R. van Dobben de Bruyn
The Brauer–Manin obstruction on curves
Mémoire, July 12, 2013
Supervisor: Prof. J.-L. Colliot-Thélène
Université Paris-Sud XI
Preface
This work aims to define the Brauer–Manin obstruction and the finite descent
obstructions of [22], at a pace suitable for graduate students. We will give an
almost complete proof (following [loc. cit.]) that the abelian descent obstruction
(and a fortiori the Brauer–Manin obstruction) is the only one on curves that
map non-trivially into an abelian variety of algebraic rank 0 such that the Tate–
Shafarevich group contains no nonzero divisible elements. On the way, we will
develop all the theory necessary, including Selmer groups, étale cohomology,
torsors, and Brauer groups of schemes.
2
Contents
1 Algebraic geometry
1.1 Étale morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Two results on proper varieties . . . . . . . . . . . . . . . . . . .
1.3 Adelic points . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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11
17
2 Group schemes
2.1 Group schemes . . . . . . . . . .
2.2 Abelian varieties . . . . . . . . .
2.3 Selmer groups . . . . . . . . . . .
2.4 Adelic points of abelian varieties
2.5 Jacobians . . . . . . . . . . . . .
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20
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3 Torsors
3.1 First cohomology groups
3.2 Nonabelian cohomology
3.3 Torsors . . . . . . . . .
3.4 Descent data . . . . . .
3.5 Hilbert’s theorem 90 . .
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44
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4 Brauer groups
4.1 Azumaya algebras . . . . . . . .
4.2 The Skolem–Noether theorem . .
4.3 Brauer groups of Henselian rings
4.4 Cohomological Brauer group . .
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58
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5 Obstructions for the existence of rational
5.1 Descent obstructions . . . . . . . . . . . .
5.2 The Brauer–Manin obstruction . . . . . .
5.3 Obstructions on abelian varieties . . . . .
5.4 Obstructions on curves . . . . . . . . . . .
points
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74
74
77
80
82
A Category theory
A.1 Representable functors
A.2 Limits . . . . . . . . .
A.3 Functors on limits . .
A.4 Groups in categories .
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86
86
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B Étale cohomology
B.1 Sites and sheaves . . .
B.2 Čech cohomology . . .
B.3 Sheafification . . . . .
B.4 The étale site . . . . .
B.5 Change of site . . . . .
B.6 Cohomology . . . . . .
B.7 Examples of sheaves .
B.8 The étale site of a field
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100
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131
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References
134
4
Introduction
It is in general a difficult problem to decide whether a variety X over a number
field K has any rational points. On the other hand, finding Kv -points for
the various completions of K is relatively easy. Through the Hasse principle,
for certain classes of varieties the question of whether XpKq is nonempty has
become equivalent to the question of whether XpKv q is nonempty for each
completion of K.
However, there are many classes of varieties known for which there is no Hasse
principle, i.e. that have points everywhere locally, but not globally. The proof
that they have no global point usually requires some cohomological argument.
One of the constructions one can carry out is the formation of the Brauer–Manin
set
XpAK qBr X .
It is a subset of XpAK q containing XpKq, and if one can show that XpAK qBr X “
∅, then in particular X has no rational points.
One of the aims of this thesis is to define the Brauer–Manin set and show
some of its main properties. At the same time, we will provide certain other
obstructions to the existence of rational points. The main theorem (Corollary
5.4.6) is that the Brauer–Manin obstruction is the only obstruction for the
existence of rational points on curves C that map non-trivially into an abelian
variety A of algebraic rank 0 whose Tate–Shafarevich group contains no nontrivial divisible elements.
We develop most of the theory needed to define all the obstructions involved.
In particular, we have a lengthy and almost self-contained appendix on étale
cohomology. We assume the reader has familiarity with the language of schemes,
to a level equivalent to chapters II and III of Hartshorne [10]. Moreover, the
reader is assumed to have some knowledge of algebraic number theory, including
Galois cohomology. We will at one point use a theorem of global class field theory
(Theorem 5.2.5). The language of category theory will be used freely, but we
have included an appendix stating some (but possibly not all) of the results we
need.
This thesis is for a large part based on an article by M. Stoll [22]. We aim
at a pace suitable for graduate students in arithmetic geometry, assuming no
knowledge of étale cohomology. The treatment of étale cohomology in Appendix
B is mostly based on [15]. We tried to minimise the number of external results
needed, but sometimes giving the full proof takes us too far afield.
