that the category of locally compact abelian groups is not
abelian. While it is possible to overcome this technical difficulty, we restrict
our considerations here to the case of cohomology in dimension zero:
As before, let A be a discrete GS -module which is finitely generated as a
ZZ-module. For U ⊆ GS open, we endow the groups CS (A)U with a natural
(locally compact) topology: For sufficiently small V we have AV = A and
therefore
CS (A)V = Hom(A, CSV ) .
Considering A as a group with the discrete topology, Hom(A, CSV ) endowed
with the compact open topology is locally compact. For an arbitrary open
subgroup U ⊆ GS , the group Hom(A, CS )U is a closed subgroup of the locally
compact group Hom(A, CS )V , where V is chosen sufficiently small. For
U ⊆ GS the norm map NGS /U : CS (A)U → CS (A)GS is continuous. If U is
normal in GS , we endow Ĥ 0 (GS /U, CS (A)U ) with the quotient topology from
H 0 (GS /U, CS (A)U ) = CS (A)GS . Finally, we endow
Ĥ 0 (GS , CS (A)) = lim Ĥ 0 (GS /U, CS (A)U )
←−
U
with the projective limit topology. Note that the natural homomorphism
H 0 (GS , CS (A)) −→ Ĥ 0 (GS , CS (A))
is continuous and has dense image.
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§4. The Global Duality Theorem
(8.4.5) Proposition. The algebraic isomorphisms
∼ Ĥ 0 (G , C (A)),
Ĥ 0 (GS , CS0 (A)) −→
S
S
given by (8.4.2) if k is a number field, and
∼ Ĥ 0 (G , C (A)),
Ĥ 0 (GS , CS (A)) −→
S
S
given by (8.4.3) if k is a function field, are homeomorphisms. The composite
∼ H 2 (G , A)∨
CS (A)GS −→ Ĥ 0 (GS , CS (A)) −→
S
is continuous and has dense image. It is surjective if k is a number field or if
A is finite.
Proof: Let K = kSU be a finite Galois extension of k inside kS such that
AU = A. We first show that the norm map NGS /U : CS (A)U → CS (A)GS has
closed image. From this follows that Ĥ 0 (GS , CS (A)) is Hausdorff.
We start with the number field case. Let v be an archimedean prime of k and
let Γk be the image of the subgroup of positive real numbers under the injection
kv× ,→ CS (k). Let ΓK be the image of Γk in CS (K). Then we have an algebraic
decomposition of CS (A)U into a product of closed G(K|k)-submodules
CS (A)U ∼
= C 0 (A)U × Hom(A, ΓK ) .
S
CS0 (A)U
As
is compact and the norm is continuous, NGS /U CS0 (A)U is a
compact subgroup of CS (A)GS . The norm map NGS /U : Hom(A, ΓK ) →
HomGS (A, ΓK ) is a homomorphism of finite dimensional IR-vector spaces and
has closed image. Therefore
NGS /U CS (A)U = NGS /U CS0 (A)U × NGS /U Hom(A, ΓK )
is a closed subgroup of CS (A)GS .
In the function field case, we proceed in a similar way. Let CS0 ⊆ CS be the
subgroup of idèle classes of degree zero. CS0 is a level-compact GS -module.
Let x ∈ CS (k) be a pre-image of 1 under the degree map deg : CS (k) ZZ,
and let Γk ∼
= ZZ be the discrete subgroup in CS (k) generated by x. Denote by
ΓK the image of Γk in CS (K). Then
CS0 (A)U × Hom(A, ΓK )
is an open subgroup of finite index in CS (A)U . The norm group NGS /U CS0 (A)U
is compact and NGS /U Hom(A, ΓK ) is discrete as a subgroup of the discrete
group HomGS (A, Γk ). We conclude that
NGS /U CS0 (A)U × NGS /U Hom(A, ΓK ) ⊆ Hom(A, CS )GS
is closed, as it is the product of a compact subgroup and a discrete subgroup.
Therefore NGS /U CS (A)U , containing a closed subgroup of finite index, is
closed.
In the number field case, we conclude that the isomorphism
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(∗)
Chapter VIII. Cohomology of Global Fields
∼ Ĥ 0 (G(K|k), Hom(A, C (K)))
Ĥ 0 (G(K|k), Hom(A, CS0 (K))) −→
S
obtained in the proof of (8.4.2) is a continuous bijection from a compact group
onto a Hausdorff group, and therefore a homeomorphism.
Now let k be a function field. In the proof of (8.4.3) we obtained the
continuous bijection
(∗∗)
∼ Ĥ 0 (G(K|k), Hom(A,
Ĥ 0 (G(K|k), Hom(A, CS (K))) −→
C S (K))).
The group CS (K)/CS0 (K) is a trivial G(K|k)-module isomorphic to ZZ. Therefore the image of the natural homomorphism
Ĥ 0 (G(K|k), Hom(A, CS0 (K))) −→ Ĥ 0 (G(K|k), Hom(A, CS (K)))
is a compact subgroup of finite index. Hence Ĥ 0 (G(K|k), Hom(A, CS (K))) is
compact and (∗∗) is a homeomorphism.
Passing to the projective limit over all K in (∗) or (∗∗), respectively, we obtain
∼ H 2 (G , A)∨
the first result, and the composite CS (A)GS → Ĥ 0 (GS , CS (A)) →
S
is continuous and has dense image.
If k is number field, then the module Hom(A, CS0 ) is level-compact, and
so CS0 (A)GS → Ĥ 0 (GS , Hom(A, CS0 )) is surjective by (1.9.11). It follows that
CS (A)GS → Ĥ 0 (GS , CS (A)) is surjective. If k is a function field and A is
finite, then the module Hom(A, CS ) = Hom(A, C S ) is level-compact, and
CS (A)GS → Ĥ 0 (GS , CS (A)) is also surjective in this case.
2
§5. Local Cohomology of Global Galois Modules
As a bridge from local to global cohomology, we shall now introduce an
idèlic cohomology for GS -modules, GS again being the Galois group G(kS |k).
We start with some general observations concerning the passage from global
to local Galois modules.
Let k be a global field, p a prime of k, K|k a (possibly infinite) Galois
extension and P a prime of K above p. The field
KP =
[
kp0 0 ,
k0
0
where k runs through the finite extensions of k inside K and p0 = P ∩ k 0 , is
a (possibly infinite) Galois extension of the local field kp . The Galois group
G(KP |kp ) is naturally isomorphic to the decomposition group GP (K|k) of P
in G(K|k). We fix a separable closure K P of KP (which is also a separable
closure of kp ).
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§5. Local Cohomology of Global Galois Modules
Now let A be a discrete G(K|k)-module. We consider A as a G(K P |kp )module via the natural projection
G(K P |kp ) GP (K|k).
As any two extensions P1 and P2 of p to K are conjugate, and as inner automorphisms of groups act trivially on group cohomology, the groups
H i (G(K P1 |kp ), A) and H i (G(K P2 |kp ), A) are canonically isomorphic for all i.
We denote them by
H i (kp , A),
hiding all choices from the notation. We also write the inclusion of K to K P
in the form K ,→ k̄p . There are canonical homomorphisms
H i (G(K|k), A) −→ H i (kp , A)
for all i ≥ 0. If A is trivialized by a finite extension k 0 of k, which happens
for example if A is finitely generated as a ZZ-module, then A is an unramified
Gkp -module for all primes p which are unramified in k 0 |k. In particular, A is
an unramified Gp -module for almost all primes p of k.
We will apply the above considerations to the case K = kS . Recall the
notation
×
}.
IN(S) = {n ∈ IN | n ∈ Ok,S
(8.5.1) Definition. We denote by ModS (GS ) the category of discrete GS modules A which are finitely generated as ZZ-modules and whose torsion has
order #tor(A) ∈ IN(S).
For a finite A ∈ ModS (GS ), we define the dual GS -module of A by
A0 = Hom(A, OS× ) = Hom(A, µ),
where OS× is the group of S-units of kS and µ is the group of roots of unity in
kS . As n = #A is in IN(S) by assumption, kS contains all n-th roots of unity.
Therefore A0 is finite of the same order as A and contained in ModS (GS ).
Furthermore, we have a canonical identification A00 = A.
We set
P i (GS , A) =
Y
H i (kp , A) ,
i ≥ 0,
p∈S
i
where the restricted product is taken with respect to the subgroups Hnr
(kp , A) =
i
i
i
im(H (k̃p |kp , A) → H (kp , A)) of H (kp , A), where k̃p is the maximal unramified extension of kp , see (7.2.14). In view of the duality theorem (7.2.17) for
the groups Ĥ i (kp , A) at the archimedean primes p, we make the following
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Chapter VIII. Cohomology of Global Fields
Convention: In the case i = 0 we agree that at archimedean primes the groups
H 0 (kp , A) denote the modified cohomology groups Ĥ 0 (kp , A).
Since the groups H i (kp , A) are all finite, we see that
P 0 (GS , A) =
P 1 (GS , A) =
P 2 (GS , A) =
P i (GS , A) =
Y
p∈ S
M
Ĥ 0 (kp , A) is compact,
p∈S∞
p∈Y
S\S∞
∈S
p
M
Y
H 0 (kp , A) ×
H 1 (kp , A)
is locally compact,
H 2 (kp , A)
is discrete,
H i (kp , A)
is finite for i ≥ 3.
p∈S∞
The direct sums in the last two lines occur because cd G(k̃p |kp ) = cd ẐZ = 1 and
cd Gkp = 2. If S is finite, all groups above are finite and the restricted product
is just the product.
The local duality theorem (7.2.9) has the following effect on the groups P i .
Let A ∈ ModS (GS ) be finite. Then for p - ∞, the pairing
invp
∪
H i (kp , A) × H 2−i (kp , A0 ) −→ H 2 (kp , k̄p× ) −→ Q/ZZ,
0 ≤ i ≤ 2, yields an isomorphism
Ξip : H i (kp , A) −→ H 2−i (kp , A0 )∨ .
(∗)
For p | ∞ we have to replace H 0 by Ĥ 0 . If A is unramified at p - ∞ and
2−i
i
p - #A, then, by (7.2.15), the groups Hnr
(kp , A0 ) and Hnr
(kp , A) are exact
i
orthogonal complements under the pairing, i.e. Hnr (kp , A) is mapped by Ξip
isomorphically onto the subgroup
2−i
(H 2−i (kp , A0 )/Hnr
(kp , A0 ))∨
⊆
H 2−i (kp , A0 )∨ .
For the restricted product of the right-hand groups with respect to the left-hand
subgroups we have the equality (cf. (1.1.13))
Y
H 2−i (kp , A0 )∨ = (
p∈S
Y
H 2−i (kp , A0 ))∨ .
p∈S
Since A is unramified for almost all p, we obtain the
(8.5.2) Proposition. If A
morphic isomorphisms
∈
ModS (GS ) is finite, the maps Ξip define homeo-
Ξi : P i (GS , A) −→ P 2−i (GS , A0 )∨
for 0 ≤ i ≤ 2.
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§5. Local Cohomology of Global Galois Modules
Now let A ∈ ModS (GS ) be arbitrary. We again define the dual GS -module
of A by
A0 = Hom(A, OS× ).
However, this is not completely satisfactory if A is not finite. Firstly, A0 is not
necessarily in ModS (GS ), and in particular, the situation is asymmetric; also
an equality like A00 = A is no longer true. Looking at this situation from the
point of view of sheaves, we are interested in the dual sheaf, which is no longer
locally constant, i.e. is not represented by a Galois module. That is why we
make the following notational convention:
For every prime p we write
def
H i (kp , A0 ) = H i (kp , Hom(A, k̄p× )),
i.e. we consider A as a local module first, and then take the dual in the local
sense. Note that we have an injection A0 ,→ Hom(A, k̄p× ) of G(k̄p |kp )-modules,
which induces maps for all i
H i (GS , A0 ) −→ H i (kp , A0 ).
If A (and hence also A0 ) is finite, both possible interpretations of H i (kp , A0 )
coincide. Now assume that A is unramified at the finite prime p. Then we
define
def
i
i
Hnr
(kp , A0 ) = Hnr
(kp , Ad ),
where Ad = Hom(A, Õp× ) (see VII §2). Again, no confusion occurs if A is
finite. If M is A or A0 , we set, as in the case of finite modules,
P i (GS , M ) =
Y
H i (kp , M ) ,
i ≥ 0,
p∈S
i
(kp , M )
where the restricted product is taken with respect to the subgroups Hnr
i
0
of H (kp , M ), and we keep our convention concerning Ĥ at archimedean
primes.
However, we have to be careful about the topologies: the cohomology
groups of A are considered as topological groups with the discrete topology
and the same holds for H i (kp , A0 ), i > 0. But for i = 0 the group H 0 (kp , A0 )
inherits the topology of Hom(A, k̄p× ) and is not discrete. Now for the topology
of P i (GS , M ), where M = A or A0 , a basis of neighbourhoods of the identity
is given by the subgroups
Y
p∈T
Xp ×
Y
i
(kp , M ),
Hnr
p∈S\T
where T varies over the finite subsets of S and Xp varies over a basis of
neighbourhoods of the identity in H i (kp , M ); so if i > 0 or M = A, we may
take Xp = 1.
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476
Chapter VIII. Cohomology of Global Fields
Therefore if S and A are infinite, among the groups P i (GS , A) only
P (GS , A) is locally compact because H 1 (kp , A) is finite by (7.2.9). Con0
sidering the cohomology of A0 , we see that Hnr
(kp , A0 ) = H 0 (kp , Ad ) is an
open compact subgroup of the locally compact group H 0 (kp , A0 ). By (1.1.13),
we conclude that P 0 (GS , A0 ) is locally compact. Since H 1 (kp , A0 ) is finite,
P 1 (GS , A0 ) is also locally compact.
1
Let us finally mention the special case when A is ZZ-free. Then, for all p
such that A is an unramified Gkp -module, we have
1
Hnr
(kp , A) = H 1 (kp , A)
and
i
Hnr
(kp , A0 ) = 0, i ≥ 1,
see (7.2.16). Therefore, if A is ZZ-free, P 1 (GS , A) is a product of finite groups,
hence compact, and P i (GS , A0 ), i ≥ 1, is discrete, being a direct sum of discrete
groups.
Again, let A ∈ ModS (GS ) be arbitrary. For each p, the local duality theorem
(7.2.9) together with (7.2.10) induces a continuous composite homomorphism
for i = 1, 2
H 2−i (kp , A0 ) −→ H 2−i (kp , AD ) −→ H i (kp , A)∨ .
This is an isomorphism of finite groups for i = 1, and for i = 2 it is an injection
with dense image of a locally compact group into a compact group. Dualizing,
we obtain continuous homomorphisms for i = 1, 2
H i (kp , A) −→ H 2−i (kp , A0 )∨ .
This is an isomorphism of finite groups for i = 1. For i = 2 it is an injection
with dense image of a discrete group into a locally compact group. If A
is unramified at p, then, by (7.2.15), this map induces a homomorphism of
subgroups
i
2−i
Hnr
(kp , A) −→(H 2−i (kp , A0 )/Hnr
(kp , A0 ))∨ ,
which is an isomorphism for i = 1. By the compatibility of Pontryagin
duality with restricted products, cf. (1.1.13), we therefore obtain continuous
homomorphisms for i = 1, 2.
Ξi : P i (GS , A) −→ P 2−i (GS , A0 )∨ .
From the above considerations, we obtain the following
(8.5.3) Proposition. Let A ∈ ModS (GS ). Then for i = 1, 2 the homomorphism
Ξi is continuous and
(i) Ξ1 is an isomorphism of locally compact groups,
(ii) Ξ2 is a continuous injection with dense image.
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477
§5. Local Cohomology of Global Galois Modules
We will now compare the groups P i (GS , A0 ) for A ∈ ModS (GS ) with the
cohomology groups H i (GS , IS (A)) of the GS -module
IS (A) = Hom(A, IS ),
where IS is the S-idèle group of kS .
Let Gp = G(kS,p |k) be the decomposition group of p in G(kS |k) with respect
to some prolongation p̄ of p to kS , see the discussion at the beginning of
this section. If K|k is a finite Galois subextension of kS with Galois group
G = G(K|k), then let GP = G(KP |kp ) where P = p̄|K . Finally, let
×
IP (A) = Hom(A, KP
)
UP (A) = Hom(A, UK,P ),
and
where UK,P is the group of units in KP . In the limit we use the notation
×
Ip (A) = Hom(A, kS,p
) and Up (A) = Hom(A, UkS ,p ).
Now let K|k be large enough so that G(kS |K) acts trivially on A. Then
IS (A)G(kS |K) = Hom(A, IS )G(kS |K) = Hom(A, IK,S )
and
Y
G
×
IndḠP (KP
).
IK,S =
p∈S
The functor Hom(A, −) commutes with , since it commutes with and with
lim
, recalling that A is finitely generated, and hence with restricted products.
−→
Thus we obtain a bijective homomorphism
Y
×
∼
ΨK : H i (G, Hom(A, IK,S )) −→
H i (GP , Hom(A, KP
)),
Q
Q
p∈S
where the restricted product is taken with respect to the subgroups
im H i (GP , UP (A)) −→ H i (GP , IP (A)) .
Note that ΨK is continuous. For i = 0 it is homeomorphic, but not for i > 0, as
H i (G, Hom(A, IK,S )) carries the discrete topology by definition. Taking the
direct limit over K, we obtain a commutative diagram
y{|z
lim H i (G, Hom(A,
IK,S ))
−→
K
∼
lim
−→
K p∈S
×
H i (GP , Hom(A, KP
))
(infp )p∈S
o inf
H i (GS , IS (A))
Y
Y
H i (kp , A0 ) = P i (GS , A0 ) .
p∈S
As any finite extension K|k is unramified at almost all primes, the restricted
Q
×
product p∈S H i (GP , Hom(A, KP
)) remains unchanged if we take it with
respect to the subgroups
im H i (GP /H P , UP (A)H P ) −→ H i (GP , IP (A)) ,
where H P is the inertia subgroup of GP . Therefore the right vertical arrow
exists and is continuous. As the left vertical map is a topological isomorphism,
we obtain the dotted continuous map from H i (GS , IS (A)) into P i (GS , A0 ).
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478
Chapter VIII. Cohomology of Global Fields
(8.5.4) Definition. The continuous homomorphisms
shi : H i (GS , IS (A)) −→ P i (GS , A0 ) ,
i ≥ 0,
are called Shapiro maps.
(8.5.5) Proposition. Let A ∈ ModS (GS ). Then the following hold:
(i) sh0 is a bijection for function fields. If k is a number field, then sh0 is
surjective with kernel
Y
ker sh0 =
Nk̄p |kp Hom(A, C× ) .
p∈S∞ (k)
(ii) sh1 is injective, and bijective if A is ZZ-free.
(iii) If A is ZZ-free, then sh2 is injective and for each ` ∈ IN(S) the induced
map on `-torsion subgroups
H 2 (GS , IS (A))(`) −→ P 2 (GS , A0 )(`)
is an isomorphism.
Remarks: 1. For ZZ-free A, the bijection sh1 of (ii) is a homeomorphism, as
both groups are discrete. The same is true for the isomorphism stated in (iii).
2. In (10.11.9) we will show the following additional facts for an arbitrary
A ∈ ModS (GS ):
• sh2 is injective and induces an isomorphism on the `-torsion part for all
` ∈ IN(S),
• if S is finite, then sh1 is bijective.
This will follow with the same proof as below using the additional fact that the
local fields (kS )p are `-closed for ` ∈ IN(S) and p ∈ S(kS ); see (10.11.6).
Proof: We start with the case i = 0. Let K|k be a finite Galois subextension
of kS such that G(kS |K) acts trivially on A. Putting G = G(K|k), we obtain
H 0 (GS , IS (A)) = Hom(A, IK,S )G
=
Y
=
Y
G
×
P
IndG
G Hom(A, KP )
p∈S
× GP
Hom(A, KP
)
p∈S
=
Y
Ip (A)GP =
p∈S
Y
H 0 (Gp , Ip (A)) .
p∈S
Let Np = G(k̄p |kS,p ), i.e. Gp = Gkp /Np . Then
× Gp
H 0 (kp , A0 ) = (Hom(A, k̄p× )Np )Gp = Hom(A, kS,p
) = H 0 (Gp , Ip (A)).
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479
§5. Local Cohomology of Global Galois Modules
If A is unramified at p - ∞, i.e. the inertia group with respect to p acts trivially
0
0
on A, then we get Hnr
(kp , A0 ) = Hnr
(Gp , Ip (A)) in the same way. Therefore
0
sh is surjective, and recalling the difference between H 0 and Ĥ 0 for p ∈ S∞ ,
we obtain the description of ker sh0 .
Since
lim H i (G(K|k), Hom(A, IK,S )) = H i (GS , IS (A))
−→
K
and (inf p )p∈S is injective for i = 1, we immediately obtain that sh1 is injective,
i.e. the first assertion of (ii).