6
Notation
Throughout this text, K will denote a field, with separable closure K̄ and absolute Galois group ΓK . We will sometimes, by abuse of notation, write K for
the scheme Spec K.
If K is a number field, then ΩK will denote its set of places. It consists of the
set of finite places ΩfK and the set of infinite places Ω8
K . If S Ď ΩK is a finite
subset containing the infinite places, then AK,S will denote the S-adèles, i.e.
ź
ź
Kv ˆ
Ov .
AK,S “
vPS
vRS
The ring of adèles of K is denoted AK :
AK “ colim
AK,S Ď
ÝÑ
S finite
ź
Kv ,
vPΩK
where the limit is taken over increasing finite sets S containing Ω8
K . If S Ď ΩK
is any subset, then ASK denotes the adèles with support in S:
ź
ASK “ AK X
Kv ,
vPS
where the intersection is the one taken in vPΩK Kv . If S is the set of finite or
S
the set of infinite places, then we will write AfK and A8
K respectively for AK .
ś
All rings are assumed Noetherian, and all schemes are assumed to be locally
Noetherian. We will tacitly assume that morphisms of schemes are locally of
finite type, except in the cases where this is obviously false (most notably, a
morphism Spec Kv Ñ X for a completion Kv of K, or the map Spec K̄ Ñ
Spec K).
A variety X over a field K is a geometrically reduced, separated scheme of finite
type over K. The scheme X ˆK K̄ is denoted X̄; it is a variety over K̄.
Recall that a point x P X is nonsingular (or regular) if OX,x is a regular local
ring, and X is smooth at x if pΩX{K qx is free of rank dimpOX,x q. When K
is algebraically closed, the two are equivalent, but this is not in general true.
Note that x P X is smooth if and only if the corresponding point x̄ P X̄ is
nonsingular.
A curve over a field K will be a smooth, proper, and geometrically connected
(hence geometrically integral) variety over K of dimension 1. A standard result
shows that it is in fact projective.
If A is an abelian group, then Adiv denotes the subgroup of divisible elements.
This need not be a divisible group, as we show in Remark 2.3.16.
7
1
Algebraic geometry
We will assume basic familiarity with the language of schemes, for instance
following Hartshorne [10]. In this chapter we will prove some additional results
that we will need later on. In the final section of this chapter, we introduce
some notions that are useful for comparing the K-rational and adelic points for
varieties over number fields.
1.1
Étale morphisms
Definition 1.1.1. Let f : X Ñ Y be a morphism of schemes that is locally of
finite type. Then f is unramified at x P X if, for y “ f pxq, the ideal in Ox
generated by my is mx , and the field extension kpyq Ñ kpxq is separable. If f is
unramified at all x P X, then f is unramified.
Lemma 1.1.2. Let f : X Ñ Y be a morphism of schemes that is locally of finite
type. Then f is unramified if and only if ΩX{Y “ 0.
Proof. Note that ΩX{Y “ 0 if and only if pΩX{Y qx “ 0 for all x P X. Let x P X
be given, and set y “ f pxq.
Let V – Spec A be an affine open neighbourhood of y, and let U – Spec B be
an affine open neighbourhood of x contained in f ´1 V . Firstly, note that
ˇ
pΩX{Y qˇU – pΩB{A q˜
(by Hartshorne [10], Remark II.8.9.2). Hence, pΩX{Y qx is none other than
pΩB{A qmx . By Matsumura [13], Exercise 25.4, this is the same as ΩBx {Ay . By
[loc. cit.], it holds that
ΩBx {Ay bAy kpyq “ ΩpBx {my Bx q{kpyq .
Moreover, we know that ΩBx {Ay is a finitely generated Bx -module (Hartshorne
[10], Corollary II.8.5). Hence, by Nakayama’s lemma, it is zero if and only if
ΩpBx {my Bx q{kpyq “ 0.
But it is a standard result that a k-algebra of finite type R satisfies ΩR{k “ 0 if
and only if R is a finite product of finite separable field extensions of k. Since
Bx {my Bx is also a local ring, this can only be the case if my Bx “ mx and
kpxq Ñ kpyq is separable.
Hence, f is unramified at x if and only if pΩX{Y qx “ 0. The result follows by
considering these conditions for all x P X.
Definition 1.1.3. Let f : X Ñ Y be a morphism of schemes. Then f is étale
if f is locally of finite type, flat, and unramified.
Remark 1.1.4. If Y “ Spec K is a point, then X Ñ Y is étale if and only
if X is a (possibly infinite) disjoint union of spectra of finite separable field
extensions L{K.
8
Lemma 1.1.5. Open immersions are étale.
Proof. Open immersions are clearly flat, unramified, and locally of finite type.
Lemma 1.1.6. Let f : X Ñ Y and g : Y Ñ Z be étale. Then the composite
morphism g ˝ f is étale.
Proof. Clearly, the composition of morphisms that are locally of finite type is
locally of finite type, and the composition of flat morphisms is flat. Moreover,
we have an exact sequence of sheaves on X:
f ˚ ΩY {Z Ñ ΩX{Z Ñ ΩX{Y Ñ 0
(see Hartshorne [10], Prop. II.8.11). Since the first and third terms vanish, so
does the middle term.
Lemma 1.1.7. Let f : X Ñ Y be an étale morphism, and let Y 1 Ñ Y be any
morphism. Then the base change f 1 : X 1 Ñ Y 1 of f along Y 1 Ñ Y is étale.
Proof. It is once again clear that the base change of a flat morphism that is
locally of finite type is flat and locally of finite type. Moreover, by Hartshorne
[10], Prop. II.8.10, we have
ΩX 1 {Y 1 “ g ˚ ΩX{Y ,
where g : X 1 Ñ X is the base change of Y 1 Ñ Y along f . Hence, ΩX 1 {Y 1 is zero,
since ΩX{Y is.
Lemma 1.1.8. Let f : X Ñ Y be locally of finite type. Then ΩX{Y “ 0 if and
only if the diagonal morphism ∆ : X Ñ X ˆY X is an open immersion.
Proof. Recall from Hartshorne [10], Section II.8 that the diagonal morphism
factors as X Ñ W Ñ X ˆY X, where W Ď X ˆY X is an open subscheme and
X Ñ W is a closed immersion. Then ΩX{Y is the sheaf ∆˚ pI {I 2 q, where I
is the sheaf of ideals corresponding to the closed immersion X Ñ W .
Hence, it is clear that if X Ñ X ˆY X is an open immersion, then the restriction
of I to ∆pXq is zero, hence ΩX{Y “ 0.
Conversely, if ΩX{Y “ 0, then Ix {Ix2 “ 0 for all x P X. But Ix Ď mx , since
I is the ideal defining X Ď W . Hence, Ix “ 0 by Nakayama’s lemma.
Hence,
X is inside the open set V “ W z Supp I . Since X is defined by I and
ˇ
I ˇV “ 0, this makes X Ñ V both an open and closed immersion, hence an
isomorphism onto its image. Hence, X Ñ V Ñ W Ñ X ˆY X is a composition
of open immersions, hence an open immersion.
Corollary 1.1.9. Let f : X Ñ Y be any morphism, and let g : Y Ñ Z be
unramified. If gf is étale, then so is f .
9
Proof. Since g is unramified, the diagonal Y Ñ Y ˆZ Y is an open immersion.
One easily sees that the square
1ˆf
X
X ˆZ Y
f
f ˆ1
∆Y
Y
Y ˆZ Y
is a pullback. Hence, 1ˆf is an open immersion, so in particular it is étale. But
by definition the square
π2
X ˆZ Y
Y
g
π1
X
g˝f
Z
is a pullback. Hence, π2 is étale, since g ˝ f is. Hence, f “ π2 ˝ p1ˆf q is étale,
since it is the composition of two étale morphisms.
Proposition 1.1.10. Let f : X Ñ Y be a closed immersion that is flat (hence
étale). Then f is an open immersion.
Proof. Since flat morphisms are open (Hartshorne [10], Exercise III.9.1), we can
assume that f is surjective. If V Ď Y is an affine open (say V – Spec A), then
U “ f ´1 pV q is Spec A{I for some ideal I Ď A. Now Spec A{I Ñ Spec A is
surjective, so by Atiyah–MacDonald [4], Exercise 3.16, aec “ a for all ideals
a Ď A. In particular, setting a “ 0 we find that A Ñ A{I is injective, i.e. I “ 0.