Now let A be ZZ-free. Since A ∈ ModS (GS ), we have an isomorphism
A∼
= ZZr for some r, of trivial G(k̄p |(kS )p )-modules. Using Hilbert’s Satz 90,
we obtain
H 1 (G(k̄p |(kS )p ), Hom(A, k̄ × )) ∼
= H 1 (G(k̄p |(kS )p ), k̄ × )r = 0.
p
p
The Hochschild-Serre spectral sequence induces an isomorphism
1
×
∼
H 1 (G((kS )p |kp ), Hom(A, (kS )×
p )) → H (G(k̄p |kp ), Hom(A, k̄p ))
and an injection
H 2 (G((kS )p |kp ), Hom(A, k̄p× )) ,→ H 2 (G(k̄p |kp ), Hom(A, k̄p× )).
Using the commutative diagram
M
H 2 (G((kS )p |kp}~€ ), Hom(A, (kS )×
p ))
M
p∈S
H 2 (G(k̄p |kp ), Hom(A, k̄p× ))
p∈S
2
sh
H 2 (GS , IS (A))
P 2 (GS , A0 ),
we obtain the first assertion of (iii). Now let ` ∈ IN(S). Then `∞ | G((kS )p |kp )
for all nonarchimedean primes p ∈ S. As kS is totally imaginary, we obtain
H i (k̄p |(kS )p , A0 )(`) = 0 for all p ∈ S and i ≥ 1. Hence the horizontal injection
in the last diagram is an isomorphism on the `-torsion subgroups. This finishes
the proof of (iii).
Finally, we show (ii). For an arbitrary A ∈ ModS (GS ) we consider the
commutative diagram
Y
H 1 (G((kS )p |kp‚ƒ„ ), Hom(A, (kS )×
p ))
Y
p∈S
H 1 (G(k̄p |kp ), Hom(A, k̄p× ))
p∈S
1
sh
H 1 (GS , IS (A))
P 1 (GS , A0 ),
which shows that sh1 is injective. If A is ZZ-free, then P 1 (GS , A0 ) maps isoL
morphically to the direct sum p∈S H 1 (G(k̄p |kp ), Hom(A, k̄p× )) and the same
is true for H 1 (GS , IS (A)). Hence sh1 is an isomorphism if A is ZZ-free. This
finishes the proof of (ii).
2
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480
Chapter VIII. Cohomology of Global Fields
§6. Poitou-Tate Duality
We proved in (7.2.9) a cohomological duality theorem for local fields and
in (8.4.4) an analogous duality theorem for global fields. The latter, however,
does not play the same important role for global fields as its local prototype
for local fields. But both together lead to the main result of arithmetic Galois
cohomology: a duality principle for a local-global statement. The present
section is devoted to the formulation and the proof of this main theorem.
Recall that GS = G(kS |k) is the Galois group of the maximal extension kS |k
which is unramified outside the given nonempty set of primes S, where S∞ is
contained in S if k is a number field. ModS (GS ) is the category of discrete
GS -modules which are finitely generated as ZZ-modules and whose torsion has
order #tor(A) ∈ IN(S), and
A0 = Hom(A, OS× )
is the dual GS -module of A ∈ ModS (GS ), where OS× is the group of S-units
of kS . Further recall the convention that the cohomology groups H ∗ (kp , A0 )
are defined by first considering A as a local Galois module and then taking the
dual in the local sense, i.e. applying Hom(−, k̄p× ).
(8.6.1) Proposition. Let A ∈ ModS (GS ) and let M be equal to A or A0 . Then
the homomorphism
res i : H i (GS , M ) →
Y
H i (kp , M )
p∈S
maps H i (GS , M ) into P i (GS , M ) =
Y
H i (kp , M ).
p∈S
Proof: First let M = A. Since A is finitely generated, it becomes a trivial Galois module over a finite Galois subextension LA |k of kS |k. If x ∈ H i (GS , A),
then there exists a finite Galois subextension K|k of kS |k such that x is the
image of an element y ∈ H i (K|k, AGK ) under the inflation map. It follows
that for all nonarchimedean primes p of S which are unramified in KLA |k, the
i
class xp = resp x is contained in the subgroup Hnr
(kp , A) of H i (kp , A).
Now let M = A0 and x ∈ H i (GS , A0 ). As above we have x = inf y with
y ∈ H i (K|k, (A0 )GK ) for a finite Galois subextension K|k of kS |k. For all
nonarchimedean primes p of S which are unramified in KLA |k, the extension Kp |kp is a subextension of the maximal unramified extension k̃p of kp .
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481
§6. Poitou-Tate Duality
Therefore the class xp = resp x is contained in the image of
H i (k̃p |kp , Hom(A, OS× )Gk̃p ) = H i (k̃p |kp , Hom(A, (OS× )Gk̃p ))
i
i
i
in H i (kp , A0 ), i.e. in Hnr
(kp , A0 ) = Hnr
(kp , Ad ) = Hnr
(kp , Hom(A, Õp× )). This
proves the proposition.
2
(8.6.2) Definition. The homomorphisms
λi : H i (GS , M ) −→ P i (GS , M )
for M = A and M = A0 , which exist by (8.6.1), are called the localization
maps. We set
Xi (GS , M ) = ker(λi ) . ∗)
It is evident that for M = A0 , the map λi is the composition
shi
H i (GS , A0 ) −→ H i (GS , IS (A)) −→ P i (GS , A0 ),
where the first arrow is induced by A0 = Hom(A, OS× ) → Hom(A, IS ) = IS (A),
and the second is the Shapiro map.
The groups Xi (GS , M ) are of great interest. Their vanishing would imply a
strict “local-global principle”. As a rule, the groups Xi (GS , M ) are non-zero,
but we shall see that they are always finite.
(8.6.3) Lemma. For m ∈ IN(S), we have natural isomorphisms
X1 (GS , ZZ/mZZ) ∼
= Hom(ClS (k), ZZ/mZZ) ,
∼
= ClS (k) .
X1 (GS , ZZ0 )
Proof: By class field theory, ClS (k) is naturally isomorphic to the Galois
group of the maximal abelian extension of k inside kS in which all primes of S
split completely. Hence
X1 (GS , ZZ/mZZ) = ker H 1 (GS , ZZ/mZZ) −→
Y
H 1 (kp , ZZ/mZZ)
p∈S
∼
= Hom(ClS (k), ZZ/mZZ) .
Furthermore, X1 (GS , ZZ0 ) = H 1 (GS , OS× ) ∼
= ClS (k) by Hilbert’s Satz 90 and
(8.3.11)(ii).
2
∗) If A is the G -module A(k ) of k -rational points of an abelian variety A over k, then
S
S
S
this kernel is classically called the Šafarevič-Tate group. This explains the Russian letter X
(sha).
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482
Chapter VIII. Cohomology of Global Fields
From the exact sequence
0 −→ OS× −→ IS −→ CS −→ 0,
we obtain a sequence of GS -modules
0 −→ A0 −→ IS (A) −→ CS (A) −→ 0 ,
which is exact as Ext1ZZ (A, OS× ) = 0 because OS× is #tor(A)-divisible by (8.3.4).
(8.6.4) Theorem. Let A ∈ ModS (GS ). Then the boundary map
δ
CS (A)GS −→ H 1 (GS , A0 )
is continuous and has finite image. It maps a norm group NG/U CS (A)U to zero
for some open normal subgroup U ⊆ GS . The group X1 (GS , A0 ) is finite.
Proof: First note that im(δ) = X1 (GS , A0 ), since
sh1 : H 1 (GS , IS (A)) −→ P 1 (GS , A0 )
is injective by (8.5.5)(ii). Let K = kSH be a finite Galois extension of k
inside kS which trivializes A and µn , where n = #tor(A). Furthermore, let
K1 = kSH1 be a finite Galois extension of k inside kS which contains the
maximal unramified abelian extension of K in which all primes p ∈ S(K) are
completely decomposed. As an H-module, A is isomorphic to a direct sum of
modules of the form ZZ or µm with m|n. We have X1 (H, ZZ0 ) = ClS (K) and
X1 (H, (µm )0 ) = Hom(ClS (K), ZZ/mZZ)
by (8.6.3), and we also have the corresponding statement for the group H1 . By
class field theory and the principal ideal theorem, we conclude that the maps
H1
1
1
1
1
0
0
0
0
res H
H1 : X (H, A ) → X (H1 , A ), cor H : X (H1 , A ) → X (H, A )
are zero. The commutative diagram
Ž‰Š‡ˆ…†Œ‹‘’“ H1
IS (A)
NH/H1
IS (A)H
NGS /H
IS (A)GS
CS (A)H1
δH1
NH/H1
CS (A)H
0
0
δH
X1 (H, A0 )
0
cor H
G
NGS /H
CS (A)GS
X1 (H1 , A0 )
S
δ
X1 (GS , A0 )
shows that δ annihilates the subgroup NGS /H1 CS (A)H1
more, the diagram
⊆
0
CS (A)GS . Further-
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483
§6. Poitou-Tate Duality
”•ž›˜™–— ¡¢Ÿœš H1
IS (A)
res H
H1
CS (A)H1
δH 1
X1 (H1 , A0 )
res H
H1
IS (A)H
CS (A)H
0
δH
G
G
IS (A)GS
CS (A)GS
X1 (H, A0 )
0
G
res HS
res HS
0
res HS
δ
X1 (GS , A0 )
0
shows that CS (A)GS is a subgroup of the group
B := im(IS (A)H1 −→ CS (A)H1 ).
By (8.3.5), the image CK1 ,S of the homomorphism IK1 ,S → CS (K1 ) is open of
×
finite index. Its kernel OK
is a discrete subgroup of IK1 ,S . For m ∈ IN(S),
1 ,S
Hom(µm , IK1 ,S ) is compact. As X1 (H1 , ZZ/mZZ) = Hom(ClS (K1 ), ZZ/mZZ) is
finite, the image of the homomorphism
Hom(µm , IK1 ,S ) −→ Hom(µm , CS (K1 ))
is a closed subgroup of finite index, hence open. The kernel of this homomor×
phism is the finite group Hom(µm , OK
) = ZZ/mZZ. Therefore B ⊆ CS (A)H1
1 ,S
is open of finite index and
0 −→ A0
H1
−→ IS (A)H1 −→ B −→ 0
is an exact sequence of locally compact groups with continuous GS /H1 -action.
Furthermore, there exists a GS /H1 -invariant open neighbourhood U of 1 in
IS (A)H1 which maps homeomorphically onto its image U 0 in B. Let V ⊆ U
be a G/H1 -invariant open neighbourhood of 1 with V · V −1 ⊆ U and let V 0
be the homeomorphic image of V in B. The subset V 0 ∩ CS (A)GS is an open
neighbourhood of 1 in CS (A)GS .
Let x ∈ V 0 ∩ CS (A)GS . There exists a unique pre-image y ∈ V of x. The
inner derivation GS /H1 → IS (A)H1 , g 7→ y −1 gy, takes values in A0 H1 and
induces a derivation cy : GS /H1 → A0 H1 . The class c̄y ∈ H 1 (GS /H1 , A0 H1 )
of cy maps to δ(x) ∈ H 1 (GS , A0 ) under inflation. For g ∈ GS /H1 , we have
y −1 gy ∈ U ∩ A0 H1 = {1}. Hence δ(x) = 0. This shows that the subgroup
ker(δ) contains an open neighbourhood of 1 in CS (A)GS and is therefore
open. Consequently, δ is locally constant. Furthermore, δ factors through the
compact quotient CS (A)GS /NGS /H1 CS (A)H1 . Therefore im(δ) = X1 (GS , A0 )
is finite.
2
In the following we want to construct a canonical non-degenerate pairing
between the groups X1 (GS , A0 ) and X2 (GS , A) for each finitely generated
GS -module A ∈ ModS (GS ). We need two lemmas. As before, write
A0 = Hom(A, OS× ) , IS (A) = Hom(A, IS ) , CS (A) = Hom(A, CS ).
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Chapter VIII. Cohomology of Global Fields
(8.6.5) Lemma. Let A ∈ ModS (GS ) and i = 1 or i = 2. Then the homomorphism
λi
Ξi
ξ i : H i (GS , A) −→ P i (GS , A) −→ P 2−i (GS , A0 )∨ ,
where Ξi was defined in (8.5.2), has kernel ker ξ i = Xi (GS , A).
2
Proof: By (8.5.3), the homomorphisms Ξi , i = 1, 2, are injective.
Recall that P 2−i (GS , A0 )∨ is a locally compact group for i = 1, 2. Therefore
we can dualize the composite map ξ i = Ξi ◦ λi for i = 1, 2 (but not Ξ2 , since
“P 2 (GS , A)∨ ” does not exist at least if S and A are infinite).
(8.6.6) Lemma. For i = 0 and i = 1, we have a commutative diagram of
topological groups
H i (GS ,¦¥¤£ IS (A))
can
H i (GS , CS (A))
∆i
shi
P i (GS , A0 )
(ξ 2−i )∨
H 2−i (GS , A)∨ ,
where ∆i is induced by the cup-product pairing
∪
inv
H i (GS , CS (A)) × H 2−i (GS , A) −→ H 2 (GS , CS ) −→ Q/ZZ.
The map (ξ 2−i )∨ on the bottom is the dual to the map ξ 2−i obtained in (8.6.5).
Proof: Let p ∈ S be a prime. We make the convention that H 0 (kp , −) denotes
Ĥ 0 (kp , −) if p is archimedean. Using the notation of the preceding section, we
have the commutative diagram
±°²¯®¬­«ª©§¨
H i (kp , Hom(A,
k̄p× )) × H 2−i (kp , A)
∪
inf
×
H i (Gp , Hom(A, kS,p
)) × H 2−i (Gp , A)
resp
invp
Q/ZZ
inf
∪
resp
H i (GS , Hom(A, IS )) × H 2−i (GS , A)
H 2 (kp , k̄p× )
×
H 2 (Gp , kS,p
)
resp
∪
H 2 (GS , IS )
inv
Q/ZZ .
For x ∈ H i (GS , IS (A)), we therefore obtain a commutative diagram
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§6. Poitou-Tate Duality
º·¶¸¹³´µ S , A)
H 2−i (G
Y
Y
H 2−i (kp , A)
p∈S
H i (kp , A0 )∨
p∈S
∪shi (x)
∪x
⊕invp
H 2 (GS , IS )
M
Q/ZZ
p∈S
Σ
inv
H 2 (GS , CS )
Q/ZZ .
The map from H 2−i (GS , A) to Q/ZZ via the upper right corner is exactly
(ξ 2−i )∨ ◦ shi (x) ∈ H 2−i (GS , A)∨ , while the map via the lower left corner is
∆i ◦ can(x). This shows that the diagram of the lemma commutes.
2
Now we are ready to prove one of the main results of this section.
(8.6.7) Theorem (Poitou-Tate Duality). Let A be a finitely generated GS module with #tor(A) ∈ IN(S). Then there is a perfect pairing ∗)
X1 (GS , A0 ) × X2 (GS , A) −→ Q/ZZ
of finite groups, which is induced by the cup-product, i.e. the diagram
H 0 (GS ,»¼½¾¿ CS (A)) × H 2 (GS , A)
X1 (GS , A0 )
∪
H 2 (GS , CS )
Q/ZZ
× X2 (GS , A)
commutes.
Proof: Consider the diagram with exact upper row
ÉÈÂÇÅÆÄÃÁÀ GS
IS (A)
(1)
γ
sh0
P 0 (GS , A0 )
CS (A)GS
δ
H 1 (GS , A0 )
i
∆0
(ξ 2 )∨
H 2 (GS , A)∨
H 1 (GS , IS (A))
sh1
ε
H 1 (GS , A0 )
λ1
P 1 (GS , A0 ) .
∼
∼
∗) A pairing A × B → Q/ZZ is perfect if it induces isomorphisms A →
B ∨ and B →
A∨ .
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Chapter VIII. Cohomology of Global Fields
The map ∆0 is the composition of the global duality map (8.4.4), (8.4.5),
∼ H 2 (G , A)∨ ,
∆ˆ 0 : Ĥ 0 (GS , CS (A)) −→
S
with the canonical homomorphism H 0 (GS , CS (A)) → Ĥ 0 (GS , CS (A)) (which
has dense image). The left-hand square is commutative by (8.6.6) and the righthand one by the remark following (8.6.2). By (8.6.4), the homomorphism δ
maps a norm group NGS /U CS (A)U with U sufficiently small to zero, and we
obtain a commutative diagram
ÏÎÌÍÊË GS
CS (A)
∆ˆ 0
∼
Ĥ 0 (GS , CS (A))
H 2 (GS , A)∨
π
ε
CS (A)GS /NGS /U CS (A)U
δ
H 1 (GS , A0 )
where ε = δ ◦ π ◦ (∆ˆ 0 )−1 . This map ε completes diagram (1). We have
im(ε) = im(δ) = ker(i) = ker(λ1 ) = X1 (GS , A0 ) (observe that sh1 is injective
by (8.5.5)(ii)). By (8.5.5)(i), the map sh0 is surjective. Since
ε(ξ 2 )∨ sh0 = δπ(∆ˆ 0 )−1 (ξ 2 )∨ sh0 = δπ(∆ˆ 0 )−1 ∆0 γ = δγ = 0 ,
we obtain ε(ξ 2 )∨ = 0, i.e. we have an inclusion im((ξ 2 )∨ ) ⊆ ker(ε). Since
ε is continuous and has finite image by (8.6.4) and (8.4.4), ker(ε) is an open
subgroup in H 2 (GS , A)∨ . In particular, ker(ε) contains the closure of im((ξ 2 )∨ )
(note that im((ξ 2 )∨ ) is closed if A is finite or if k is a number field).
Let x ∈ ker(ε) and let V ⊆ ker(ε) be an arbitrarily chosen open subgroup. As ∆0 has dense image by (8.4.5), there exists y ∈ CS (A)GS with
∆0 (y) ∈ x + V . Hence δ(y) = ε∆0 (y) = 0. Thus there exists an element
z ∈ IS (A)GS such that γ(z) = y. It follows that ∆0 (y) = (ξ 2 )∨ (sh0 (z)), and so
x + V contains an element of im((ξ 2 )∨ ). As H 2 (GS , A)∨ is profinite, it has a
basis of neighbourhoods of zero consisting of subgroups. Therefore im((ξ 2 )∨ )
is dense in ker(ε) and the induced map ker(ε)∨ → P 0 (GS , A0 )∨ is injective.
We conclude that ξ 2 factors as
ξ 2 : H 2ÐÑ (GS , A)
ker(ε)∨
P 0 (GS , A0 )∨ .
This implies that there is an isomorphism
X1 (GS , A0 )∨ = im(ε)∨ ∼
= ker(ξ 2 ) = X2 (GS , A),
which obviously is induced from the diagram
ÕÓÔÒ
H 0 (GS , CS (A)) × H 2 (GS , A)
∪
Q/ZZ
δ
X1 (GS , A0 )
× X2 (GS , A) .
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487
§6. Poitou-Tate Duality
For finite modules we can describe the pairing obtained in (8.6.7) explicitly.
Let A ∈ ModS (GS ) be finite and let us consider the “new” pairing
t
X1 (GS , A0 ) × X2 (GS , A) −→ Q/ZZ
which is defined as follows: let x and x0 be cocycles representing classes
[x] ∈ X2 (GS , A) and [x0 ] ∈ X1 (GS , A0 ). As H 3 (GS , OS× )(p) = 0 for every
prime number p ∈ IN(S) (see (8.3.11)), there is a cochain z ∈ C 2 (GS , OS× ) such
that
x0 ∪ x = ∂z.
Moreover, for every p there are cochains yp ∈ C 1 (kp , A) and yp0
such that for the components xp = λp (x), x0p = λp (x0 ) we have
xp = ∂yp
and
∈
C 0 (kp , A0 )
x0p = ∂yp0 .
Then yp0 ∪ xp − zp and x0p ∪ yp − zp are 2-cocycles with values in k̄p× which
differ by a coboundary. In fact,
∂(yp0 ∪ xp ) = ∂yp0 ∪ xp = x0p ∪ xp = ∂zp
and analogously ∂(x0p ∪ yp ) = ∂zp , and moreover
∂(yp0 ∪ yp ) = x0p ∪ yp − yp0 ∪ xp .
The “new” pairing is now defined by
[x0 ] t [x] =
X
invp [yp0 ∪ xp − zp ] =
p∈S
X
invp [x0p ∪ yp − zp ].
p∈ S
One checks in a straightforward manner that this definition does not depend on
the choice of the representing cocycles x and x0 and the choice of the cochains
z, yp and yp0 .
(8.6.8) Proposition. The pairing defined above coincides with the pairing
obtained in (8.6.7).
Proof: Let [x0 ] ∈ X1 (GS , A0 ) and [x] ∈ X2 (GS , A). The set of 0cochains yp0 ∈ H 0 (kp , A0 ) with ∂yp0 = x0p , p ∈ S, can be interpreted as
a 0-cochain in H 0 (GS , IS (A)). Under the projection IS → CS the element
(yp0 ) ∈ H 0 (GS , IS (A)) becomes a 0-cochain y 0 ∈ H 0 (GS , CS (A)) such that
δy 0 = [x0 ] and the value of the pairing in (8.6.7) is inv[y 0 ∪ x]. Consider the
commutative diagram (8.6.6)
H 0 (GS ,Ö×ØÙ IS (A))
H 0 (GS , CS (A))
P 0 (GS , A0 )
H 2 (GS , A)∨ .