Hence, f |U : U Ñ V is an isomorphism. Since V was arbitrary, this shows that
f is an isomorphism.
Corollary 1.1.11. Let f : X Ñ Y be étale and separated, and suppose Y is
connected. Then any section s of f is an isomorphism onto an open connected
component.
Proof. The base change of ∆X : X Ñ X ˆY X along s is the map
1ˆs : Y ÝÑ Y ˆY X,
where the structure morphism of Y (on the right hand side) as Y -scheme is
via f s, which is just 1Y since s is a section. Hence, the second projection
π2 : Y ˆY X Ñ X is an isomorphism, which identifies 1ˆs with the map
s : Y Ñ X.
Since f is separated, ∆X is a closed immersion, hence so is 1ˆs “ s. Hence,
s is unramified. Since f and f s “ 1Y are étale, Corollary 1.1.9 shows that s is
étale. Hence, by the proposition above it follows that s is an isomorphism onto
a clopen set. This set must be a connected component since Y is connected.
10
Corollary 1.1.12. Let f : X Ñ Y be étale and separated, and suppose Y is
connected. Then two sections s1 , s2 of f for which the topological maps agree
on a point must be the same.
Proof. Both s1 and s2 are isomorphisms onto an open connected component, and
since their topological maps agree on a point, this must be the same connected
component U . But then both s1 and s2 are two-sided inverses of f |U , hence
they are the same.
Corollary 1.1.13. Let f, g : X Ñ Y be S-morphisms, where X is a connected
S-scheme and Y {S is étale and separated. Suppose x P X is such that f pxq “
gpxq “ y, and that the maps kpyq Ñ kpxq induced by f and g are the same.
Then f “ g.
Proof. The maps
1ˆf : X ÝÑ X ˆS Y
1ˆg : X ÝÑ X ˆS Y
are sections to the first projection π1 : X ˆS Y Ñ X. Moreover, π1 is étale and
separated since Y Ñ S is. Since f pxq “ gpxq “ y, the compositions
ÝÑ
ÝÑ
txu Ñ X
X ˆS Y
(1.1)
induced by f and g both factor as
txu ÝÑ
ÝÑ txu ˆS tyu Ñ X ˆS Y,
(1.2)
and the assumption on the maps kpyq Ñ kpxq implies that the maps in (1.2)
coincide for f and g. Hence, so do the compositions in (1.1), so x maps to the
same point under 1 ˆ f and 1 ˆ g. The result now follows from the preceding
corollary.
1.2
Two results on proper varieties
We will prove two well-known theorems about proper varieties (Theorem 1.2.7
and Theorem 1.2.14 below).
Lemma 1.2.1. Let f : X Ñ Y be surjective, and let Y 1 Ñ Y be any morphism.
Then the base change X 1 Ñ Y 1 of f along Y 1 Ñ Y is surjective.
Proof. Let y 1 P Y 1 be given, and let y P Y be its image. There is a commutative
cube
X1
X
Xy1 1
Xy
Y1
y
1
Y.
y
11
The left, right and back squares are pullbacks, hence so is the front square.
Moreover, Xy is nonempty since X Ñ Y is surjective.
Hence, if U Ď Xy is some nonempty affine open, say U – Spec A, then the
inverse image of U in Xy1 1 is SpecpA bkpyq kpy 1 qq. Since U is nonempty, A is
not the zero ring. Hence, A bkpyq kpy 1 q is not the zero ring either, since field
extensions are faithfully flat. Hence, Xy1 1 contains a nonempty open subset,
hence is nonempty.
Since y 1 was arbitrary, we see that all fibres of X 1 Ñ Y 1 are nonempty. Hence,
X 1 Ñ Y 1 is surjective.
Proposition 1.2.2. Let f : X Ñ Y and g : Y Ñ Z be morphisms of schemes.
If g is separated and gf proper, then f is proper. If moreover f is surjective
and g is of finite type, then g is proper.
Proof. The first statement is Hartshorne [10], Corollary II.4.8(e). For the second, we only have to show that g is universally closed.
Let Z 1 be any scheme over Z. Write X 1 “ X ˆZ Z 1 and Y “ Y ˆZ Z 1 . Then
we have maps
f1
g1
X 1 ÝÑ Y 1 ÝÑ Z 1 ,
and f 1 and g 1 f 1 are closed. Moreover, f 1 is surjective, since surjectivity is stable
under base change (by the lemma above).
But if V Ď Y 1 is closed, then f 1 pf 1´1 pV qq “ V by surjectivity of f 1 . Hence, the
image of V under g 1 is the image of the closed set f 1´1 pV q under g 1 f 1 , which is
closed.
Corollary 1.2.3. Let f : X Ñ Y be a morphism of separated schemes of finite
type over a base scheme S. Let V be a closed subscheme of X that is proper
over S, and let Z be its scheme theoretic image in Y . Then Z is proper over S.
Proof. Since Z Ñ Y is a closed immersion, it is separated. Since Y Ñ S is
separated as well, so is Z Ñ S. By the first part of the proposition above, the
morphism V Ñ Z is proper. Moreover, by definition of the scheme theoretic
image, the morphism V Ñ Z is dominant, hence it is surjective. The result now
follows from the second part of the proposition.
Lemma 1.2.4. Let φ : A Ñ B be an injective ring homomorphism, and assume
that the induced morphism Spec B Ñ Spec A is closed. Then φ´1 pB ˆ q “ Aˆ .
Proof. The morphism Spec B Ñ Spec A is dominant since φ is injective. Since
it is also closed, it is surjective. We clearly have
Aˆ Ď φ´1 pB ˆ q.
Now let a P φ´1 pB ˆ q. If a P p for some prime ideal p Ď A, let q be a prime
of B such that φ´1 pqq “ p. Then φpaq P q, contradicting the assumption that
a P φ´1 pB ˆ q. Hence, a is not in any prime ideal, so it is invertible.
12
Lemma 1.2.5. Let φ : A Ñ B be an injective ring homomorphism. Write ψ
for the homomorphism ArT s Ñ BrT s, and suppose that the induced morphism
A1B Ñ A1A is closed. Then φ is integral.
Proof. Let b P B be given, and consider the map BrT s Ñ Br 1b s. Let C be the
image of the composition
“ ‰
ArT s Ñ BrT s Ñ B 1b .
Then the morphism Spec Br 1b s Ñ Spec C is the restriction of A1B Ñ A1A to
certain closed subschemes, hence is also a closed map. Moreover, the ring homomorphism C Ñ Br 1b s is injective.
Hence, by the lemma above, the image of T in C is invertible, since the image
of T in Br 1b s is invertible. Hence, b P C, so we can write
ˆ ˙i
1
b“
ai
b
i“0
n
ÿ
ř
for certain a0 , . . . , an P A. Then bn`1 ´ ai bn´i “ 0 in Br 1b s, so there exists
m P Zě0 such that
˜
¸
n
ÿ
bm bn`1 ´
ai bn´i “ 0
i“0
in B. Hence, b is integral over A.
Corollary 1.2.6. Let f : Spec B Ñ Spec A be a proper morphism of affine
schemes. Then f is finite.
Proof. Let I be the kernel of A Ñ B. Then Spec A{I Ñ Spec A is a closed
immersion, hence it is finite. Moreover, A{I Ñ B is injective and g : Spec B Ñ
Spec A{I is proper, so by the lemma above, g is integral. Since it is also of finite
type, it is finite. The composition of two finite morphisms is finite.
Theorem 1.2.7. Let f : X Ñ Y be a morphism of varieties over a field K. If
X is proper and connected, and is Y affine, then f is constant (i.e. f pXq is a
point).
Proof. By our definition of varieties, X and Y are separated and of finite type
over K. By Corollary 1.2.3, the scheme theoretic image Z of f is proper over K.
Since Y is affine, so is Z, so by the corollary above, Z is finite over K. Finally,
Z is connected since X is, so it is a point.
Lemma 1.2.8. Let S be a scheme, and let Spec R be an affine scheme over S,
where R is a domain with field of fractions K. Let X{S be separated, and let
f, g : Spec R Ñ X be two S-morphisms such that the compositions
Spec K ÝÑ Spec R ÝÑ
ÝÑ X
coincide. Then f “ g.
13
Proof. Write η for the generic point in Spec R, i.e. the image of Spec K Ñ
Spec R. Let U “ Spec A be an affine open neighbourhood of f pηq “ gpηq, and
let V Ď f ´1 pU q X g ´1 pU q be an affine open neighbourhood of η of the form
V “ Dpxq “ Spec Rr x1 s for some x P R.