With the notation of the “new” pairing, we obtain
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Chapter VIII. Cohomology of Global Fields
[x0 ] t [x] =
X
invp [yp0 ∪ xp − zp ] = inv[y 0 ∪ x] ,
p∈S
because the image of the 2-cochain (zp ) ∈ C 2 (GS , OS× ) in H 2 (GS , CS ) is zero.
2
For A ∈ ModS (GS ), we have constructed a natural perfect pairing between
X1 (GS , A0 ) and X2 (GS , A). If A is finite, we can interchange the roles of
A and A0 to obtain a perfect pairing between X1 (GS , A) and X2 (GS , A0 ) as
well. If A is infinite, the situation is no longer symmetric. If A is ZZ-free, i.e.
A0 is a Spec(Ok,S )-torus, we have the following
(8.6.9) Theorem. Let A ∈ ModS (GS ) be torsion-free. Then there is a perfect
pairing of finite groups
X1 (GS , A) × X2 (GS , A0 ) −→ Q/ZZ ,
which is induced by the cup-product, i.e. the diagram
H 1 (GÛÜÝÞÚ S , A) × H 1 (GS , CS (A))
X1 (GS , A) ×
∪
H 2 (GS , CS )
Q/ZZ
X2 (GS , A0 )
commutes.
Proof: Since H 1 (GS , A) is a finite group by (8.3.20)(ii), its subgroup
X1 (GS , A) is also finite. By (8.6.6), the diagram
H 1 (GS ,èçæäåãáâßà IS (A))
H 1 (GS , CS (A))
sh1
δ
H 2 (GS , A0 )
H 2 (GS , IS (A))
∆1
P 1 (GS , A0 )
(ξ 1 )∨
H 1 (GS , A)∨
sh2
ϕ
H 2 (GS , A0 )
λ2
P 2 (GS , A0 )
commutes. The map ∆1 is an isomorphism by (8.4.4), and ϕ is defined so that
the square in the middle commutes. Obviously, the upper sequence is exact.
By (8.5.5)(ii), (iii) the map sh1 is bijective and sh2 is injective. Therefore
the lower sequence in the commutative diagram above is also exact. Since
coker (ξ 1 )∨ = X1 (GS , A)∨ by (8.6.5), we obtain X1 (GS , A)∨ ∼
= X2 (GS , A0 ).
Finally, the associated pairing to Q/ZZ is induced by the cup-product, as stated.
This finishes the proof of the theorem.
2
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489
§6. Poitou-Tate Duality
Remarks: 1. If S is a finite set containing the set Sp for some prime number p
and A ∈ ModS (GS ) is arbitrary, then one can prove the duality between
X1 (GS , A) and X2 (GS , A0 ) at least for the p-components of these groups;
see (10.11.10).
2. The duality between Xi (GS , A) and X3−i (GS , A0 ), i = 1, 2, proven above
for torsion-free A (resp. for a torus), has a vast generalization if S is the set of
all primes: it holds for 1-motives; see [76].
3. If S is the set of all places, then (8.6.9) holds for an arbitrary finitely generated Gk -module A. This can be seen as follows (statement and proof are due
to J.-L. COLLIOT-THÉLÈNE): There exists an exact sequence 0 → C → B →
L
A → 0 with B torsionfree and C ∼
= ni=1 IndUGik ZZ, where Ui ⊆ Gk are open subgroups; see [CS], Lemma 0.6. As the groups H 1 (k, C), H 1 (k, C 0 ), X2 (k, C)
and X2 (k, C 0 ) vanish, we obtain isomorphisms X1 (k, B) ∼
= X1 (k, A) and
X2 (k, B 0 ) ∼
= X2 (k, A0 ), and so the statement for general A follows from that
in the torsionfree case. This argument extends to the case when S has Dirichlet
density 1.
We proved a duality theorem for global fields in (8.4.4) which, together
with its analogue (7.2.6) for local fields, leads to the main result of arithmetic
Galois cohomology: a 9-term exact sequence connecting the local and global
cohomology groups.
(8.6.10) Long Exact Sequence of Poitou-Tate. Let S be a nonempty set of
primes of a global field k and assume that S ⊇ S∞ if k is a number field. Let
A be a finite GS -module of order #A ∈ IN(S).
(i) There is a canonical exact sequence of topological groups
0éêëìíîïðñòóô
H 0 (GS , A)
P 0 (GS , A)
H 2 (GS , A0 )∨
H 1 (GS , A)
(finite)
(compact)
(compact)
(discrete)
(locally compact)
0
H 0 (GS , A0 )∨
P 2 (GS , A)
H 2 (GS , A)
H 1 (GS , A0 )∨
(finite)
(discrete)
(discrete)
(compact)
P 1 (GS , A)
(ii) For i ≥ 3, the restriction map
∼
H i (GS , A) −→
M
H i (kp , A)
p∈SIR
is an isomorphism.
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Chapter VIII. Cohomology of Global Fields
Remark: Some people consider it more suggestive to arrange the terms of the
exact 9-term sequence in the following form.
0õö÷øùúûüýþÿ
H 0 (GS , A)
P 0 (GS , A)
H 2 (GS , A0 )∨
H 1 (GS , A)
P 1 (GS , A)
H 1 (GS , A0 )∨
H 2 (GS , A)
P 2 (GS , A)
H 0 (GS , A0 )∨
0.
Proof of (8.6.10)(ii): For function fields the assertion is obviously true by
(8.3.17), so we may assume that k is a number field. If A = 0 the statement is
trivial. Therefore we may assume that S ⊇ Sp for at least one prime number p.
In particular, kS is totally imaginary.
As `∞ | #GS for all ` ∈ IN(S), (1.9.15) implies Ĥ −1 (GS , A0 ) = 0. From the
exact sequence
0 −→ A0 −→ IS (A) −→ CS (A) −→ 0
of level-compact GS -modules (observe that CS (A) = Hom(A, CS0 )) we obtain
the exact sequence
0 → Ĥ −1 (GS , IS (A)) → Ĥ −1 (GS , CS (A)) → Ĥ 0 (GS , A0 ) → Ĥ 0 (GS , IS (A))
by (1.9.13). Again by (1.9.15), we have Ĥ 0 (GS , A0 ) = A0GS . For p ∈ S put
Gp = G(kS,p |kp ). For p ∈ S r S∞ and ` ∈ IN(S) we have `∞ | #Gp and (1.9.15)
implies Ĥ 0 (Gp , A0 ) = A0Gp . We obtain
Y
Y
Ĥ 0 (GS , IS (A)) ∼
A0Gp ×
Ĥ 0 (Gp , A0 ).
=
p∈S\S∞
p∈S∞
Therefore the map Ĥ 0 (GS , A0 ) → Ĥ 0 (GS , IS (A)) is injective, yielding the isomorphism
∼ Ĥ −1 (G , C (A)).
Ĥ −1 (GS , IS (A)) −→
S
S
By (1.9.15), we have Ĥ −1 (Gp , A0 ) = 0 for all p ∈ S r S∞ and therefore a
canonical isomorphism
Y
M
Ĥ −1 (GS , IS (A)) ∼
Ĥ −1 (kp , A0 ).
Ĥ −1 (Gp , A0 ) ∼
=
=
p∈S
p∈SIR
Using the duality theorem for archimedean primes (7.2.17)
Ĥ −1 (kp , A0 ) ∼
= H 3 (kp , A)∨ ,
we obtain the commutative diagram
Ĥ −1 (GS , IS (A))
M
H 3 (kp , A)∨
Ĥ −1 (GS , CS (A))
(λ3 )∨
H 3 (GS , A)∨
p∈SIR
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491
§6. Poitou-Tate Duality
where the vertical isomorphism on the right-hand side is given by (8.4.4).
Hence
M
∼
λ3 : H 3 (GS , A) −→
H 3 (kp , A).
p∈SIR
is an isomorphism.
In order to prove the assertion for all i ≥ 3 we use the exact sequence
0 −→ A −→ IndG A −→ B −→ 0,
where G = G(K|k), for K a finite totally imaginary extension of k inside
kS such that A is a trivial G(kS |K)-module, and the finite module B is defined as the quotient of IndG A by A. For i ≥ 3 we have H i (GS , IndG A) =
H i (G(kS |K), A) = 0 and
M
H i (kp , IndG A) =
H i (KP , A) = 0, p ∈ SIR .
P|p
Therefore the horizontal maps in the commutative diagram
H i (G S , B)
H i+1 (GS , A)
λi
M
H i (kp , B)
p∈SIR
λi+1
M
H i+1 (kp , A)
p∈SIR
are isomorphisms for i ≥ 3. Now the result follows by induction.
2
In order to prove part (i) of theorem (8.6.10), we start with the following
observations.
(8.6.11) Proposition. For A ∈ ModS (GS ), the localization map
is proper.∗)
λ1 : H 1 (GS , A) −→ P 1 (GS , A)
Proof: Since A is finitely generated, A is a GT -module for all sufficiently
large finite subsets T ⊆ S. The groups
PT = P 1 (GT , A) ×
Y
1
Hnr
(kp , A)
p∈S\T
are compact, and each compact neighbourhood of 1 in P 1 (GS , A) is contained in some PT for T ⊆ S finite. Therefore we have to show that
(λ1 )−1 (PT ) ⊆ H 1 (GS , A) is compact and, since H 1 (GS , A) is discrete, this
comes down to showing that (λ1 )−1 (PT ) is finite. Consider the canonical
injections
∗) A map of topological spaces is called proper if the pre-images of compact subsets are
compact.
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Chapter VIII. Cohomology of Global Fields
M
1
H 1 (kp , A)/Hnr
(kp , A) ,→
p∈S\T (k)
diag
M
Y
H 1 (Tp , A) ,→
p∈S\T (k)
H 1 (TP , A),
P∈S\T (kT )
where Tp denotes the inertia group of the local group G(k̄p |kp ) for a prime p.
We obtain the commutative and exact diagram
H 1 (G(kS
|kT ), A)
Y
H 1 (TP , A)
P∈S\T (kT )
H 1 (GS , A)
P 1 (GS , A)
H 1 (GT , A)
Y
P 1 (GT , A) ×
1
(kp , A) = PT .
Hnr
p∈S\T (k)
Since A is a trivial G(kS |kT )-module and since the images of the inertia groups
TP , P ∈ S r T (kT ), generate G(kS |kT ) as a normal subgroup, the upper
horizontal map is injective. Thus
(λ1 )−1 (PT ) ⊆ H 1 (GT , A) .
We will show that H 1 (GT , A) is a finite group. Recalling (8.3.20), we may
assume that A is torsion-free. Then the exact sequence
0 −→ H 1 (GT /H, A) −→ H 1 (GT , A) −→ H 1 (H, A) = 0,
where H is an open normal subgroup of GT acting trivially on A, shows that
H 1 (GT , A) is finite.
2
(8.6.12) Lemma. Let A
exact diagram
Y
Nk̄p |kp Hom(A,
C× )
ModS (GS ) be finite. Then there is a commutative
∈
ϕ
NGS CS (A)
p∈S∞
H 0 (GS , IS (A))
H 0 (GS , CS (A))
P 0 (GS , A0 )
H 2 (GS , A)∨
δ
ε
H 1 (GS , A0 )
H 1 (GS , IS (A))
H 1 (GS , A0 )
P 1 (GS , A0 )
where the map ϕ is an isomorphism. We have a natural exact sequence
0 → H 0 (GS , A0 ) → P 0 (GS , A0 ) → H 2 (GS , A)∨ → X1 (GS , A0 ) → 0 .
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493
§6. Poitou-Tate Duality
Proof: As we can represent kS as the union of a countable tower of finite
subextensions, we can apply the results of II §7 to projective systems indexed
by the set of finite extensions K of k inside kS . Consider the projective system
which attaches to each K the group H 0 (G(kS |K), A0 ) with the norm maps as
transition maps. If K trivializes A0 and L|K is a finite extension, then the
norm
NL|K : A0 = H 0 (G(kS |L), A0 ) −→ A0 = H 0 (G(kS |K), A0 )
is just multiplication by [L : K]. For all prime numbers ` | #A0 ∈ IN(S),
`∞ divides the order of GS (observe that k(µ`∞ ) ⊆ kS ). Therefore the above
projective system is a ML-zero system, and (2.7.3) implies
Ri (lim
)H 0 (G(kS |K), A0 ) = 0,
←−
i≥0.
We let K run through the extensions of k inside kS which trivialize A0 and
consider the exact sequences
0 −→ A0 −→ Hom(A, IK,S ) −→ Hom(A, IK,S )/A0 −→ 0.
Passing to the projective limit over K and using the ML-zero property, we
obtain an isomorphism
0
∼ lim Hom(A, I
lim
Hom(A, IK,S ) −→
K,S )/A .
←−
←−
K
K
1
By (8.5.5) the map sh is injective, hence
X1 (G(kS |K), A0 ) = ker H 1 (G(kS |K), A0 ) −→ H 1 (G(kS |K), IS (A) .
Therefore the exact sequence
0 −→ A0 −→ IS (A) −→ CS (A) −→ 0
implies the exact sequence
0 −→ Hom(A, IK,S )/A0 −→ Hom(A, CS (K)) −→ X1 (G(kS |K), A0 ) −→ 0 .
Since
0
1
lim X (G(kS |K), A ) =
←−
K
2
∨
lim X (G(kS |K), A)
−→
K
= 0,
∼ lim
we obtain the isomorphism lim
Hom(A, IK,S ) →
Hom(A, CK,S ). Con←− K
←− K
sider the commutative diagram
lim
Hom(A,
IK,S )
←−
K
lim
Hom(A, CK,S )
←−
K
NGS IS (A)
NGS CS (A).
As the GS -module CS (A) = CS0 (A) is level-compact, the right vertical map is
surjective, and hence the natural map
NGS IS (A) −→ NGS CS (A)
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Chapter VIII. Cohomology of Global Fields
is also surjective. Let K be an extension of k inside kS trivializing A0 and
let p ∈ S r S∞ (K). For a sufficiently large finite extension L of K inside kS
(adjoin roots of unity) the norm map
NL|K :
Y
×
Hom(A, L×
P ) −→ Hom(A, Kp )
P|p
is the zero map. Therefore the universal norms of IS (A) have trivial components at the nonarchimedean primes and we obtain an isomorphism
Y
∼ N
Nk̄p |kp Hom(A, C× ) −→
GS IS (A) .
p∈S∞
We define ϕ as the composite of this isomorphism with the natural map
NGS IS (A) −→ NGS CS (A), which is surjective, as shown above.
Let us explain the other objects and maps in the diagram of the lemma.
The second row of the diagram is part of the long exact cohomology sequence
associated to the short exact sequence A0 ,→ IS (A) CS (A). The first arrow
in the third row is the dual of the localization map λ2 (A), while the third
arrow is the localization map λ1 (A0 ). For the commutativity of the left- and
right-hand lower squares, see (8.6.6) and the definition of the localization map
(8.6.2). The map ε is given by the perfect pairing between X1 (GS , A0 ) and
X2 (GS , A); see (8.6.7). In particular, the lower row is exact. The square in the
middle commutes by the construction of ε. The exactness of the two columns
follows from (8.5.5)(i) and (8.4.4).
Consider the commutative diagram
()*+,!"#$%&'
Y
Nk̄p |kp Hom(A, C× )
ϕ
NGS CS (A)
p∈S∞
0
H 0 (GS , A0 )
H 0 (GS , IS (A))
H 0 (GS , CS (A))
0
H 0 (GS , A0 )
P 0 (GS , A0 )
H 2 (GS , A)∨
The third row is a complex; the middle row and the columns are exact. A
diagram chase shows that ϕ is an isomorphism and that the lower row is exact.
This finishes the proof.
2
In order to prove the exactness of the 9-term sequence, we apply (8.6.12) to
A and A0 and get the two horizontal exact sequences in the diagram (where we
write G for GS )
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§6. Poitou-Tate Duality
0567843-./012
H 0 (G, A)
H 2 (G, A0 )∨
P 0 (G, A)
H 1 (G, A)
P 1 (G, A)
(∗)
Ξ1
ψ
H 0 (G, A0 )∨
0
P 2 (G, A)
H 1 (G, A0 )∨
H 2 (G, A)
λ∨
P 1 (G, A0 )∨ .
We define the map ψ as the composition λ∨ ◦ Ξ1 . The pairings
H 1 (G;:9A?=><@ S , A) × H 1 (GS , A0 )
λ
∪
H 2 (GS , OS× )
∪
H 2 (GS , IS )
λ0
P 1 (GS , A) × P 1 (GS , A0 )
⊕ invp
M
Q/ZZ
p∈S
ι
H 2 (GS , CS )
Σ
inv
Q/ZZ
show that im(λ) and im(λ0 ) annihilate each other. Therefore the map ψ induces
a 9-term complex which we denote by
P T (k, S, A),
and we know that it is exact at all places but the middle one, which remains
to be shown. If the set S is finite, then all groups occurring are finite and
the complex is exact if and only if the alternating product of the orders of the
groups in the complex is equal to 1. For a finite GS -module A, we define the
second partial Euler-Poincaré characteristic by
χ2 (GS , A) =
#H 0 (GS , A) · #H 2 (GS , A)
,
#H 1 (GS , A)
and put
ϕ(GS , A) = χ2 (GS , A)/
Y
p∈S∞
#H 0 (kp , A)
,
||#A||p
where for function fields the set S∞ is empty by definition.
(8.6.13) Lemma. Let S be a finite nonempty set of primes of the global field k
containing S∞ if k is a number field. Let A be a finite GS -module of order
#A ∈ IN(S). Then
(i) ϕ(GS , A) · ϕ(GS , A0 ) = 1 if and only if the complex P T (k, S, A) is exact.
(ii) ϕ(GS , −) is multiplicative on short exact sequences of finite modules in
ModS (GS ).
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Chapter VIII. Cohomology of Global Fields
Proof: Using the product formula (7.3.5) and (7.3.1), we obtain
#H 0 (kp , A) · #H 0 (kp , A0 )
=
(||#A||p )2
p∈S∞
Y
=
Y
p∈S∞
Y
#Ĥ 0 (kp , A)
||#A||p
||#A||p
p∈S\S∞
=
Y
||#A||p
p∈S\S∞
=
Y
#Ĥ 0 (kp , A)
p∈S∞
Y #Ĥ 0 (kp , A) · #Ĥ 2 (kp , A)
p∈S∞
#Ĥ 1 (kp , A)
#P 0 (GS , A) · #P 2 (GS , A)
#P 1 (GS , A)
It follows that
ϕ(GS , A) · ϕ(GS , A0 ) = χ2 (GS , A) · χ2 (GS , A0 )/
#P 0 (GS , A) · #P 2 (GS , A)
,
#P 1 (GS , A)
thus showing (i). In order to prove (ii), let
0 −→ A1 −→ A2 −→ A3 −→ 0
be an exact sequence of finite modules in ModS (GS ). From the exact cohomology sequence and (8.6.10)(ii) we obtain
#P 3 (GS , A2 )
χ2 (GS , A1 )χ2 (GS , A3 )
=
χ2 (GS , A2 )
#P 3 (GS , A1 ) · #P 3 (GS , A3 )
·
#P 4 (GS , A1 ) · #P 4 (GS , A3 )
· #δP 4 (GS , A3 )
#P 4 (GS , A2 )
= #δP 4 (GS , A3 )
because the Herbrand index of finite modules is equal to 1 by (1.7.6), and so
#P 3 (GS , A) = #P 4 (GS , A). For p ∈ S∞ the sequence
0 −→ H 0 (kp , A1 ) −→ H 0 (kp , A2 ) −→ H 0 (kp , A3 ) −→ δH 0 (kp , A3 ) −→ 0
is exact and
#δH 0 (kp , A3 ) = #δ Ĥ 0 (kp , A3 ) = #δH 4 (kp , A3 ).
Therefore
Y #H 0 (kp , A1 ) · #H 0 (kp , A3 )
χ2 (GS , A1 ) · χ2 (GS , A3 )
=
,
χ2 (GS , A2 )
#H 0 (kp , A2 )
p∈S∞ (k)
and using ||#A2 ||p = ||#A1 ||p · ||#A3 ||p , this proves assertion (ii).
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497
§6. Poitou-Tate Duality
Proof of (8.6.10)(i): We proceed with several reduction steps.
Step 1. P T (k, S, A) is exact for A = ZZ/pZZ and A = µp , p ∈ IN(S).
Proof of step 1: As the Poitou-Tate sequence is self-dual, it suffices to deal
with the case A = µp . Using (8.3.4), we find the isomorphism
∼ C /p
IS /p −→
S
and the exact sequence
0 −→ µp −→ p IS −→ p CS −→ 0 .
Taking cohomology, we obtain the exact sequence
(1)
H 1 (GS , µp ) −→ H 1 (GS , p IS ) −→ H 1 (GS , p CS ).