Now Rr x1 s is also a domain with fraction field K, so Rr x1 s Ñ K is monic.
Moreover, f |V and g|V are given by certain ring homomorphisms
“1‰
A ÝÑ
ÝÑ R x ,
and the compositions with Rr x1 s Ñ K coincide. This forces f |V “ g|V .
Now V is dense in Spec R since Spec R is irreducible. Since Spec R is reduced
and X is separated, this forces f “ g (cf. Hartshorne [10], Exercise II.4.2).
Definition 1.2.9. A scheme C is called a Dedekind scheme if it is integral,
normal, noetherian, and of dimension 1.
Example 1.2.10. Let R be a ring. Then Spec R is a Dedekind scheme if and
only if R is a Dedekind domain.
Example 1.2.11. Let K be a field, and X a variety over K. Then X is a
Dedekind scheme if and only if X is a nonsingular, connected (hence integral)
variety of dimension 1. In particular, this holds for all X ˆK L (L{K finite)
when X is a curve.
Proposition 1.2.12. Let C be an S-scheme that is a Dedekind scheme, and let
X be a proper S-scheme. Let U Ď C be a nonempty open subset, and f : U Ñ X
an S-morphism. Then f extends uniquely to a morphism on C.
Proof. Since C has dimension 1, the complement of U is finite. By induction,
we can assume that it consists of a single point P . If Q is any closed point on
C, then OC,Q is a discrete valuation ring, since it is a normal local noetherian
domain of dimension 1. Its fraction field is OC,η , where η is the generic point
of C.
By the valuative criterion of properness, the map Spec OC,η Ñ X coming from
U Ñ X extends uniquely to a map Spec OC,Q Ñ X, making commutative the
diagram
Spec OC,η
X
Spec OC,Q
S.
Since X is of finite type over S, such a morphism factors as
Spec OC,Q Ñ V Ñ X
for some open set V Ď C containing Q (each generator in an affine of X maps
into some OV , and we take the intersection over finitely many generators).
14
Similarly, any two factorisations
Spec OC,Q Ñ V
ÝÑ
ÝÑ
X
have to coincide on some open W Ď V containing Q.
Applying this to Q “ P , we find that there is an open V Ď C containing P
and a morphism g : V Ñ X inducing the unique map Spec OC,P Ñ X induced
by Spec OC,η Ñ X. Moreover, for any Q P U X V , there exists an open W
containing Q such that the compositions
W ÑU XV
ÝÑ
ÝÑ
X
induced by f and g coincide. Hence, the maps f, g : U X V Ñ X coincide, so
they glue uniquely to a morphism
U Y V Ñ X.
But V contains the sole point outside U , hence U YV “ C. This shows existence,
and uniqueness is clear.
Lemma 1.2.13. Let R be a Dedekind domain with fraction field K. Let Z be
an integral scheme, and let
f
g
Spec K ÝÑ Z ÝÑ Spec R
be morphisms such that their composition is the morphism given by R Ñ K. If
f is dominant and g is proper and surjective, then g has a section.
Proof. Let ηZ and η be the generic points of Z and Spec R respectively. Since
f is dominant, its image is tηZ u, so gpηZ q “ η. Comparing the respective local
rings shows that K Ď OZ,ηZ Ď K, so in fact equality holds.
In particular, g is generically finite. Hence, by Exercise II.3.7 of Hartshorne [10],
there exists an open dense subset U Ď Spec R such that g ´1 pU q Ñ U is finite.
We can take U to be Spec Ra for some a P R. Then Ra is integrally closed since
R is, and g ´1 pU q “ Spec B for some ring B.
Since OZ,η “ K, the field of fractions of B is K. Since the composite map
Ra Ñ B Ñ K is injective, so is Ra Ñ B. Hence, B is an Ra -subalgebra of K.
It is finite over Ra , hence it equals Ra since Ra is integrally closed. That is,
„
g : g ´1 pU q ÝÑ U.
Hence, we have a section h : U Ñ Z on U . Since R is a Dedekind domain and
Z Ñ Spec R is proper, the lemma above shows that h extends uniquely to a
section h : Spec R Ñ Z of g.
Theorem 1.2.14. Let S be a scheme, and let Spec R be an affine scheme over
S, where R is a Dedekind domain with field of fractions K. Let X{S be proper.
Then any morphism Spec K Ñ X factors uniquely as
Spec K
X.