Since H 1 (GS , CS ) = 0, we get from the exact sequences
0 → p CS → CS →(CS )p → 0
and
0 →(CS )p → CS → CS /p → 0
the commutative and exact diagram
H 0 (GSDECBFGHIJK , CS )
p
H 0 (GS , (CS )p )
δ
H 1 (GS , p CS )
0
ϑ
H 0 (GS , CS )
p
H 0 (GS , CS )
CS (k)/CS (k)p
0
H 0 (GS , CS /p)
and therefore an exact sequence
(2)
ϑ
0 −→ H 1 (GS , p CS ) −→ CS (k)/p −→ H 0 (GS , CS /p).
Replacing CS by IS , we obtain an analogous sequence
(3)
ϑ
0 −→ H 1 (GS , p IS ) −→ IS (k)/p −→ H 0 (GS , IS /p),
and all three sequences (1)–(3) fit into the commutative and exact diagram
H 1 (GRSTQLMNOP S , µp )
H 1 (GS , p IS )
ϑ
ϑ̃
H 1 (GS , p CS )
ϑ
IS (k)/p
CS (k)/p
H 0 (GS , IS /p)
H 0 (GS , CS /p).
We get an exact sequence
ϑ̃
H 1 (GS , µp ) −→ IS (k)/p −→ CS (k)/p .
By (7.2.13), the diagram
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498
Chapter VIII. Cohomology of Global Fields
H 1 (GYXUWV S , µp )
ϑ̃
IS (k)/p
rec
λ1
P 1 (GS , µp )
Ξ1
Y
P 1 (GS , ZZ/pZZ)∨
Gab
kp /p .
p∈S
is anti-commutative. As DS (k) is divisible by (8.3.13), the reciprocity map
induces an isomorphism
∼ Gab /p .
rec : CS (k)/p −→
S
We obtain the commutative diagram
H 1 (Gab_`^[\]Z S , µp )
ϑ̃
IS (k)/p
CS (k)/p
rec
−Ξ1 ·λ1
Y
rec
Gab
kp /p
Gab
S /p
p∈S
(λ1 )∨
H 1 (GS , ZZ/pZZ)∨ .
The sequence with the diagonal arrows is up to sign the central part of
P T (k, S, µp ), which is therefore exact.
Step 2. P T (k, S, A) is exact if P T (k, T, A) is exact for all finite subsets T
Proof of step 2: We have seen that im λ0S ⊆ (im λS )⊥ , where
λS : H 1 (GS , A) −→ P 1 (GS , A)
and
⊆
S.
λ0S : H 1 (GS , A0 ) −→ P 1 (GS , A0 ),
and the orthogonal complement ⊥ is taken with respect to the pairing
∪
P 1 (GS , A) × P 1 (GS , A0 ) −→ H 2 (GS , IS ) −→
M
Σ
Q/ZZ −→ Q/ZZ.
p∈S
Assume that we have proved
im λ0T = (im λT )⊥
for all finite subsets T of S with #A ∈ IN(T ) (and S∞ ⊆ T in the number
field case) for the corresponding localization maps λ0T and λT . Let πT :
P 1 (GS , A0 ) P 1 (GT , A0 ) be the canonical projection. Then the commutative
diagram
inf
H 1 (Gdcfe T , A0 )
H 1 (GS , A0 )
λ0T
P 1 (GT , A0 )
λ0S
πT
P 1 (GS , A0 )
shows that
im λ0T = πT (im λ0S ).
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499
§6. Poitou-Tate Duality
In order to prove the inclusion (im λS )⊥ ⊆ im λ0S , let x ∈ (im λS )⊥ . Then for
all sufficiently large finite subsets T ⊆ S, we have πT (x) ∈ (im λT )⊥ = im λ0T ,
so that
x∈
\
T
πT−1 (im λ0T ) =
\
πT−1 πT (im λ0S ) = im λ0S .
T
Since λS is proper by (8.6.11), im λS is closed in P 1 (GS , A) and analogously
im λ0S is closed in P 1 (GS , A0 ), and so x ∈ im λ0S = im λ0S . We obtain
im λ0S = (im λS )⊥ .
Hence, in order to prove the exactness of P T (k, S, A), we may assume that S
is finite.
Step 3. Let 0 → A → B → C → 0 be an exact sequence of finite modules
in ModS (GS ). If P T (k, S, A) and P T (k, S, C) are exact, then P T (k, S, B) is
also exact.
Proof of step 3: Since we may assume that S is finite, this follows directly
from (8.6.13).
Step 4: P T (k, S, A) is exact if pA = 0 and A is trivialized by a finite p-extension
of k inside kS , where p is any prime number.
Proof of step 4: By (1.6.13), A has a composition series whose graded pieces
are isomorphic to ZZ/pZZ. Therefore the exactness of P T (k, S, A) follows from
steps 1 and 3.
Step 5: P T (k, S, A) is exact for any finite A ∈ ModS (GS ).
Proof of step 5: By step 3, we may assume that A is a simple module; in
particular, pA = 0 for some prime number p. Let K|k be a finite Galois
subextension of kS |k which trivializes A. Let Gp be a p-Sylow subgroup
of G = G(K|k) and Kp ⊆ K the subfield associated to Gp . By step 4,
P T (Kp , S, A) is exact. The composition of cor and res between the complexes
P T (k, S, A) and P T (Kp , S, A) is multiplication by [Kp : k], which is prime
to p. Hence P T (k, S, A) can be identified with a direct summand of the exact
complex P T (Kp , S, A) and is therefore exact. This finishes the proof.
2
There is an alternative way to prove the exactness of the Poitou-Tate sequence
which avoids the reduction step to finite S and counting the orders of all
cohomology groups occurring. The following proof of step 3 uses some
calculations with double complexes and is taken from [169] (dévissage). We
start with the following
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500
Chapter VIII. Cohomology of Global Fields
Claim: Let (A•• , d0 , d00 ) be a (commutative or anti-commutative) double complex of abelian groups. For (i, j) ∈ ZZ × ZZ we put
Z ij = ker(d0 : Aij → Ai+1,j ) ∩ ker(d00 : Aij → Ai,j+1 ),
B ij = im(d00 ◦ d0 : Ai−1,j−1 → Aij )
and
H ij = Z ij /B ij .
If for (i, j) ∈ ZZ × ZZ the groups H i (A•,j ) and H i+1 (A•,j ) vanish, then we have
an exact sequence
h
H ij −→ H j (Ai,• ) −→ H i+1,j −→ H i,j+1 −→ H j+1 (Ai,• ) .
Proof: The map H j (Ai,• ) → H i+1,j is induced by d0 . The map h is defined
as follows. Let x ∈ H i+1,j and let a ∈ Ai+1,j be a representing element. We
choose b ∈ Aij with d0 (b) = a and define h(x) ∈ H i,j+1 as the class of d00 (b). Our
assumptions imply that h is well-defined. The verification of the exactness of
the sequence is a straightforward diagram chase.
Now we build a double complex A•• in the following way. In column i =
−1 + 3k, k ∈ ZZ, we put P T (k, S, A) shifted by 3k positions downwards. In
column i = 3k, k ∈ ZZ, we put P T (k, S, B) shifted by 3k positions downwards
and in column i = 1 + 3k, k ∈ ZZ, we put P T (k, S, C) shifted by 3k positions
downwards. Let us look at a part of this unbounded double complex.
ghijklmnopqrstuvwxyz{|}~€‚ƒ„…†‡ˆ‰Š‹ŒŽ
H 1 (A0 )∨
H 1 (B 0 )∨
H 1 (C 0 )∨
H 0 (A0 )∨
P 1 (A)
P 1 (B)
P 1 (C)
P 2 (A)
H 1 (A)
H 1 (B)
H 1 (C)
H 2 (A)
H 2 (A0 )∨
H 2 (B 0 )∨
H 2 (C 0 )∨
H 1 (A0 )∨
.
1
The object P (B) is situated at position (0, 0) and the double complex is
periodic with period (3, −3). The rows are the long exact sequences (truncated
if k is not totally imaginary) attached to the short exact sequence 0 → A →
B → C → 0. The columns are exact, except the i = 0 column at j = 0, where
we want to show the exactness.
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501
§6. Poitou-Tate Duality
Applying the claim, we obtain an exact sequence
h0
h
H 1,−1 −→ H 0,0 −→ H 0 (A0,• ) −→ H 1,0 −→ H 0,1
and homomorphisms
h
∼ H 1,−1 ,→ H 0,0 →
∼ H −1,1 ,
H 2,−2 →
h0
∼ H 1,0 H 0,1 →
∼ H −1,2 .
H 2,−1 →
Since H 2,−1 = H −1,2 is a subquotient of X2 (GS , A), it is finite. Hence
h0 is an isomorphism. As H 1,−1 is a subquotient of X1 (GS , C), the group
H 2,−2 = H −1,1 is finite, and so h is an isomorphism. The exact sequence above
shows that H 0 (A0,• ) = 0, hence P T (k, S, B) is exact.
For finitely generated modules we have the following statement which is
weaker than (8.6.10)(i).
(8.6.14) Theorem. Let S be a nonempty set of primes of a global field k and
assume that S ⊇ S∞ if k is a number field. Let A ∈ ModS (GS ). Then we have
an exact sequence
H 1 (GS , A) → P 1 (GS , A) → H 1 (GS , A0 )∨ → H 2 (GS , A) → P 2 (GS , A) .
Proof: The existence and exactness of the partial sequence
P 1 (GS , A) → H 1 (GS , A0 )∨ → H 2 (GS , A) → P 2 (GS , A)
follows from (8.6.7). The existence of the complex
H 1 (GS , A) → P 1 (GS , A) → H 1 (GS , A0 )∨
follows in exactly the same way as for finite A, cf. the proof of (8.6.10)(i). We
denote by AI the cokernel of the map sh1 : H 1 (GS , IS (A)) → P 1 (GS , A0 ).
Recall that sh1 is injective by (8.5.5). Denote by AC the cokernel of the
map α : H 1 (GS , CS (A)) → H 1 (GS , A)∨ induced by the cup-product and the
invariant map H 2 (GS , CS ) → Q/ZZ. By (8.4.4), α is an isomorphism if A is
torsion-free. The commutative diagram
H 1 (GS ,‘’ IS (A))
P 1 (GS , A0 )
H 1 (GS , CS (A))
H 1 (GS , A)∨
induces a homomorphism AI → AC , and our next aim is to show that it is
injective.
Let T ⊆ A be the torsion submodule of A and let F = A/T . The statement of
the theorem is true for A = T by (8.6.10). Therefore we have the commutative
exact diagram
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502
Chapter VIII. Cohomology of Global Fields
H 1 (G™š›œžŸ˜“”•–— S , T 0 )
H 1 (GS , IS (T ))
H 1 (GS , CS (T ))
αT
sh1
H 1 (GS , T 0 )
H 2 (GS , T 0 )
P 1 (GS , T 0 )
H 1 (GS , T )∨
TI
TC
H 2 (GS , T 0 )
We conclude that the maps αT : H 1 (GS , CS (T )) → H 1 (GS , T )∨ and TI → TC
are injective. We proceed by considering the commutative and exact diagram
H 1 (GS ,§¨©ª«¬¥¤ ¡¢£¦ IS (F ))
sh1
P 1 (GS , F 0 )
H 1 (GS , IS (A))
sh1
H 1 (GS , IS (T ))
sh1
P 1 (GS , A0 )
P 1 (GS , T 0 )
AI
TI
H 2 (GS , IS (F ))
sh2
P 2 (GS , F 0 )
By (8.5.5) the map sh2 on the right-hand side induces an isomorphism on the
`-torsion part for all prime numbers ` | #T . Therefore the map AI → TI is
injective and the commutative diagram
A­®¯° I
TI
AC
TC
shows that the map AI → AC is also injective. Next we consider the commutative and exact diagram
H 0 (GS ,±²³´µ¶·¸¹º CS (T ))
H 1 (GS , CS (F ))
H 1 (GS , CS (A))
αA
H 2 (GS , T )∨
H 1 (GS , F )∨
H 1 (GS , A)∨
H 1 (GS , CS (T ))
αT
H 1 (GS , T )∨ .
The left-hand vertical arrow is surjective by (8.4.5). We conclude that the map
αA is injective. Finally, consider the diagram
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503
§7. The Global Euler-Poincaré Characteristic
H 1 (G»¼½¾¿ÀÁÂÃÄ S , A0 )
H 1 (GS , IS (A))
H 1 (GS , CS (A))
H 1 (GS , A0 )
P 1 (GS , A0 )
H 1 (GS , A)∨
AI
AC .
All sequences but the middle row are exact, hence the middle row is also exact.
As Ξ1 is an isomorphism by (8.5.3), we obtain the statement of the theorem by
dualizing the middle row.
2
Theorem (8.6.10) was announced by J. TATE in [229], but he never published
a proof. G. POITOU gave a proof in the case that S is the set of all places in
[169], and we have used several of his arguments here. The extension to finitely
generated modules in (8.6.14) is due to T. TAKAHASHI [226].
§7. The Global Euler-Poincaré Characteristic
In this section we calculate the Euler-Poincaré characteristic of a finite
module A ∈ ModS (GS ), where S is finite. We start by proving an assertion
×
for a finite Galois extension
concerning the Galois module structure of OK,S
K|k of number fields.
(8.7.1) Lemma. Let G be a finite group and E|F an extension of fields. Let
M1 and M2 be finite dimensional F [G]-modules such that M1 ⊗ E and M2 ⊗ E
are isomorphic as E[G]-modules. Then M1 and M2 are F [G]-isomorphic.
Proof: ∗) For finite dimensional F [G]-modules M1 and M2 consider the
isomorphism
HomF [G] (M1 , M2 ) ⊗ E ∼
= HomE[G] (M1 ⊗ E, M2 ⊗ E) ,
∗) The essential part of the proof is taken from [7], chap.IV, §8, lemma.
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Chapter VIII. Cohomology of Global Fields
which is induced from the canonical isomorphism HomF (M1 , M2 ) ⊗ E ∼
=
HomE (M1 ⊗E, M2 ⊗E), cf. [17], chap.II, §5, prop.7(ii), by taking G-invariants.
If M1 ⊗ E ∼
= M2 ⊗ E, then, in particular, dimF M1 = dimF M2 and we can
speak of the determinant of a homomorphism between M1 and M2 (by choosing
bases of M1 and M2 ). Let ηi , i = 1, . . . , m, be an F -basis of HomF [G] (M1 , M2 ).
Then {ηi } is also an E-basis of the vector space HomE[G] (M1 ⊗ E, M2 ⊗ E).
Since M1 ⊗ E and M2 ⊗ E are isomorphic by assumption, there exist ai ∈ E
P
such that det( ai ηi ) =/ 0. Let
X
f (t) = det(
ti ηi ) ∈ F [t1 , . . . , tm ].
f (t) is not the zero polynomial because f (a1 , . . . , am ) =/ 0. Then there exists
a finite extension F 0 of F and an element b = (b1 , . . . , bm ) ∈ F 0 m such that
P
f (b) =/ 0 (if F is infinite, then we can take F 0 = F ). It follows that bi ηi is
an F 0 [G]-isomorphism of M1 ⊗ F 0 onto M2 ⊗ F 0 . Considered as an F [G]∼ M n , where n = [F 0 : F ]. By
homomorphism, this is an isomorphism M1n →
2
the Krull-Schmidt theorem, see [32], §6, thm. 6.12, we obtain M1 ∼
= M2 as
F [G]-modules.
2
(8.7.2) Proposition. Let K|k be a finite Galois extension of number fields
with Galois group G = G(K|k). Let r1 and r2 be the number of real and
complex primes of k, respectively, and let r10 ≤ r1 be the cardinality of the set
0
S∞
of real primes of k which become complex in K. Finally, let S ⊇ S∞ be a
finite set of primes of k and let S f = S r S∞ . Then there are isomorphisms of
Q[G]-modules
M
×
Q ⊕ OK,S
⊗Q ∼
Q
=
P∈S(K)
∼
=
M
G
Ind GP Q
p∈S(k)
0
∼
= Q[G]r2 +r1 −r1 ⊕
M
G
Ind GP Q .
0 ∪S f )(k)
p∈(S∞
Proof: Consider the G-invariant map
×
−→
Lg = (∆, log) : ZZ ⊕ OK,S
M
IR
P∈S(K)
where ∆ is the diagonal embedding of the trivial G-module ZZ and log(a) =
(log |a|P )P∈S(K) . Then by Dirichlet’s unit theorem ker Lg = µ(K) and the
L
image of Lg is a G-lattice Γ of rank s = #S(K) in P∈S(K) IR, cf. [160],
chap.VI, (1.1). It follows that
M
×
⊗ IR ∼
IR
IR ⊕ OK,S
=
P∈S(K)
as IR[G]-modules, hence by (8.7.1) for F = Q and E = IR, we obtain the
desired result.
2
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505
§7. The Global Euler-Poincaré Characteristic
(8.7.3) Corollary. Let K|k be a finite extension of number fields with Galois
group G = G(K|k) and let p be a prime number not dividing the order of G.
Then there exists a ZZp [G]-isomorphism
×
⊗ ZZp ∼
ZZp ⊕ OK,S
= µp∞ (K) ⊕
M
ZZp
P∈S(K)
where S ⊇ S∞ is a finite set of primes and G acts on the sum on the right via
its action on S(K).
Proof: Consider the ZZp [G]-lattices
×
Γ1 = ZZp ⊕ (OK,S
⊗ ZZp )/µp∞ (K)
and
M
Γ2 =
ZZp .
P∈S(K)
Since the order of G is prime to p, every finitely generated, ZZp -free ZZp [G]module is projective by (2.6.11). From the Qp [G]-isomorphism Γ1 ⊗ Qp ∼
=
Γ2 ⊗ Qp obtained in (8.7.2), it follows from representation theory (5.6.10)(ii)
that Γ1 ∼
= Γ2 . Therefore we obtain an exact G-invariant sequence
×
0 −→ µp∞ (K) −→ ZZp ⊕ OK,S
⊗ ZZp −→
M
ZZp −→ 0
P∈S(K)
which splits, since Γ2 is ZZp [G]-projective.
2
Recall that the second partial Euler-Poincaré characteristic of a finite GS module A is defined by
χ2 (GS , A) =
#H 0 (GS , A) · #H 2 (GS , A)
.
#H 1 (GS , A)
(8.7.4) Global Euler-Poincaré Characteristic Formula. Let S be a finite
nonempty set of primes of the global field k containing S∞ if k is a number
field. Let A be a finite GS -module of order #A ∈ IN(S). Then
χ2 (GS , A) =
Y
p∈S∞
Y #Ĥ 0 (kp , A0 )
#H 0 (kp , A)
=
.
0
0
||#A||p
p∈S∞ #H (kp , A )
If k is a function field, this simply says χ2 (GS , A) = 1.
Remarks: 1. Recall that ||#A||p is #A if p is real and (#A)2 if p is complex.
2. The second equality in the statement (8.7.4) follows from (7.3.5).
3. If k is a function field or a totally imaginary number field and A a finite
module in ModS (GS ), then χ(GS , A) = χ2 (GS , A) by (8.3.17) and (8.3.18).
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Chapter VIII. Cohomology of Global Fields
Proof: First we prove the formula for the Euler-Poincaré characteristic for
the GS -module A = µp , where p ∈ IN(S). Let K = k(µp ) and let G = G(K|k).
We consider the Grothendieck group K00 (IFp [G]) of finitely generated IFp [G]modules M and we denote the corresponding class of M by [M ]. We obtain
(i)
[H 0 (kS |K, µp )] = [µp ],
(ii)
×
[H 1 (kS |K, µp )] = [OK,S
/p] + [p ClS (K)],
(iii) [H 2 (kS |K, µp )] = [ClS (K)/p] − [IFp ]
+[
M
M
IFp ] + [
Ĥ 0 (Gp , ZZ/pZZ)],
p∈S∞ (K)
p∈S\S∞ (K)
where S∞ is redundant in the function field case. (i) is obvious and (ii) follows
from the exact Kummer sequence (8.3.4), which induces the exact sequence
×
0 −→ OK,S
/p −→ H 1 (kS |K, µp ) −→ p ClS (K) −→ 0
where we used (8.3.11)(ii). From (8.6.12) we obtain the exact sequence
0 → H 0 (kS |K, ZZ/pZZ) → P 0 (kS |K, ZZ/pZZ) → H 2 (kS |K, µp )∨
→ X1 (G(kS |K), ZZ/pZZ) → 0 .
We have
X1 (G(kS |K), ZZ/pZZ)∨ = ClS (K)/p
by (8.6.3), and since
M
P 0 (kS |K, ZZ/pZZ) =
M
IFp ⊕
Ĥ 0 (Gp , ZZ/pZZ) ,
p∈S∞ (K)
p∈S\S∞ (K)
we obtain (iii).
If k is a function field, then
×
[OK,S
/p ⊕ IFp ] = [
M
IFp ] + [µp ]
p∈S(K)
and we obtain
2
X
(−1)i [H i (GS , µp )] = 0
i=0
since [p ClS (K)] = [ClS (K)/p]. This proves the formula in the function field
case for A = µp .
We proceed with the proof for number fields in the case that A = µp . From
(8.7.3) we obtain
×
[OK,S
/p] = [
M
IFp ] + [µp ] − [IFp ].
p∈S(K)
Combining this with (i)−(iii), we obtain
2
X
M
i=0
p∈S∞ (K)
(−1)i [H i (kS |K, µp )] = [
Ĥ 0 (Gp , ZZ/pZZ)] − [
M
IFp ].
p∈S∞ (K)
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507
§7. The Global Euler-Poincaré Characteristic
Now we consider the homomorphism
Θ : K00 (IFp [G]) −→ ZZ , [M ] 7−→ dimIFp M G
(observe that #G is equal to 1 for p = 2 and is prime to p for p =/ 2, so that
M 7→ M G is an exact functor). Obviously,
Θ
2
P
χ2 (GS , M ) = p
(−1)i [H i (G(kS |K),M )]
.
i=0
Therefore
χ2 (GS , µp ) = 1
for function fields and
χ2 (GS , µp ) =
#Ĥ 0 (kp , IFp )
0
p∈S∞ (k) #H (kp , IFp )
Y
for number fields. This finishes the proof in the case that A = µp .
Now we will prove the formula for the Euler-Poincaré characteristic for an
arbitrary finite GS -module whose order is in IN(S). Recall that the function
ϕ(GS , −) defined by
ϕ(GS , A) = χ2 (GS , A)/
Q
= χ2 (GS , A)/
Q
p∈S∞
#H 0 (kp ,A)
||#A||p
#Ĥ 0 (kp ,A0 )
p∈S∞ #H 0 (kp ,A0 )
.
is multiplicative on short exact sequences, see (8.6.13)(ii). A second property
of ϕ is
ϕ(GS , IndUGS A) = ϕ(U, A)
for U ⊆ GS open .
For the χ2 -part of ϕ, this follows from Shapiro’s lemma. Furthermore,
||#IndUGS A||p =
Y
||#A||P ,
P|p
because #IndUGS A = (#A)(GS :U ) , and
#H 0 (kp , IndUGS A) =
Y
#H 0 (KP , A)
P|p
for p ∈ S∞ (k), where K denotes the fixed field of U . The last equality is trivial
for complex primes, and if p is real, we have by I §5 ex. 5
H 0 (Gp , IndUGS A) =
Y
H 0 (GσP0 , Aσ ).
σ ∈Gp \GS /U
Thus the desired property of ϕ for induced modules is proved.
The reduction from the general statement (8.7.4) to the case A = µp now
goes as follows. Let K|k be a finite subextension of kS |k trivializing A and
A0 with Galois group G = G(K|k). Since ϕ is multiplicative, we may assume
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508
Chapter VIII. Cohomology of Global Fields
that pA = 0 for some prime number p ∈ IN(S), and we consider ϕ as a
homomorphism from the Grothendieck group K00 (IFp [G]) of finitely generated
IFp [Ḡ]-modules into Q×
+:
ϕ : K00 (IFp [G]) −→ Q×
+ .
We want to prove that ϕ is identical to 1. The group Q×
+ is torsion-free. Thus,
C
N ) = ϕ(C, N ) = 1,
using lemma (7.3.4), we have to show that ϕ(GS , IndḠ
where C is a cyclic subgroup of Ḡ of order prime to p, C is the open subgroup
of GS given by C = C/G(kS |K) and N is a finitely generated IFp [C̄]-module.
Hence we may assume that K|k is a cyclic extension of degree prime to p
trivializing A and A0 . We have
H i (kS |k, A) ∼
for all i ≥ 0.
= H i (kS |K, A)G
Let
χ0 : K00 (IFp [G]) −→ K00 (IFp [G]) , [M ] 7−→
2
X
(−1)i [H i (kS |K, M )] ,
i=0
and
Θ : K00 (IFp [G]) −→ ZZ , [M ] 7−→ dimIFp M G ;
then
0
χ2 (GS , M ) = pΘχ ([M ])
(observe that M →
7 M G is an exact functor). ∗)
Claim: For a finite IFp [G]-module M , we have
χ0 ([M ]) = [M 0∨ ] χ0 ([µp ]) .
Proof: The pairing
µp × Hom(M 0 , IFp ) −→ Hom(M 0 , µp ) = M ,
(ζ, f ) 7→ (x 7→ ζ f (x) ) ,
defines G-isomorphisms via the cup-product
∼ H i (k |K, M )
H i (kS |K, µp ) ⊗ Hom(M 0 , IFp ) −→
S
(recall that µp and M are trivial G(kS |K)-modules). This proves the claim.
Since χ0 ([µp ]) = 0 for function fields, we obtain χ2 (GS , A) = 1 in this case.
For the rest of the proof, let k be a number field. For p ∈ S∞ (K) we define
ψp0 : K00 (IFp [G]) −→ K00 (IFp [G]) , [M ] 7−→
[Ĥ 0 (KP , M )]−[H 0 (KP , M 0 )] ,
X
P|p
and
ψ 0 : K00 (IFp [G]) −→ K00 (IFp [G]) , [M ] 7−→
X
ψp0 ([M ]) .
p∈S∞ (k)
∗) Θ is not a ring-homomorphism with respect to the ring structure of K 0 (IF [G]) given by
p
0
the tensor product.
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509
§7. The Global Euler-Poincaré Characteristic
Obviously,
ψp0 ([M ]) = [M 0 ] ψp0 ([µp ]).
Using the claim and the assertion χ0 ([µp ]) = ψ 0 ([µp ]) proven at the beginning
of this part, we find for the finite GS -module A that
0
0
ϕ(GS , A) = pΘχ ([A])−Θψ ([A])
=
Y
pΘ([A
0∨ ]ψ 0 ([µ ])−[A0 ]ψ 0 ([µ ]))
p
p
p
p
.
p∈S∞ (k)
It remains to prove that for p ∈ S∞ (k) and for an IFp [G]-module A,
Θ([A∨ ]ψp0 ([µp ])) = Θ([A]ψp0 ([µp ])) .
It is easy to see that
0,
(
ψp0 ([µp ])
=
P
−[IndG
G IFp ] ,
p = 2, p is real and remains real in K,
otherwise.
This proves the equality above if p = 2 and p is real and remains real in K.
Therefore we assume that we are in one of the other cases. Then it follows
G
P
from IndḠP IFp ⊗ A ∼
= IndG
G A and
0
P
H 0 (G, IndG
G A) = H (GP , A)
that
p
Θ([A]ψp0 ([µp ])) = Θ(−[IndG
G A])
G
p
= − dimIFp (IndG
G A)
= − dimIFp AGP .
Since
dimIFp AGP = dimIFp AGP = dimIFp (A∨ )GP ,
the equality holds in all cases. This finishes the proof.
2
(8.7.5) Corollary. Let p be a prime number and let S be a finite set of primes
of the number field k containing S∞ and all primes above p. If p =/ 2, then for
all j ∈ ZZ,
2
X
(
i
i
(−1) dimIFp H (GS , ZZ/pZZ(j)) =
i=0
−r1 (k) − r2 (k) for j odd,
−r2 (k)
for j even,
and if p = 2, then
2
X
(−1)i dimIF2 H i (GS , ZZ/2ZZ) = −r2 (k) .
i=0
Here r1 (k) and r2 (k) denote the number of real and complex places of the
number field k. In the function field case, this alternating sum of dimensions
is zero.
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Chapter VIII. Cohomology of Global Fields
Proof: If p is odd, then for a number field k and an archimedean prime p,
the valuation ||p||p is equal to p resp. p2 if p is real resp. complex, and the
order of H 0 (kp , ZZ/pZZ(j)) is 1 for p real, j odd, and p in all other cases. In
the case p = 2, observe that all twists are equal, and we obtain the result as
above.
2
(8.7.6) Corollary. With the notation of (8.7.5), the following equalities hold
for p ≥ 2 and all j ∈ ZZ:
2
X
(
i
(−1) rank ZZp Hi (GS , ZZp (j)) =
i=0
−r1 (k) − r2 (k) for j odd,
−r2 (k)
for j even,
and
2
X
(
(−1)
i
i
rank ZZp Hcts
(GS , ZZp (j))
=
i=0
−r1 (k) − r2 (k) for j odd,
−r2 (k)
for j even.
Proof: The second equality follows from the first using (2.7.12), and with
exactly the same arguments as in the proof of (7.3.8) (replacing Gk by GS ),
we obtain
2
X
i
(−1) rank ZZp Hi (GS , ZZp (j)) =
2
X
(−1)i dimIFp H i (GS , ZZ/pZZ(−j))
i=0
i=0
−dimIFp p H2 (GS , ZZp (j)) .
Now the corollary follows from (8.7.5) and the following lemma.
2
(8.7.7) Lemma. Let k be a global field, let p =/ char(k) be a prime number
and let S be a finite nonempty set of primes of k containing S∞ and all primes
above p if k is a number field.
If p is odd, then for all j ∈ ZZ, the homology group H2 (GS , ZZp (j)) is ZZp torsion-free, and if p = 2, then
(
tor(H2 (GS , ZZ2 (j))) =
(ZZ/2ZZ)r1 (k) for j odd,
0
for j even.
Proof: We prove the dual statement, i.e. H 2 (GS , Qp /ZZp (−j)) is p-divisible
if p is odd or j is even, and its co-torsion is equal to (ZZ/2ZZ)r1 (k) otherwise.
This is clear for function fields by (8.3.17) and in the number field case for
p =/ 2 since cdp GS ≤ 2 in these cases. We have the commutative diagram
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511
§7. The Global Euler-Poincaré Characteristic
ÅÆÇÈ m ZZ(−j))
H 3 (GS , ZZ/p
pm H
H 3 (kp , ZZ/pm ZZ(−j))
Y
p∈S∞ (k)
Y
3
(GS , Qp /ZZp (−j))
pm H
3
(kp , Qp /ZZp (−j))
p∈S∞ (k)
where the vertical maps are isomorphisms by (8.6.10)(ii). Thus
∼
H 2 (GS , Qp /ZZp (−j))/pm −→
Y
H 2 (kp , Qp /ZZp (−j))/pm .
p∈S∞ (k)
Since we have
H (kp , Q2 /ZZ2 (−j)) ∼
= Ĥ 0 (kp , Q2 /ZZ2 (−j)) ∼
=
2
(
ZZ/2ZZ for j odd,
0
for j even,
2
for a real prime p ∈ S∞ (k), this yields the lemma.
A very useful application of the Poitou-Tate duality theorem and the global
Euler characteristic formula is a duality statement for certain subgroups of
H 1 (Gk , A) which are defined by local conditions. In particular, this is important
for the theory of elliptic curves where one considers the so-called Selmer group
of an elliptic curve.
(8.7.8) Definition. Let k be a global field and A a finite Gk -module of order
prime to the characteristic of k . A collection L of local conditions for A is a
family Lp of subgroups of H 1 (Gp , A) such that
Lp = H 1 (Gp /Tp , ATp ) = ker(H 1 (Gp , A) → H 1 (Tp , A))
for almost all p, where Tp denotes the inertia subgroup of Gp .
If L is a collection of local conditions for A, then LD is a collection of local
conditions for the Gk -module A0 = Hom(A, µ) defined by
LD
p := the orthogonal complement of Lp with respect to the pairing
∪
H 1 (Gp , A) × H 1 (Gp , A0 ) −→ Q/ZZ .
The corresponding global groups are defined by
HL1 (k, A) := ker H 1 (Gk , A) −→
Y
H 1 (Gp , A)/Lp ,
p
HL1 D (k, A0 ) := ker H 1 (Gk , A0 ) −→
Y
H 1 (Gp , A0 )/LD
p .
p
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512
Chapter VIII. Cohomology of Global Fields
If L is a collection of local conditions for the finite Gk -module A, then we
choose a finite nonempty set S of primes of k containing all primes p such
that ATp =/ A or Lp =/ H 1 (Gp /Tp , ATp ), and S∞ and all divisors of #A if k is
number field. Hence A and A0 are GS -modules and we have
HL1 (k, A) = ker H 1 (GS , A) −→
Y
H 1 (Gp , A)/Lp ,
p∈S
HL1 D (k, A0 ) = ker H 1 (GS , A0 ) −→
Y
H 1 (Gp , A0 )/LD
p .
p∈S
The following theorem is due to A. WILES.
(8.7.9) Theorem. Let k be a global field and A a finite Gk -module of order
prime to char(k), and let L be a collection of local conditions for A. Then
#HL1 (k, A)
#H 0 (k, A) Y
#Lp
=
·
.
1
0
0
0
0
#HLD (k, A ) #H (k, A ) p #H (kp , A)
Proof: Let S be a finite set of primes of k chosen as above. Recall, by
convention we have S∞ = ∅ in the function field case. From the commutative
exact diagram
ÉÊËÌÍÎÏÐÑÒÓ
Y
Y
H 1 (kp , A)/Lp
p∈S
X1 (GS , A)
H 1 (kp , A)/Lp
HL1 D (k, A0 )∨
p∈S
Y
H 1 (GS , A)
H 1 (kp , A)
H 1 (GS , A0 )∨
p∈S
Y
HL1 (k, A)
Lp
p∈S
Y
∨
(H 1 (kp , A0 )/LD
p ) ,
p∈S
we obtain the exact sequence
0 → X1 (GS , A) → HL1 (k, A) →
Y
Lp → H 1 (GS , A0 )∨ → HL1 D (k, A0 )∨ → 0,
p∈S
and so the exact sequence
0 → H 0 (GS , A) → P 0 (GS , A) → H 2 (GS , A0 )∨
→ HL1 (k, A) →
Y
Lp → H 1 (GS , A0 )∨ → HL1 D (k, A0 )∨ → 0.
p∈S
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513
§8. Duality for Unramified and Tamely Ramified Extensions
Recall that in the number field case, for p ∈ S∞ , the group H 0 (Gp , A) is
replaced by Ĥ 0 (Gp , A) in the term P 0 (GS , A). Using (8.7.4), we get
Y
Y
#HL1 (k, A)
#H 0 (GS , A)
#Lp
#Lp
=
·χ2 (GS , A0 )·
·
1
0
0
0
0
0
#HLD (k, A ) #H (GS , A )
p∈S∞ #Ĥ (kp , A) p∈S\S∞ #H (kp , A)
=
Y
#H 0 (k, A) Y
#Lp
#Lp
·
·
.
0
0
0
0
#H (k, A ) p∈S∞ #H (kp , A) p∈S\S∞ #H (kp , A)
But this is the desired result, since for p ∈/ S the group Lp is equal to
H 1 (Gp /Tp , ATp ) and for all p we have #H 1 (Gp /Tp , ATp ) = #H 0 (Gp , A) because of the exact sequence
0ÔÕÖ×Ø
H 0 (Gp , A)
ATp
1−Frobp
ATp
H 1 (Gp /Tp , ATp )
0. 2
§8. Duality for Unramified and Tamely Ramified
Extensions
In this section we prove duality theorems for unramified and tamely ramified
extensions of a global field.
Let us fix some notation. Let k be a global field and let S and T be sets of
primes of k. Further, let c be a full class of finite groups. We set
kST
kST (c)
the maximal extension of k unramified outside S in which all
primes of T are completely decomposed,
the maximal c-extension of k inside kST .
From now on we assume that T is nonempty and contains S∞ if k is a number
field. Furthermore, we assume that S ∩ T = ∅; in particular, S contains only
nonarchimedean primes. We set
kST,tr
the maximal extension of k unramified outside S in which all
primes of T are completely decomposed and all primes of S
are at most tamely ramified,
kST,tr (c)
the maximal c-extension of k inside kST,tr .
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514
Chapter VIII. Cohomology of Global Fields
T,tr
∅
and the field k∅
is the maximal unramified
Remarks: 1. We have k∅T = k∅
∅
S∞
extension of k. If k is a number field, we have k∅ = k∅
.
2. Let k be a number field and let S be a set of finite primes of k such that
S ∩ Sp = ∅ for all prime numbers p with ZZ/pZZ ∈ c. Then a c-extension is
automatically tamely ramified at the primes in S, and we have kSS∞ ,tr (c) = kS (c).
For any extension K of k inside kST,tr we put
S
EK,T
= ker
×
OK,T
−→
Y
1
UK,p /UK,p
,
p∈S(K)
S
i.e. EK,T
is the group of T -units which are principal units at all places of S. If
S
S
T = S∞ , we write EK
= EK,S
.
∞
The main result of this section is the following theorem (8.8.1). It generalizes
a duality theorem for unramified extensions contained in [257] and a duality
theorem for tamely ramified extensions obtained by G. WIESE in [248].
(8.8.1) Theorem. Let k be a global field and let T be a nonempty set of primes
of k which contains S∞ if k is a number field. Let S be a finite set of primes
disjoint from T and let c be any full class of finite groups. Let A be a discrete
G(kST,tr (c)|k)-module which is finitely generated and free as a ZZ-module. Then
there are canonical topological isomorphisms
T,tr
Ĥ i (G(kST,tr (c)|k), Hom(A, EkST,tr (c),T )) ∼
= Ĥ 3−i (G(kS (c)|k), A)∨
S
for all i ∈ ZZ.
Before proving the theorem, we deduce some corollaries.
(8.8.2) Corollary. Let k be a number field and let T be a set of primes of k
containing S∞ . For the maximal unramified extension k∅T in which all primes
of T are completely decomposed there are natural isomorphisms
Ĥ i (G(k∅T |k), Ok×T ,T ) ∼
= Ĥ 3−i (G(k∅T |k), ZZ)∨
∅
for all i ∈ ZZ. In particular, H 2 (G(k∅T |k), Ok×T ,T ) = 0.
∅
Proof: This follows from (8.8.1) by setting c to be the class of all finite
groups, S = ∅ and A = ZZ.
2
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515
§8. Duality for Unramified and Tamely Ramified Extensions
(8.8.3) Corollary. Let k be a number field and let S be a finite set of nonarchimedean primes of k. Let p be a prime number such that S ∩ Sp = ∅. Then
there are natural isomorphisms for all i ∈ ZZ
Ĥ i (G(kS (p)|k), EkSS (p) ) ∼
= Ĥ 3−i (G(kS (p)|k), ZZ)∨ .
In particular, H 2 (G(kS (p)|k), EkSS (p) ) = 0.
Proof: By remark 2 above, we have kSS∞ ,tr (p) = kS (p). Therefore, applying
(8.8.1) to the class of finite p-groups with T = S∞ and A = ZZ, we obtain the
result.
2
In order to prove (8.8.1), we introduce some further objects. Let K be a
finite extension of k inside kST,tr . We consider the group
S
UK,T
=
Y
UK,p ×
Y
Kp× ×
p∈T
/(S∪T )
p∈
Y
1
UK,p
,
p∈S
which is a subgroup of the idèle group IK . Let ClTS (K) be the cokernel of
S
the composite map UK,T
−→ IK −→ CK . The notation is motivated by the
∅
identification ClT (K) = ClT (K). The reciprocity homomorphism of global
class field theory induces an isomorphism of finite abelian groups
∼ G(k T,tr |K)ab .
ClTS (K) −→
S
For an infinite extension K of k in kST,tr we define the above groups as the
direct limit over the finite layers. For any K, we obtain an exact sequence
S
S
0 −→ EK,T
−→ UK,T
−→ CK −→ ClTS (K) −→ 0.
We have ClTS (kST,tr (c))(c) = 0 by the principal ideal theorem.
In the following we use the notation Ĥ i (K|k, −) for Ĥ i (G(K|k), −). We
start with a lemma.
(8.8.4) Lemma. We have a natural isomorphism
H 1 (kST,tr (c)|k, EkST,tr (c),T ) ∼
= ClTS (k)(c).
S
Proof: Let ZZ(c) be the subring of Q obtained from ZZ by inverting all primes p
with ZZ/pZZ ∈/ c. For an abelian torsion group A, we have a natural isomorphism
∼ A⊗ Z
A(c) →
Z(c) . Since the extension kST,tr (c)|k is unramified outside S, tamely
ramified at S and completely decomposed at T , the G(kST,tr (c)|k)-module
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Chapter VIII. Cohomology of Global Fields
UkST,tr (c),T is cohomologically trivial; see (7.1.2). Since ClTS (kST,tr (c))(c) = 0,
S
we have an exact sequence
0 → EkST,tr (c),T ⊗ ZZ(c) → UkST,tr (c),T ⊗ ZZ(c) → CkT,tr (c) ⊗ ZZ(c) → 0 ,
S
S
S
which induces a natural isomorphism
∼ coker U S → C
H 1 (kST,tr (c)|k, EkST,tr (c),T ) −→
Z(c) ∼
= ClTS (k)(c).
k ⊗Z
k,T
S
2
This shows the statement.
Proof of (8.8.1): Applying (8.4.1) to K = kST,tr (c), we obtain natural isomorphisms
T,tr
Ĥ i (k T,tr (c)|k, Hom(A, C T,tr )) ∼
= Ĥ 2−i (k (c)|k, A)∨ , i ∈ ZZ.
S
kS
(c)
S
Let K|k be a finite Galois extension inside kST,tr (c) which trivializes A. We
consider the exact sequence
(∗)
S
S
0 −→ EK,T
−→ UK,T
−→ CK −→ ClTS (K) −→ 0.
The association K 7→ ClTS (K) is merely a G(kST,tr (c)|k)-modulation but not
a module, as we are lacking Galois descent; see (1.5.10) and page 56. However, for each fixed K, the group G(K|k) acts on ClTS (K), and we will consider ClTS (K) as a G(K|k)-module. Thus (∗) is an exact sequence of disS
crete G(K|k)-modules. The group UK,T
is a cohomologically trivial G(K|k)module. Applying Hom(A, −) and taking cohomology, we therefore obtain
isomorphisms
Ĥ i+1 (K|k, Hom(A, E S )) ∼
= Ĥ i (K|k, Hom(A, X(K))),
K,T
where X(K) denotes the kernel of the surjection CK ClTS (K), and a long
exact sequence
· · · −→ Ĥ i (K|k, Hom(A, X(K))) −→ Ĥ i (K|k, Hom(A, CK ))
−→ Ĥ i (K|k, Hom(A, ClTS (K))) −→ · · · .
As the cohomology groups of Hom(A, CK ) and of Hom(A, ClTS (K)) are finite,
the same is true for the cohomology groups of Hom(A, X(K)).
Now let K 0 be another finite Galois extension of k inside kST,tr (c) which
contains K. We consider ClTS (K) as a G(K 0 |k)-module via the projeccan
tion G(K 0 |k) G(K|k). Then the natural maps ClTS (K) −→ ClTS (K 0 ) and
norm
ClTS (K 0 ) −→ ClTS (K) are homomorphisms of G(K 0 |k)-modules and we obtain
associated homomorphisms
inf : Ĥ i (K|k, Hom(A, ClTS (K))) → Ĥ i (K 0 |k, Hom(A, ClTS (K 0 ))), i ≥ 1,
def : Ĥ i (K 0 |k, Hom(A, ClTS (K 0 ))) → Ĥ i (K|k, Hom(A, ClTS (K))), i ≤ 0.
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517
§8. Duality for Unramified and Tamely Ramified Extensions
Similarly, we obtain associated maps between the cohomology groups of
Hom(A, X(K 0 )) and Hom(A, X(K)).
Let K 0 be the maximal abelian extension of K in kST,tr (c). By class field
theory, K 0 is a finite extension of K and we have a natural isomorphism
∼ G(K 0 |K). The commutative diagram
rec : ClTS (K)(c) →
norm
ÜÛÚÙ 0 )(c)
ClTS (K
ClTS (K)(c)
rec
rec
G(kST,tr (c)|K 0 )ab can G(kST,tr (c)|K)ab
shows, since can is the zero map, that
Ý 0 )(c) norm ClS (K)(c)
ClTS (K
T
is the zero map. Noting that
Ĥ i (K|k, Hom(A, ClTS (K))) = Ĥ i (K|k, Hom(A, ClTS (K)(c))),
we obtain
lim Ĥ i (K|k, Hom(A, ClTS (K))) = 0 for i ≤ 0.
←−
K
Furthermore,
since
lim Ĥ i (K|k, Hom(A, ClTS (K))) = 0
−→
K
for i ≥ 1,
can
Þ
ClTS (K)(c)
ClTS (K 0 )(c)
is the zero map by the principal ideal theorem. Passing to the inverse, resp.
direct, limit, we obtain isomorphisms
T,tr
lim Ĥ i (K|k, Hom(A, X(K))) ∼
= Ĥ i (k (c)|k, Hom(A, C T,tr )), i ≤ 0,
S
←−
K, def
kS
(c)
T,tr
lim Ĥ i (K|k, Hom(A, X(K))) ∼
= Ĥ i (kS (c)|k, Hom(A, CkT,tr (c) )), i ≥ 1.
−→
K, inf
S
Therefore we obtain canonical isomorphisms for i ≥ 2:
S
))
Ĥ i (kST,tr (c)|k, Hom(A, ETS )) ∼
H i (K|k, Hom(A, EK,T
= lim
−→ K
∼
= lim Ĥ i−1 (K|k, Hom(A, X(K)))
−→ K
T,tr
∼
= Ĥ i−1 (kS (c)|k, Hom(A, CkT,tr (c) ))
S
3−i T,tr
∨
∼
Ĥ
(k
(c)|k,
A)
,
=
S
and for i ≤ 0:
S
Ĥ i (kST,tr (c)|k, Hom(A, ETS )) ∼
H i (K|k, Hom(A, EK,T
))
= lim
←− K
∼
Ĥ i−1 (K|k, Hom(A, X(K)))
= lim
←− K
T,tr
∼
= Ĥ i−1 (k (c)|k, Hom(A, C T,tr ))
S
T,tr
∼
= Ĥ 3−i (kS (c)|k, A)∨ ,
kS
(c)
where ETS = EkST,tr (c),T .
S
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518
Chapter VIII. Cohomology of Global Fields
For i = 1, we need an additional argument. Let K0 |k be a finite Galois
extension inside kST,tr (c) which trivializes A and let K be a finite Galois extension of k inside kST,tr (c) containing the maximal abelian extension K0ab of
K0 in kST,tr (c). Since H 1 (kST,tr (c)|K0 , ETS ) ∼
= ClTS (K0 )(c) by (8.8.4), we obtain
from the Hochschild-Serre spectral sequence for the group extension
1 −→ G(kST,tr (c)|K) −→ G(kST,tr (c)|K0 ) −→ G(K|K0 ) −→ 1
the exact sequence
S
0 −→ H 1 (K|K0 , EK,T
) −→ ClTS (K0 )(c) −→ ClTS (K)(c).
The map on the right is zero as K contains the field K0ab , and we get an
isomorphism
inf
S
)
H 1 (K|Kß 0 , EK,T
H 1 (kST,tr (c)|K0 , ETS ) ∼
= ClTS (K0 )(c).
Therefore we obtain an exact sequence
S
S
)) −→ H 1 (K|k, Hom(A, EK,T
)) −→
0 −→ H 1 (K0 |k, Hom(A, EK
0 ,T
S
H 0 (K0 |k, Hom(A, ClTS (K0 )(c))) −→ H 2 (K0 |k, Hom(A, EK
)),
0 ,T
which shows, by comparing it with the same sequences for all extensions of K
inside kST,tr (c), that
S
à
))
H 1 (K|k, Hom(A,
EK,T
H 1 (kST,tr (c)|k, Hom(A, ETS ))
is an isomorphism. In exactly the same way one proves that the injective map
inf
H 2 (K|k, A) → H 2 (kST,tr (c)|k, A) induces an isomorphism
inf
á
H 2 (K|k,
A)
inf
H 2 (kST,tr (c)|k, A).
S
S
)G(K|k) and
)G(K|k) , Hom(A, EK,T
Furthermore, since the groups Hom(A, UK,T
ab
1
S
H (K|k, Hom(A, EK,T )) do not depend on K ⊇ K0 , the exact sequence
S
S
0 −→ Hom(A, EK,T
)G(K|k) −→ Hom(A, UK,T
)G(K|k) −→
S
Hom(A, X(K))G(K|k) −→ H 1 (K|k, Hom(A, EK,T
)) −→ 0
shows that the natural map
0
∼ Hom(A, X(K 0 ))G(K |k)
Hom(A, X(K))G(K|k) −→
âãäåæçè
is an isomorphism for any finite Galois extension K 0 inside kST,tr (c) containing
K. If K 0 is the maximal abelian extension of K in kST,tr (c), then in the
commutative and exact diagram
0
Hom(A, X(K 0 ))G(K |k)
0
Hom(A, CK 0 )G(K |k)
0
Hom(A, ClTS (K 0 ))G(K |k)
ϕ
Hom(A, ClTS (K))G(K|k)
Hom(A, X(K))G(K|k)
Hom(A, CK )G(K|k)
the canonical map ϕ is zero on the c-part. This implies an isomorphism
∼ Hom(A, C )G(K|k) ⊗ Z
Hom(A, X(K))G(K|k) ⊗ ZZ(c) −→
Z(c) ,
K
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§8. Duality for Unramified and Tamely Ramified Extensions
519
where ZZ(c) is the subring of Q obtained from ZZ by inverting all primes p with
ZZ/pZZ ∈/ c. Now we consider the commutative and exact diagram
NK|k Hom(A, éêëìíîïX(K)) ⊗ ZZ(c)
NK|k Hom(A, CK ) ⊗ ZZ(c)
Hom(A, X(K))G(K|k) ⊗ ZZ(c)
Hom(A, CK )G(K|k) ⊗ ZZ(c)
Ĥ 0 (K|k, Hom(A, X(K)))
Ĥ 0 (K|k, Hom(A, CK )).
Replacing k by K0 (i.e. A ∼
= ZZrankA ), the lower map is an isomorphism since
S
Ĥ 0 (K|K0 , X(K)) ∼
)
= Ĥ 1 (K|K0 , EK,T
T,tr
S
∼
)
= H 1 (kS (c)|K0 , EK,T
2 T,tr
∨
∼
= Ĥ (k (c)|K0 , ZZ)
S
∼
= Ĥ 2 (K|K0 , ZZ)∨
∼
= G(K|K0 )ab
∼
= Ĥ 0 (K|K0 , CK ).
Therefore
∼ N
NK|K0 Hom(A, X(K)) ⊗ ZZ(c) −→
Z(c)
K|K0 Hom(A, CK ) ⊗ Z
is an isomorphism. As NK|k = NK0 |k ◦ NK|K0 , we obtain an isomorphism
∼ N
NK|k Hom(A, X(K)) ⊗ ZZ(c) →
Z(c) .
K|k Hom(A, CK ) ⊗ Z
This shows that
∼ Ĥ 0 (K|k, Hom(A, C ))
Ĥ 0 (K|k, Hom(A, X(K))) →
K
is an isomorphism and therefore
S
)) ∼
H 1 (K|k, Hom(A, EK,T
= Ĥ 0 (K|k, Hom(A, X(K)))
∼
= Ĥ 0 (K|k, Hom(A, CK ))
∼
= Ĥ 2 (K|k, A)∨ .
As the inflation maps
inf
S
H 1 (K|k, Hom(A, EK,T
)) −→ H 1 (kST,tr (c)|k, Hom(A, ETS ))
inf
and H 2 (K|k, A) −→ H 2 (kST,tr (c)|k, A) are isomorphisms, this shows the existence of the required isomorphism for i = 1.
It remains to show that this isomorphism is natural, i.e. independent of
the choice of K. For typesetting reasons, we prove this below in the case
c = (all finite groups), leaving the necessary modification in the general case
to the reader. We consider the perfect pairing of finite groups
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Chapter VIII. Cohomology of Global Fields
S
H 1 (K|k, Hom(A, EK,T
)) × H 2 (K|k, A) → Q/ZZ
(∗)
S
∼ H 2 (K|k, A)∨ ob)) →
associated to the isomorphism H 1 (K|k, Hom(A, EK,T
tained above. By construction, it is induced via the vertical isomorphisms in
the commutative diagram
ñðòôõó
Ĥ 0 (K|k, Hom(A,
X(K))) × H 2 (K|k, A)
∪
H 2 (K|k, X(K))
o δ
inv0
Q/ZZ
δ
S
H 1 (K|k, Hom(A, EK,T
)) × H 2 (K|k, A)
∪
S
H 3 (K|k, EK,T
),
followed by the pairing in the upper line, where inv0 denotes the composite
map
inv
H 2 (K|k, X(K)) −→ H 2 (K|k, CK ) −→ Q/ZZ.
Consider the commutative diagram
÷öýûüùúøþÿ E S )
H 3 (K|k,
K,T
δ
H 2 (K|k, CK )
inf
inf
H 3 (kST,tr |k, ETS )
H 2 (K|k, X(K))
δ
∼
H 2 (kST,tr |k, X(kST,tr ))
inv
Q/ZZ
inf
H 2 (kST,tr |k, CkT,tr )
inv
Q/ZZ.
S
Denoting the map H 3 (kST,tr |k, ETS ) → Q/ZZ induced by the lower line by inv00 ,
we conclude that the perfect pairing (∗) is the cup-product pairing
∪
S
S
H 1 (K|k, Hom(A, EK,T
)) × H 2 (K|k, A) −→ H 3 (K|k, EK,T
)
inf
inv00
S
) −→ H 3 (kST,tr |k, ETS ) −→ Q/ZZ.
composed with the composite H 3 (K|k, EK,T
It is therefore compatible via the inflation maps with the analogous diagram
for an extension K 0 of K in kST,tr . This concludes the proof.
2
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Chapter IX
The Absolute Galois Group
of a Global Field
In this chapter we will deal with two fundamental questions in algebraic
number theory. The first of these questions is the determination of all extensions
of a fixed base field k (where the most important case is k = Q), which means
exploring how these extensions are built up over each other, how they are
related and how they can be classified. In other words, we want to study the
structure of the absolute Galois group Gk of k as a profinite group. But in
contrast to the local Galois groups we are far from a complete understanding of
the global situation and there are many conjectures but only a few conceptual
results. For example, there is a famous conjecture due to I. R. ŠAFAREVIČ
which asserts that the subgroup Gk(µ) of Gk is a free profinite group, where
k(µ) is the field obtained from k by adjoining all roots of unity. This was
proved by F. POP [171] for function fields, but the conjecture is open in the
number field case.
However, this first question reflects only one side of the situation which we
consider in algebraic number theory. Namely, it is of purely algebraic nature
and can be asked for any field. But number fields are arithmetic fields, endowed
with primes, completions etc. Therefore, the second fundamental problem is
the question of the decomposition of prime ideals in an extension, and it can
be interpreted as the problem of determining the local groups Gkp with respect
to the global group Gk , i.e. how they lie inside Gk and how they interact.
As we are far from being able to answer these two questions concerning
the “algebraic structure” and the “arithmetic structure” of Gk in complete
generality, we will only give partial answers to these questions. In the first
section we deal with the so-called Hasse principle, also called the local-global
principle. This is the question for the kernel of the natural maps
res i : H i (Gk , A) −→
Y
H i (Gkp , A) ,
i = 1, 2,
p
where A is a Gk -module and res i is given by the restriction maps res ip . The
kernel is just the obstruction to a local-global principle (i.e. that the global
properties of H i (Gk , A) are determined by the corresponding local ones). One
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522
Chapter IX. The Absolute Galois Group of a Global Field
also wants to know this in the case of restricted ramification, so that one wants
to determine the kernel
Xi (kS , T, A) = ker H i (GS (k), A) −→
Y
H i (Gkp , A)
p∈T
where T
⊆
S are sets of primes of k.
In the second section we deal with the cokernels of these maps. For a finite
set T and a “large” set S, the surjectivity is (under certain assumptions) the
famous theorem of GRUNWALD-WANG. As a corollary, one obtains that given
finitely many abelian local extensions Kp |kp , there exists an abelian global
extension K|k realizing the given local ones (except for one very special case).
In §3 we extend the surjectivity results of §2 by showing the existence of
global cohomology classes which realize finitely many given local classes and
are of a very special kind. This will be used in an essential way in the proofs
of the theorems of Neukirch and Šafarevič later in this chapter.
In §4 we deal with the question of whether a finite number of local groups Gkp
can form a free product inside Gk (actually we only consider the corresponding
pro-p-factor groups for a prime number p). Here the theorem of GrunwaldWang will play an important role.
A partial answer to the conjecture of Šafarevič is given by K. IWASAWA:
the maximal prosolvable quotient of Gk(µ) is a free prosolvable group. The
same is true for the Galois group of the maximal prosolvable extension k̃
of the maximal abelian extension k ab of k. This will be proved in §5 by
considering embedding problems. In this section we also prove the theorem of
Neukirch which gives a sufficient condition under which finitely many given
local extensions can be simultaneously realized by a solvable global extension.
We conclude the chapter with a famous theorem of Šafarevič which asserts
that, for a given global field k, every finite solvable group G occurs as a Galois
group of a Galois extension K of k, i.e. G(K|k) = G.
§1. The Hasse Principle
The name Hasse principle has its origin in the classical theorem of HasseMinkowski, which asserts that a quadric has a rational point over a number
field k if and only if it has a rational point over all completions of k. Another
example of a local-global principle is the theorem of Hasse on the Brauer group
of a global field (8.1.17).
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523
§1. The Hasse Principle
Let k be a global field and S a set of primes of k. We do not assume
that S is nonempty nor that S contains S∞ if k is a number field. We recall
the convention that an archimedean prime is unramified if and only if it is
completely decomposed, i.e. we consider the extension C|IR of archimedean
local fields as ramified. Given any separable closure k̄ of k, we denote the
maximal extension of k inside k̄ which is unramified outside S by kS and we
set GS (k) = G(kS |k). The ring of S-integers of k is defined by
Ok,S = {a ∈ k | vp (a) ≥ 0
for all p ∈/ S ∪ S∞ }.
If k is a function field and S = ∅, then Ok,S is the algebraic closure of the
prime field in k. Recall the notation
×
IN(S) = {n ∈ IN | n ∈ Ok,S
}.
For a finite GS (k)-module A with #A ∈ IN(S) we set A0 = Hom(A, OS× ), where
OS× = Ok×S ,S .
We are interested in the kernel X1 (kS , T, A) given by the exact sequence
0 −→ X1 (kS , T, A) −→ H 1 (kS |k, A) −→
Y
H 1 (kp , A) ,
p∈T
where T ⊆ S and A is a finite GS (k)-module. If T = S, we also use the
notation X1 (kS , A) := X1 (kS , S, A) = X1 (GS , A).
(9.1.1) Definition. We say the Hasse principle holds (in dimension 1) for the
GS -module A and the set of primes T ⊆ S if
X1 (kS , T, A) = 0 .
In this generality the Hasse principle is obviously not true even if T = S. To
give an example, let A = ZZ/pZZ and let S ⊇ Sp ∪ S∞ be a finite set of primes
of the number field k. Then X1 (kS , S, A) is equal to the dual of the p-primary
part of the S-ideal class group ClS (k) of k which is nontrivial in general.
One might think that the Hasse principle would hold for a GS -module A
if S = T is the set of all primes of k. But for a reducible Gk -module A there
are counterexamples; see [72], 7.3, and [95]. In this section we will deduce
several criteria on S, T and A that imply the vanishing of X1 (kS , T, A).
If k(A) denotes the minimal trivializing extension of k for the module A,
i.e. if Gk(A) is the kernel of the homomorphism Gk → Aut(A) given by the
action of Gk on A, then we define, for a Galois extension K|k inside kS and
containing k(A), the group X1 (K|k, T, A) by the exactness of
res
0 −→ X1 (K|k, T, A) −→ H 1 (K|k, A) −→
Y
H 1 (Kp |kp , A).
p∈T
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Chapter IX. The Absolute Galois Group of a Global Field
As above, we put X1 (K|k, A) := X1 (K|k, S, A) if T = S. We need some
further notation.
(9.1.2) Definition. Let k be a global field and let S be a set of primes of k.
The Dirichlet-density δ(S) of S is the limit
P
−s
p∈S\S∞ N (p)
δ(S) = δk (S) = lim+ P
−s
s→1
all finite p N (p)
if it exists. We have 0
notation
≤
δ(S)
≤
1. For sets S1 and S2 of primes, we use the
S1 ⊂
∼ S2 :⇐⇒ δ(S1 r S2 ) = 0,
i.e. S1 is contained in S2 up to a set of primes of density zero. If Ω|k is a finite
separable extension of k, then we denote by
δΩ (S) = δΩ (S(Ω))
the Dirichlet density of the set S(Ω) of primes of Ω given by all extensions of
S = S(k). Furthermore, we set
cs(Ω|k) := {p a prime of k | p splits completely in Ω|k},
Ram(Ω|k) := {p a prime of k | p ramifies in Ω|k}
for a finite Galois extension Ω|k.
Let Ω|k be a finite Galois extension and let p vary over the primes of k which
are unramified in Ω|k. Then the set Frobp of the Frobenius automorphisms
with respect to the prolongations of p to Ω is a conjugacy class in G(Ω|k). For
a conjugacy class C in G(Ω|k) we put
PΩ|k (C) = {p a prime of k | Frobp = C}.
For a proof of the following theorem refer to [160], chap.VII, (13.6), and [53],
chap. 5, §4 in the function field case.
(9.1.3) Čebotarev’s Density Theorem. Let Ω|k be a finite Galois extension
with group G and let C be a conjugacy class in G. Then
#C
δ(PΩ|k (C)) =
.
#G
For a set S of primes of k and a finite Galois extension Ω|k we mention the
relation
δΩ (S) = δk S ∩ cs(Ω|k) · [Ω : k].
We start with some calculations.
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525
§1. The Hasse Principle
(9.1.4) Lemma. Let p be a prime number, m ≥ 1 a natural number and let
G ⊆ (ZZ/pm ZZ)× = Aut(ZZ/pm ZZ) be a subgroup. Let A be the G-module
which is isomorphic to ZZ/pm ZZ as an abelian group and on which G acts in the
canonical way. Then
Ĥ i (G, A) = 0 for all i ∈ ZZ,
unless p = 2, m ≥ 2 and −1 ∈ G, in which case
Ĥ i (G, A) ∼
= ZZ/2ZZ for all i ∈ ZZ.
Zp . If p is
Proof: Let α = 1 + ps u ∈ U 1 , s ≥ 1, u ∈ ZZ×
p , be a principal unit of Z
odd or if s ≥ 2, then vp (αr − 1) = s + vp (r) for all r ∈ IN.
Let us start with the case p odd. Then
G1 = ker( G → (ZZ/pZZ)× ) = G ∩ im(U 1 →(ZZ/pm ZZ)× )
is a p-Sylow subgroup of G and in order to show the first statement, we may
replace G by the cyclic group G1 . Let pm−s , s ≥ 1 be the order of G. We
use the same letter for an element α ∈ ZZ×
Z/pm ZZ)× . Fix a
p and its image in (Z
generator α ∈ U s r U s+1 of G. Then AG = ker(α − 1: A → A) = pm−s A. As
pm−s
X−1
i=0
m−s
αp
−1
= pm−s v,
α =
α−1
i
∈ Z
Z×
p,
with v
we have NG (A) = pm−s A = AG . Since finite cyclic groups have
trivial Herbrand index by (1.7.6), we obtain Ĥ i (G, A) = 0 for all i ∈ ZZ.
Now we assume p = 2 and we leave the trivial case m = 1 aside. Then there
are three possibilities for G:
a) G ∼
= {±1} × hαi with α ∈ U 2 ,
b) G ∼
= hαi with α ∈ U 2 .
c) G ∼
= hαi with −α ∈ U s r U s+1 , 2 ≤ s ≤ m − 1.
Note that −1 ∈ G only in case a). In case b) the same argument as for
odd p shows Ĥ i (G, A) = 0 for all i ∈ ZZ. In case c) we have the equality
AG = ker(α − 1: A → A) = 2m−1 A. The order of G is equal to 2m−s ≥ 2. As
2m−s
X−1
i=0
m−s
m−s
−1
α2
− 1 (−α)2
=
= 2m−1 v,
α =
α−1
α−1
i
∈ Z
Z×
2,
i
with v
we have NG (A) = 2m−1 A. In the same way as in case b) we
conclude Ĥ (G, A) = 0 for all i ∈ ZZ. Let us finally consider case a) and let
2m−s be the order of the subgroup hαi ⊆ G. Since
2m−s
X−1
i=0
i
α +
2m−s
X−1
−αi = 0,
i=0
the norm map NG : A → A is the zero map. We have AG = 2m−1 A and
IG A = 2A, hence Ĥ i (G, A) ∼
= ZZ/2ZZ for i = −1, 0. By case b), we have
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Chapter IX. The Absolute Galois Group of a Global Field
Ĥ i (hαi, A) = 0 for all i. For i ≥ 1 we obtain
#H i (G, A) = #H i ({±1}, Ahαi ) = #Ĥ 0 ({±1}, 2m−s A) = 2.
In the same way, using homology, we obtain #Ĥ i (G, A) = 2 for i ≤ −2.
2
We will use the lemma above in the following situation: Let G be a subgroup
of the Galois group G(k(µpm )|k), where k is any prime field with char(k) =/ p.
Then G acts on µpm via the cyclotomic character χcycl : G ,→ (ZZ/pm ZZ)× . In
the case when k = Q and p = 2 the group G corresponds to a field in the
diagram
Q(ζ2m )
Q(ζ2m−1 )
Q(i · η2m )
Q(ζ8 )
Q(ζ4 )
Q(η2m )
√
Q( −2)
Q(η2m−1 )
√
Q( 2)
Q
m
where η2m = ζ2m + ζ2−1
m and ζ2m is a primitive 2 -th root of unity.
(9.1.5) Definition. Let k be a field and m = 2r m0 , m0 odd, a natural number
prime to char(k). We say that we are in the special case (k, m) if r ≥ 2 and −1
is in the image of the cyclotomic character χcycl : G(k(µ2r )|k) → (ZZ/2r ZZ)× .
(9.1.6) Proposition. Let p be a prime number, r
with char(k) =/ p. Then
Ĥ i (G(k(µpr )|k), µpr ) = 0
except when p = 2, r
≥
≥
IN, and let k be any field
for all i ∈ ZZ,
2 and we are in the special case (k, 2r ). In this case
Ĥ i (G(k(µ2r )|k), µ2r ) ∼
= ZZ/2ZZ
Let p = 2 and r
∈
for all i ∈ ZZ.
2. Then the special case (k, 2r ) occurs if and only if
char(k) = 0 and Q(µ2r ) ∩ k is real,
or char(k) = ` ≡ −1 mod 2r and IF` (µ2r ) ∩ k = IF` .
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527
§1. The Hasse Principle
Remark: The special case in positive characteristic occurs if and only if
k(µ2r+1 ) = k(µ2r ) and this field is a quadratic extension of k (cf. ex. 3 below).
Proof: As the group G = G(k(µpr )|k) acts on µpr via the cyclotomic character
χcycl : G ,→ (ZZ/pr ZZ)× , we may apply (9.1.4). It therefore suffices to identify
the case −1 ∈ im(χcycl ) if p = 2, r ≥ 2. If char(k) = 0, then the cyclotomic
character factors as
∼ G(Q(µ r )|k ∩ Q(µ r )) ,→ (Z
Z/2r ZZ)×
χcycl : G −→
2
2
and, since −1 ∈ (ZZ/2r ZZ)× ∼
= G(Q(µ2r )|Q) is the complex conjugation, we
have −1 ∈ im(χcycl ) if and only if k ∩ Q(µ2r ) is real.
Now let char(k) = ` ∈/ {0, 2}. Then the cyclotomic character factors as
∼ G(IF (µ r )|k ∩ IF (µ r )) ,→ (Z
χcycl : G −→
Z/2r ZZ)× .
` 2
` 2
The group G(IF` (µ2r )|k ∩IF` (µ2r )) is cyclic, generated by the relative Frobenius
automorphism σ and
χcycl (σ) = `[IF` (µ2r )∩k:IF` ] .
Hence −1 ∈ im(χcycl ) if and only if `[IF` (µ2r )∩k:IF` ]·n ≡ −1 mod 2r for some
n ∈ IN. The last condition is equivalent to ` ≡ −1 mod 2r and IF` (µ2r )∩k = IF` .
2
(9.1.7) Definition. Let k be a global field, m = 2r m0 , m0 odd, a natural
number prime to char(k) and T a set of places of k. We say that we are in
the special case (k, m, T ) if we are in the special case (k, m) and all primes
p ∈ T decompose in k(µ2r )|k.
(9.1.8) Lemma. Let k be a global field. Assume that char(k) =/ 2, k(µ2r )|k is
cyclic and δ(T ) > 1/2. Then we are not in the special case (k, 2r , T ).
Proof: If we are in the special case (k, 2r ) and k(µ2r )|k is cyclic, then we
have [k(µ2r ) : k] = 2. By Čebotarev’s density theorem, the set of primes which
do not decompose in k(µ2r )|k has density 1/2 and therefore has a nontrivial
intersection with T .
2
Remarks: 1. The extension k(µ2r )|k can be non-cyclic only if k is a number
field, r ≥ 3 and k ∩ Q(µ2r ) is real.
2. If k(µ2r )|k is not cyclic, then all primes p - 2 are decomposed in k(µ2r )|k.
Indeed, these primes have a cyclic decomposition group since all p ∈/ S2 ∪ S∞
are unramified.
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528
Chapter IX. The Absolute Galois Group of a Global Field
3. If δ(T ) > 1/2, then, by the first two remarks and (9.1.8), the special case
(k, m, T ) is equivalent to the statement
k is a number field, m = 2r m0 , m0 odd, r ≥ 3,
k(µ2r )|k is not cyclic, and
all primes p dividing 2 in T decompose in k(µ2r )|k.
Now we can formulate some first cases where the Hasse principle holds.
(9.1.9) Theorem. Let k be a global field and let S be a set of primes of k.
The Hasse principle holds for a finite GS (k)-module A and the sets of primes
T ⊆ S of k, i.e.
X1 (kS , T, A) = 0,
in the following cases:
(i) A is a trivial GS (k)-module and δ(T ) > 1/p, where p is the smallest
prime divisor of #A.
(ii) A = µm with m = pr11 · · · prnn , where pi are pairwise different prime
numbers in IN(S), and
δ(cs(k(µpri i )|k) ∩ T ) >
1
pi · [k(µpri i ) : k]
for all i = 1, . . . , n, except we are in the special case (k, m, T ), where
X1 (kS , T, µm ) ∼
= ZZ/2ZZ.
(iii) cs(k(A)|k) ⊂
∼ T and #G(k(A)|k) = lcm{#G(k(A)p |kp ) | p ∈ T }.
Proof: (i) We may assume that A = ZZ/pr ZZ and want to show the injectivity
of the homomorphism
H 1 (kS |k, ZZ/pr ZZ) −→
Y
H 1 (kp , ZZ/pr ZZ).
p∈T
If ϕ : GS (k) → ZZ/pr ZZ is in the kernel, then ker(ϕ) corresponds to a cyclic
Galois extension L|k of degree ps , 0 ≤ s ≤ r, which is unramified outside S
and completely decomposed at T . Since δ(T ) > 1/p, Čebotarev’s density
theorem (9.1.3) implies s = 0, hence ϕ = 0.
(ii) We may assume that m = pr and therefore K := k(A) = k(µpr ). The
commutative and exact diagram
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529
§1. The Hasse Principle
H 1 (kS |K, A)
Y
H 1 (Kp , A)
T (K)
(∗)
X1 (kS , T, A)
0
H 1 (kS |k, A)
Y
H 1 (kp , A)
T (k)
X1 (K|k, T, A)
0
H 1 (K|k, A)
Y
H 1 (Kp |kp , A)
T (k)
shows, using
δK (T ) = δk T ∩ cs(K|k) · [K : k] >
1
p
and (i), that
X1 (kS , T, A) = X1 (K|k, T, A).
By (9.1.6) we have H 1 (K|k, µpr ) = 0 and therefore the group X1 (kS , T, A) is
trivial, unless we are in the special case (k, 2r ), where
H 1 (k(µ2r )|k, µ2r ) ∼
= ZZ/2ZZ.
If there exists a prime p ∈ T such that p does not decompose in k(µ2r ), then
G(k(µ2r )|k) = G(kp (µ2r )|kp ) and therefore X1 (k(µ2r )|k, T, µ2r ) = 0.
On the other hand, the homomorphism
res p
ZZ/2ZZ ∼
H 1 (kp (µ2r )|kp , µ2r )
= H 1 (k(µ
2r )|k, µ2r )
is the zero map if the prime p decomposes in k(µ2r ). Indeed, in this case
G(kp (µ2r )|kp ) is a proper subgroup of G(k(µ2r )|k). Let k 0 be its fixed field.
Then
G(k(µ r )|k0 )
),
ker res p = H 1 (k 0 |k, µ2r 2
G(k(µ r )|k0 )
= µ2 or k 0 = k(µ2t ) for some
which is isomorphic to ZZ/2ZZ since µ2r 2
t ≥ 2, in which case we apply (9.1.6) again. This shows assertion (ii).
(iii) Let, for a prime number p, ps be the maximal p-power dividing the order
of G = G(K|k), where K = k(A). By assumption there exists at least one
p ∈ T such that ps | #Gp , hence Gp = G(Kp |kp ) contains a Sylow group Gp
of G. Since
H 1 (G, A) −→
Y
H 1 (Gp , A)
p
1
is injective by (1.6.10), the kernel X (K|k, T, A) is zero. Using the diagram
(∗) of the proof of part (ii), we obtain the result.
2
Using Poitou-Tate duality, we obtain injectivity results for the second cohomology group.
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530
Chapter IX. The Absolute Galois Group of a Global Field
(9.1.10) Corollary. Let k be a global field and let S be a nonempty set of
primes such that S∞ ⊆ S if k is a number field. Let A be a finite GS (k)-module
with #A ∈ IN(S). Then the canonical homomorphism
H 2 (kS |k, A) −→
M
H 2 (kp , A)
S
is injective in the following cases:
(i) A = ZZ/mZZ with m = pr11 · · · prnn , where pi are pairwise different prime
numbers in IN(S), and
1
δ(cs(k(µpri i )|k) ∩ S) >
pi · [k(µpri i ) : k]
for all i = 1, . . . , n, except we are in the special case (k, m, S), where
X2 (kS , ZZ/mZZ) ∼
= ZZ/2ZZ.
(ii) A0 is a trivial GS (k)-module and δ(S) > 1/p, where p is the smallest
prime divisor of #A.
0
0
(iii) cs(k(A0 )|k) ⊂
∼ S and #G(k(A )|k) = lcm{#G(k(A )p |kp ) | p ∈ S}.
Proof: From the Poitou-Tate duality theorem (8.6.7), we know that
X2 (kS , A) ∼
= X1 (kS , A0 )∨ .
Now everything follows from (9.1.9) with T = S considering A0 instead of A.
2
A first application is a local-global principle for m-th powers.
(9.1.11) Theorem. Let k be a global field, m a natural number prime to char(k)
and T a set of primes of k of density δ(T ) = 1. Then the following holds.
(i) The localization homomorphism
k × /k ×m −→
Y
kp× /kp×m
p∈T
is injective, except in the following case:
k is a number field, m = 2r m0 , m0 odd, r ≥ 3,
k(µ2r )|k is not cyclic, and
all primes p dividing 2 in T decompose in k(µ2r )|k,
where the kernel is cyclic of order 2.
(ii) If α ∈ k × is a 2m-th power in kp× for all p
in k × .
∈
T , then α is an m-th power
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531
§1. The Hasse Principle
Proof: In view of k × /k ×m ∼
= H 1 (k, µm ) assertion (i) follows from (9.1.9)(ii),
where S is the set of all primes of k together with (9.1.8) and the subsequent
Remark 3.
r
≥
In order to prove (ii), we may reduce to the case
√ that m = 2 , r 3 and we
∈
/
are in the special case (k, m, T ). In particular, −1 k. Let K = k(µ2m ).
Then, by (9.1.9)(i), the localization homomorphism
K × /K ×2m −→
Y
Kp× /Kp×2m
p∈T (K)
is injective. Let α ∈ k × be a 2m-th power in kp× for all p ∈ T . Then α ∈ K ×2m .
The commutative and exact diagram
×2m
K × /K
H 1 (GK , µ2m )
k × /k ×2m
H 1 (Gk , µ2m )
H 1 (K|k, µ2m )
together with (9.1.6) shows that α2 = β 2m for some β ∈ k × . Hence α = ±β m .
Suppose α = −β m . As α in an m-th power in kp× for all p ∈ T , the same holds
√
for −1 ∈ k × . Hence the quadratic extension k( −1)|k is decomposed at a set
of primes of density 1 which contradicts the Čebotarev density theorem. We
conclude α = β m , showing (ii).
2
Now let S be an arbitrary set of primes of k. For a natural number m prime
to char(k), we consider the Kummer group∗)
VS (k, m) = ker k × /k ×m −→
Y
kp× /kp×m ×
p∈S
Y
kp× /Up kp×m .
/S
p∈
(Recall the convention Up = kp× if p is archimedean.) We denote the dual of
this group by the letter B (the Russian “B”), i.e.
BS (k, m) := VS (k, m)∨ .
∗) The group V (k, m) is a localization kernel for flat cohomology. We have
S
Q
VS (k, m) = ker Hfl1 (Spec(Ok,S ), µm ) → p∈S H 1 (kp , µm ) .
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532
Chapter IX. The Absolute Galois Group of a Global Field
(9.1.12) Proposition. Let k be a global field, m a natural number prime to
char(k) and S a set of primes. Then the following holds:
(i) The group VS (k, m) is finite.
(ii) If S has density δ(S) = 1, then VS (k, m) = 0 if we are not in the special
case (k, m, S).
(iii) Assume that m ∈ IN(S) and S ⊇ S∞ if k is a number field. Then
VS (k, m) ∼
= X1 (GS , µm ).
Proof: The isomorphism VS (k, m) ∼
= X1 (GS , µm ) stated in (iii) follows
immediately from Kummer theory. Hence, by (8.6.4), VS (k, m) is finite if S is
nonempty, m ∈ IN(S) and S ⊇ S∞ if k is a number field. In the general case,
let T be a nonempty finite set containing all primes dividing m∞. Then we
have an exact sequence
M
0 −→ VS∪T (k, m) −→ VS (k, m) −→
Up /Upm ,
p∈T \S
showing (i). Finally, (ii) follows from (9.1.11)(i) and the definition of VS (k, m).
2
Next we are going to discuss the Hasse principle for the cohomology of
simple modules.
(9.1.13) Lemma. Let G be a profinite group and let A be a finite simple
G-module. Assume there exists a normal subgroup N of G of order prime to
#A which acts nontrivially on A. Then
H i (G, A) = 0
for all
i ≥ 0.
Proof: Since A is simple and N is normal in G acting nontrivially on A,
we have AN = 0. As N is of order prime to #A, we have H i (N, A) = 0 for
i ≥ 1. Now the result follows from the Hochschild-Serre spectral sequence
E2st = H s (G/N, H t (N, A)) ⇒ H s+t (G, A).
2
The next proposition is due to W. GASCHÜTZ, see [220], lemma 1.
(9.1.14) Proposition. Let A be a simple Gk -module, let K = k(A) and suppose
that G = G(K|k) =/ 1 is solvable. Then
H i (G, A) = 0
for all
i ≥ 0.
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533
§1. The Hasse Principle
Proof: Since A is simple, it is an IFp -vector space for some prime number p.
The solvable group G has a nontrivial abelian normal subgroup N . We claim
that its p-Sylow subgroup Np is trivial. Indeed, Np is a normal subgroup in G
and if Np =/ 1, we obtain ANp = 0 or ANp = A since A is simple. In the first
case it would follow that A = 0 by (1.6.12) and in the second that k(A) was
not the minimal A-trivializing extension of k.
Thus G has a nontrivial normal subgroup N of order prime to p which acts
nontrivially on A (as k(A) is the minimal A-trivializing extension of k). The
result follows from (9.1.13).
2
Now we can prove that the Hasse principle holds for simple modules under
certain conditions.
(9.1.15) Theorem. Let k be a global field, S a set of primes of k and let A be
a finite simple GS (k)-module. Let p be the prime number with pA = 0 and let
T ⊆ S be a subset. Then
X1 (kS , T, A) = 0,
in the following cases:
(i) G(k(A)|k) is solvable and cs(k(A)|k) ⊂
∼ T.
(ii) p ∈ IN(S), G(k(A0 )|k) is solvable and cs(k(A0 )|k) ⊂
∼ T.
(iii) p ∈ IN(S) and there are finite Galois extensions K
such that p - [Ω : K], and
⊆
Ω of k inside kS
/ K and k(A0 ) ⊆ K,
µp ⊆
µp
⊆
Ω and cs(Ω|k) ⊂
∼ T.
Proof: (i) Put K = k(A). If K =/ k, then H 1 (K|k, A) = 0, by (9.1.14). If
K = k, this is trivially true. In the commutative diagram
!"#
H 1 (kS |K,
A)
Y
H 1 (Kp , A)
T (K)
H 1 (kS |k, A)
Y
H 1 (kp , A)
T
the upper map is injective by (9.1.9)(i), since cs(K|k) ⊂
∼ T and hence δK (T ) = 1.
This shows assertion (i).
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534
Chapter IX. The Absolute Galois Group of a Global Field
(ii) Note that A0 = Hom(A, µp ) is simple, since A is simple. Let K = k(A) and
K 0 = k(A0 ). Then K 0 = k(A0 ) ⊆ kS and
K 0 (µp ) = K(µp ) .
Therefore the group H 1 (K 0 (µp )|K, A) = H 1 (K(µp )|K, A) is zero. By (9.1.14),
we have H 1 (K|k, A) = 0, since A is simple and G(K|k) is solvable as
G(K(µp )|k) = G(K 0 (µp )|k) is. As cs(K 0 |k) ⊂
∼ T we have δK 0 (T ) = 1 and
δK(µp ) (T ) = 1. This explains the injectivity of the two left-hand side vertical
and the upper horizontal arrow in the commutative diagram
$%&'()* p ), A)
H 1 (kS |K(µ
Y
H 1 (K(µp )p , A)
T (K(µp ))
Y
H 1 (kS |K, A)
H 1 (Kp , A)
T (K)
Y
H 1 (kS |k, A)
H 1 (kp , A) ,
T (k)
which therefore implies the desired result.
(iii) Since k(A0 ) ⊆ K ⊆ Ω and µp ⊆ Ω, we have k(A) ⊆ Ω. In addition,
cs(Ω|k) ⊂
Therefore in the commutative and exact
∼ T implies δΩ (T ) = 1.
diagram
H 1 (kS +,-./|Ω, A)
Y
H 1 (Ωp , A)
T (Ω)
H 1 (kS |k, A)
Y
H 1 (kp , A)
T
H 1 (Ω|k, A)
the upper horizontal map is injective by (9.1.9)(i). Consider the normal subgroup N = G(Ω|K) ⊆ G(Ω|k). By assumption #N is prime to p. As
/ K, N acts nontrivially on the simple module A. This
k(A0 ) ⊆ K and µp ⊆
implies H 1 (Ω|k, A) = 0 by (9.1.13) and shows (iii).
2
Using Poitou-Tate duality, we obtain the
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535
§1. The Hasse Principle
(9.1.16) Corollary. Let k be a global field and let S be a nonempty set of
primes such that S∞ ⊆ S if k is a number field. Let A be a finite simple
GS (k)-module with #A ∈ IN(S) and let p be the prime number with pA = 0.
Then the canonical homomorphism
H 2 (kS |k, A) −→
M
H 2 (kp , A)
S
is injective in the following cases:
(i) G(k(A)|k) is solvable and cs(k(A)|k) ⊂
∼ S.
(ii) G(k(A0 )|k) is solvable and cs(k(A0 )|k) ⊂
∼ S.
(iii) There are finite Galois extensions K
p - [Ω : K],
/ K and k(A) ⊆ K,
µp ⊆
µp
⊆
⊆
Ω of k inside kS such that
Ω and cs(Ω|k) ⊂
∼ S.
Proof: Using the Poitou-Tate duality theorem (8.6.7),
X2 (kS , A) ∼
= X1 (kS , A0 )∨ ,
the assertion follows from (9.1.15) with T = S by considering A0 instead of A.
2
Exercise 1. Let k be a number field, m = 2r m0 , m0 odd, and let T be a set of primes of
k of density δ(T ) > 1/2. Let ζs be a primitive 2s -th root of unity and let ηs = ζs + ζs−1 .
Assume ηs ∈ k but ηs+1 ∈/ k. Show that the special case (k, m, T ) is equivalent to the following
properties:
1)
2)
3)
−1, 2 + ηs and − (2 + ηs ) are not squares in k,
r > s,
{p | p divides 2 and − 1, 2 + ηs , −(2 + ηs ) are not squares in kp } ∩ T = ∅ .
Hint: [6], chap.10.
√
Exercise 2.√ Show that (Q( 7), 23 , P), where P is the set of all places, is a special case, i.e.
for k = Q( 7) the map
Y
k × /k ×8 −→
kp× /kp×8
p
is not injective.
Exercise 3. Let m ≥ 2 and let k be a field of positive characteristic ` =/ 2. Show that the
following conditions are equivalent.
(i) ` ≡ −1 mod 2m and IF` (µ2m ) ∩ k = IF` .
(ii) k(µ2m+1 ) = k(µ2m ) is a quadratic extension of k.
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536
Chapter IX. The Absolute Galois Group of a Global Field
§2. The Theorem of Grunwald-Wang
T
Let, as in the last section, k be a global field and S a set of primes of k,
⊆ S a subset and A a finite GS (k)-module.
In this section we are interested in the cokernel of the restriction map res i =
res i (S, T ) = (res ip )p∈T
res i
H i (kS |k, A) −→
Y
H i (kp , A) coker i (kS , T, A)
T
for i = 1, 2. If i = 2 then the restricted product is just a direct sum and the
group coker 2 (kS , T, A) is often trivial by the following
(9.2.1) Proposition. Let k be a global field and let S be a nonempty set
of primes of k such that S∞ ⊆ S if k is a number field. Let A be a finite
GS (k)-module with #A ∈ IN(S) and let T ⊆ S be a subset. If S r T contains
a nonarchimedean prime, then the map
res 2 (S, T, A) : H 2 (kS |k, A) −→
M
H 2 (kp , A)
T
is surjective.
Proof: The long exact sequence of Poitou-Tate induces the commutative
exact diagram
H 2 (kS012345678 |k, A)
M
H 2 (kp , A)
H 0 (kS |k, A0 )∨
0
H 2 (kp , A)
coker 2 (kS , T, A)
0.
S
H 2 (kS |k, A)
M
T
We obtain the exact sequence
0 −→ coker 2 (kS , T, A)∨ −→ H 0 (kS |k, A0 ) −→
M
H 2 (kp , A)∨ .
S rT
If p ∈ S r T is nonarchimedean, then H (kS |k, A ) injects into H 0 (kp , A0 ) ∼
=
H 2 (kp , A)∨ . This shows the statement.
2
0
0
Now we investigate res 1 assuming that T is finite. If S is nonempty, contains
S∞ if k is a number field and #A ∈ IN(S), then the local and global duality
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537
9:;<=>?@ABCDEFG
§2. The Theorem of Grunwald-Wang
theorems induce the following commutative exact diagram:
0
X1 (kS , S r T, A0 )
H 1 (kS |k, A0 )
Y
H 1 (kp , A0 )
S rT
X1 (kS , A0 )
H 1 (kS |k, A0 )
Y
H 1 (kp , A0 )
H 1 (kS |k, A)∨
S
Y
Y
T
T
H 1 (kp , A0 )
H 1 (kp , A)∨
coker 1 (kS , T, A)∨
0
0.
This diagram implies the
(9.2.2) Lemma. Assume that S is nonempty and contains S∞ if k is a number
field. If T is finite and #A ∈ IN(S), then there is a canonical exact sequence
0HIJK
X1 (kS , A0 )
X1 (kS , S r T, A0 )
coker 1 (kS , T, A)∨
0.
Using the Hasse principles of the last section, we obtain the following result.
(9.2.3) Theorem. Let k be a global field, and let T ⊆ S be sets of primes of k
where T is finite and S contains S∞ if k is a number field. Let A be a finite
GS (k)-module with #A ∈ IN(S). Then the canonical homomorphism
res 1 (S, T, A) : H 1 (kS |k, A) −→
M
H 1 (kp , A)
T
is surjective in the following cases:
(i) A0 is a trivial GS (k)-module and δk (S) > 1/p, where p is the smallest
prime divisor of #A.
(ii) A = ZZ/mZZ with m = pr11 · · · prnn , where pi are pairwise different prime
numbers in IN(S), and
1
δ(cs(k(µpri i )|k) ∩ S) >
pi · [k(µpri i ) : k]
for i = 1, . . . , n, except we are in the special case (k, m, S r T ). In this
case the cokernel of res 1 (S, T, ZZ/mZZ) is of order 1 or 2.
0
0
(iii) cs(k(A0 )|k) ⊂
∼ S and #G(k(A )|k) = lcm{#G(k(A )p |kp ) | p ∈ S r T }.
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538
Chapter IX. The Absolute Galois Group of a Global Field
(iv) A is simple, G(k(A0 )|k) is solvable and cs(k(A0 )|k) ⊂
∼ S.
(v) A is simple, G(k(A)|k) is solvable and cs(k(A)|k) ⊂
∼ S.
(vi) A is simple, pA = 0 for a prime number p ∈ IN(S), and there are finite
Galois extensions K ⊆ Ω of k inside kS such that p - [Ω : K],
/ K and k(A) ⊆ K,
µp ⊆
µp
⊆
Ω and cs(Ω|k) ⊂
∼ S.
(vii) δk (S) = 1 and G(k(A0 )p |kp ) is cyclic for all p ∈ T .
Proof: By (9.2.2), we know that
coker 1 (kS , T, A) ⊆ X1 (kS , S r T, A0 )∨ .
Thus assertions (i)-(vi) follow from the corresponding statements in (9.1.9)
and (9.1.15), respectively. In order to prove (vii), we first note that, since
G0p = G(k(A0 )p |kp ) is cyclic for all p ∈ T and since δk (S) = 1, by Čebotarev’s
density theorem there exists a prime p̃ ∈ S r T for each p ∈ T such that
G0p = G0p̃ ⊆ G0 = G(k(A0 )|k). If x ∈ X1 (k(A0 )|k, S r T, A0 ), then res 1p̃ x = 0
by definition, hence res 1p x = 0 and therefore x ∈ X1 (k(A0 )|k, S, A0 ). Thus the
injection X1 (k(A0 )|k, S, A0 ) ,→ X1 (k(A0 )|k, S r T, A0 ) is an isomorphism.
Since δk(A0 ) (S r T ) = 1, we have
X1 (kS |k(A0 ), A0 ) = 0 = X1 (kS |k(A0 ), S r T, A0 )
and the commutative exact diagram
X1 (kSLMNOP |k, A0 )
X1 (kS |k, S r T, A0 )
X1 (k(A0 )|k, S, A0 )
X1 (k(A0 )|k, S r T, A0 )
coker 1 (kS , T, A)∨
2
implies (vii). This finishes the proof of the theorem.
(9.2.4) Corollary. Let k be a global field, and let T ⊆ S be sets of primes
of k where T is finite and S contains S∞ if k is a number field. Let K be a
finite Galois extension of k inside kS with Galois group G = G(K|k) and let
A = IFp [G]n , where p ∈ IN(S) is a prime number. If δ(S) = 1, then the map
res 1 (S, T, A) : H 1 (kS |k, A) −→
M
H 1 (kp , A)
p∈T
is surjective.
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539
§2. The Theorem of Grunwald-Wang
Proof: By (9.2.3)(ii), the lower restriction map in the commutative diagram
H 1 (kSSTRQ |k, A)
res
M
H 1 (kp , A)
p∈T
H 1 (kS |K, IFpn )
res
H 1 (KP , IFpn )
M
P∈T (K)
2
is surjective, hence the same is true for the upper restriction map.
The statement of (9.2.4) also holds if p = char(k). In fact, we have the
following stronger result.
(9.2.5) Theorem. Let k be a global field of characteristic p > 0 and let T ⊆ S
be sets of primes of k where T is finite and S r T =/ ∅. Let A be a p-primary
GS (k)-module. Then the canonical homomorphism
res 1 (S, T, A) : H 1 (kS |k, A) −→
M
H 1 (kp , A)
T
is surjective.
Proof: We start with the case A = IFp . By (8.3.2), we have
H 1 (GS (k), IFp ) = Ok,S /℘Ok,S ,
where ℘(x) = xp −x. For any prime p of k, (6.1.2) yields H 1 (kp , IFp ) = kp /℘kp .
Furthermore, ℘kp is open in kp by (6.1.6). Since S r T =/ ∅, the strong
approximation theorem (cf. [22], th. 15.1) shows that the natural map
Ok,S −→
M
kp /℘kp
p∈T
is surjective. This proves the assertion for A = IFp . Using the same argument as
in the proof of (9.2.4), we obtain the result for modules of the form A = IFp [G]n ,
where G is any finite quotient of GS (k).
Now let 0 → A0 → A → A00 → 0 be an exact sequence of p-primary
GS (k)-modules. Since cdp GS (k) = 1 by (8.3.3), the exact and commutative
diagram
H 1 (kS UVWXYZ[\]|k, A0 )
M
T
H 1 (kp , A0 )
H 1 (kS |k, A)
H 1 (kS |k, A00 )
0
M
M
H 1 (kp , A00 )
0
T
H 1 (kp , A)
T
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540
Chapter IX. The Absolute Galois Group of a Global Field
shows that the surjectivity statement for A implies that for A00 , and that surjectivity for A0 and A00 shows surjectivity for A. Any finite GS (k)-module A
with pA = 0 is a quotient of a module of the form IFp [G]n . Hence the result is
true for such modules. Any finite p-primary GS (k)-module has a composition
series with graded pieces annihilated by p. Hence the result is true for all finite
p-primary GS (k)-modules. Finally, the general case follows by passing to the
inductive limit over all finite submodules.
2
(9.2.6) Corollary. Let k be a global field of characteristic p > 0 and let T
be a finite set of primes of k. If k T is the maximal extension of k which is
completely decomposed at the primes of T , then
cdp G(k T |k) = 1 .
Proof: Since the Galois group GkT is generated by the decomposition groups
GkP , P|p and p ∈ T , the map res 2 in the commutative diagram
c_`ab^ A)
H 1 (k,
res 1
M
H 1 (kp , A)
p∈ T
res 2
T
H 1 (k, A)
H 1 (k T , A)G(k |k)
H 2 (k T |k, A)
0
T
is injective; here A is a p-primary G(k |k)-module, also considered as a Gk module. The zero in the lower exact sequence is a consequence of (6.5.10).
Since res 1 is surjective by the theorem above, it follows that res 2 is an isomorphism, and therefore H 2 (k T |k, A) = 0. Hence cdp G(k T |k) ≤ 1 by (3.3.2). As
the p-Sylow subgroups of G(k T |k) are nontrivial, we obtain the result.
2
For the trivial module A = ZZ/pr ZZ we are going to prove a more far reaching
surjectivity result.
(9.2.7) Theorem. Let k be a global field, p a prime number and S a set of
primes of k with δ(S) = 1. Let T0 and T be disjoint subsets of S such that T0
is finite and δ(T ) = 0.
If k is a function field or if k is a number field and we are not in the special
case (k, pr , S r(T0 ∪ T )), then the canonical homomorphism
H 1 (GS (k), ZZ/pr ZZ) −→
M
p∈T0
H 1 (kp , ZZ/pr ZZ) ⊕
M
H 1 (Tp , ZZ/pr ZZ)Gp
p∈T
is surjective, where Gp = G(k̄p |kp ) and Tp ⊆ Gp is the inertia group.
If k is a number field and we are in the special case (k, pr , S r(T0 ∪ T )),
then the cokernel of this map is of order 1 or 2.
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541
§2. The Theorem of Grunwald-Wang
Proof: By convention, Tp = Gp , hence H 1 (Tp , ZZ/pr ZZ)Gp = H 1 (kp , ZZ/pr ZZ),
if p is archimedean. Therefore we may move all archimedean primes of T to
T0 and assume T ∩ S∞ = ∅. We first assume that S ⊇ Sp ∪ S∞ if k is a number
field. In the following we write H i (−) for H i (−, ZZ/pr ZZ).
Let T1 ⊆ T be a finite subset. Noting that H 1 (kp ) → H 1 (Tp )Gp is surjective,
(9.2.3)(ii) and (9.2.5) imply the result for the natural map
H 1 (G(S r T )∪T1 (k)) −→
M
H 1 (kp ) ⊕
p∈T0
For T1
⊆
T2
⊆
M
H 1 (Tp )Gp .
p∈T1
T , we obtain the commutative diagram
H 1 (G(S defg r T )∪T2 )
M
H 1 (kp ) ⊕
p∈T0
H 1 (G(S r T )∪T1 )
H 1 (Tp )Gp
M
p∈T2
M
H 1 (kp ) ⊕
H 1 (Tp )Gp .
M
p∈T1
p∈T0
Passing to the limit over all finite subsets of T , we obtain the desired result.
It remains to consider the case when k is a number field and S does not
contain Sp ∪ S∞ . Let V = (Sp ∪ S∞ ) r S. By what we just proved, the map
H 1 (GS∪V (k)) −→
M
H 1 (kp ) ⊕
p∈T0 ∪V
M
H 1 (Tp )Gp ,
p∈T
is surjective, unless we are in the special case (k, pr , S r(T0 ∪ T )) where the
L
cokernel is of order 1 or 2. Since H 1 (kS∪V |kS )GS (k) ,→ p∈V H 1 (Tp )Gp is
injective, the same statement as above holds true also for the map
H 1 (GS (k)) −→
M
H 1 (kp ) ⊕
p∈T0
M
p∈V
1
Hnr
(Gp ) ⊕
M
H 1 (Tp )Gp .
p∈T
This implies the desired result.
2
The following assertion is known as the theorem of GRUNWALD-WANG,
who, however, only considered the case of a cyclic group A.
(9.2.8) Theorem. Let S be a finite set of primes of a global field k and let
A be a finite abelian group. Let for all p ∈ S finite abelian extensions Kp |kp
be given such that G(Kp |kp ) may be embedded into A. If k is a number field
assume we are not in the special case (k, exp(A), P(k) r S), where P(k) is the
set of all places of k and exp(A) denotes the exponent of A.
Then there exists a global abelian extension K|k with Galois group A such
that K has the given completions Kp for all p ∈ S.
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542
Chapter IX. The Absolute Galois Group of a Global Field
Proof: We show the following stronger statement: the map
M
Epi(Gk , A) −→
Hom(Gkp , A)
S
is surjective, where Epi(Gk , A) denotes the set of surjective homomorphisms
from Gk onto A.
Let q1 , . . . , qr be primes not in S (and not dividing 2 in the number field
case), and let
ϕqi : Gkqi −→ A
be homomorphisms such that the images of the ϕqi generate the group A. For
each prime p ∈ S, let ϕp : G(Kp |kp ) ,→ A be an embedding of the local group
into A. Denote by S 0 the union S ∪ {q1 , . . . , qr } and let T and T 0 be the
complements of S and S 0 in P(k).
Since δ(T ) = δ(T 0 ) = 1, (9.1.8) implies that the special cases (k, 2r , T ) and
(k, 2r , T 0 ) can only occur if k is a number field and k(µ2r )|k is not cyclic.
Then all primes p - 2 decompose in k(µ2r )|k and therefore (k, 2r , T ) and
(k, 2r , T 0 ) are equivalent. Furthermore (k, 2r , T ) implies (k, 2r+1 , T ). Decomposing A into a product of cyclic groups, (9.2.3) (ii) and (9.2.5) imply the
L
surjectivity of the map H 1 (k, A) −→ S 0 H 1 (kp , A) if we are not in the special case (k, exp(A), T 0 ) = (k, exp(A), T ). Now a pre-image (ϕ : Gk → A) ∈
Hom(Gk , A) = H 1 (k, A) of
(ϕq1 , . . . , ϕqr , ϕp , p ∈ S) ∈
M
Hom(Gkp , A) =
S0
M
H 1 (kp , A)
S0
realizes the local extensions Kp |kp and it is surjective by the choice of the
homomorphisms ϕqi .
2
Remark: The conclusion of (9.2.8) may hold even if we are in the special case
(k, exp(A), P(k) r S). By (9.2.2) this is exactly the case when
X1 (k̄, P(k), A0 ) ∼
= X1 (k̄, P(k) r S, A0 ).
Finally, we give an application to embedding problems with induced Gmodules as kernel.
(9.2.9) Proposition. Let K|k be a finite Galois extension of global fields with
Galois group G = G(K|k) and let A = IFp [G]n , where p is a prime number.
Then the embedding problem
hijkl
Gk
1
A
E
G
1
is properly solvable.
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543
§3. Construction of Cohomology Classes
Proof: Since H 2 (G, A) vanishes, the embedding problem has a solution
ψ0 : Gk −→ E, see (3.5.9). Let p1 , . . . , pr be primes of k which split completely in K and let ϕi : Gkpi −→ A be homomorphisms such that their images
generate A. By (9.2.4) and (9.2.5) the natural map
H 1 (k, A) −→
r
M
H 1 (kpi , A)
i=1
is surjective. Let [x] ∈ H 1 (k, A) be a 1-class such that res i [x] = ϕi − ψ0 |Gkp
i
for i = 1, . . . , r. Then ψ = x·ψ0 : Gk −→ E is a new solution of the embedding
problem; this is proper, since ψ|Gkp = ϕi for i = 1, . . . , r, and so ψ(GK ) = A.
i
2
Exercise 1. Let k be a global field. Prove that every finite abelian group A occurs as a Galois
group of a finite abelian extension of k.
Exercise 2. Prove the following generalization of the theorem of GRUNWALD-WANG.
Let S be a finite set of primes of a global field k and let A be a finite abelian group. Let
for all p ∈ S finite abelian extensions Kp |kp be given such that G(Kp |kp ) may be embedded
into A. Let T be a set of primes of k of density zero which is disjoint to S. Assume we are
not in the special case (k, exp(A), P(k) r(S ∪ T )).
Then there exists a global abelian extension K|k with Galois group A such that K has the
given completions Kp for all p ∈ S and K|k is unramified at all p ∈ T .
§3. Construction of Cohomology Classes
The aim of this section is to establish the existence of global cohomology classes with given local behaviour. More precisely, given classes xp ∈
H 1 (kp , A) for all p in a finite set of primes T , we are looking for a global class
x ∈ H 1 (k, A) which maps to xp for all p ∈ T and satisfies additional conditions
at all other primes. We start by introducing some notation.
(9.3.1) Definition. Let kp be a local field and let A be a Gkp -module.
(i) We call a class xp ∈ H 1 (kp , A) cyclic if it is split by a cyclic extension
of kp , i.e. if there exists a cyclic extension Kp |kp such that xp lies in the
kernel of the restriction map H 1 (kp , A) → H 1 (Kp , A).
(ii) We call xp unramified if it is split by an unramified extension, i.e. if it is
1
contained in the unramified part Hnr
(kp , A) of H 1 (kp , A). In particular,
if xp is unramified, then it is cyclic.
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544
Chapter IX. The Absolute Galois Group of a Global Field
For A = µp the following theorem is equivalent, via Kummer theory, to
Šafarevič’s theorem about the existence of certain algebraic numbers ([188])
and we shall make use of the ideas of Šafarevič’s proof.
(9.3.2) Theorem. Let k be a global field and let p be a prime number different
to char(k). Let K ⊆ Ω be finite Ga