Spec R
15
Proof. Uniqueness is given by Lemma 1.2.8. For existence, it suffices to prove
the result for the base change X ˆS Spec R, as Spec R-scheme (note that it is
still proper, since properness is stable under base change). That is, we will
assume that S “ Spec R.
Let Z be the scheme theoretic image of Spec K Ñ X. Since Spec K is reduced,
it is just the reduced induced structure on the closure of the image (Hartshorne
[10], Exercise II.3.11(d)). Since Z is the closure of a point, it is irreducible,
hence integral since it is reduced.
Since X Ñ Spec R is proper, it is a closed map. Hence, the image of the closed
set Z is closed in Spec R. Since it contains the generic point, it must be equal
to Spec R. That is, the map Z Ñ Spec R is surjective. Since Z Ñ X and
X Ñ Spec R are proper, so is Z Ñ Spec R.
Hence, by Lemma 1.2.13, the map Z Ñ Spec R has a section. Then the composition
Spec R Ñ Z Ñ X
gives the required map.
Corollary 1.2.15. Let R be a Dedekind domain with fraction field K with a
map Spec R Ñ S. Let X{S be proper. Then
XpKq “ XpRq.
Proof. This is a reformulation of the theorem.
Remark 1.2.16. Similarly, Lemma 1.2.8 says that the map
XpRq Ñ XpKq
is an inclusion whenever X is separated over S (for any domain R).
Remark 1.2.17. Theorem 1.2.14 does not hold for a general (not necessarily
normal) domain of dimension 1, even if we restrict to projective schemes over
S “ Spec R.
For example, if k is a field and R “ krT 2 , T 3 s – krX, Y s{pX 3 ´ Y 2 q, then
K “ kpT q. If we take X “ Spec krT s, then the map
X Ñ Spec R
is finite, hence projective. However, the canonical K-point of X induced by
krT s Ñ kpT q can never factor through Spec R, since the ring homomorphism
krT s Ñ kpT q does not factor through R “ krT 2 , T 3 s.
Similarly, the assumption that R has dimension 1 cannot be dropped. For
instance, the blow-up X of A2K in the origin has the same function field as A2K ,
yet there is no section of X Ñ A2K .
16
1.3
Adelic points
Throughout this section, K will denote a number field, and AK its ring of adèles.
We state some basic properties about the AK -points of a K-variety X. For a
more complete treatment, see [5].
Proposition 1.3.1. Let X{K be a variety.
(1) There exists a finite set S Ď ΩK containing the infinite places and a
scheme XS over OK,S such that
XS ˆOK,S K – X.
(2) If Y is another variety and YS a model over OK,S , then
colim HomOK,T pXT , YT q “ HomK pX, Y q,
ÝÑ
where XT denotes XS ˆOK,S OK,T for S Ď T (and similarly for Y ).
(3) If X{K is separated, proper, flat, smooth, affine, or finite, then XT {OK,T
has the same property for some finite set T containing S.
(4) If XS and XS1 1 are two such models, then they become isomorphic on some
finite set T containing S and S 1 . Moreover, any two such isomorphisms
become the same for T large enough.
Proof. This is Theorem 3.4 of [5].
Proposition 1.3.2. Let XS be a model of X over OK,S . There is a natural
identification
ˇ
#
+
ˇ
ź
ˇ
X pAK q “ pxv q P
XpKv q ˇ xv P XS pOv q for almost all v .
ˇ
vPΩ
K
Proof. This follows from Theorem 3.6 of [5].
Remark 1.3.3. It is clear from Proposition 1.3.1 (4) that the set XS pOv q (in
the definition) does not depend on S or on the model XS chosen.
Definition 1.3.4. Let X{K be a variety. Then we define a topology on XpAK q
as the restricted product topology, via the identification of the proposition.
Proposition 1.3.5. Let X{K be a variety.
(1) XpAK q is a locally compact Hausdorff space,
(2) If X is isomorphic to the affine line, then the topology on XpAK q is the
same as the usual topology on AK ,
(3) If X – X1 ˆK X2 , then the topology on XpAK q “ X1 pAK q ˆ X2 pAK q is
the product topology,
(4) If X Ñ Y is a morphism of K-varieties, then the map XpAK q Ñ Y pAK q
is continuous,
(5) If X Ñ Y is a closed immersion, then XpAK q Ñ Y pAK q is a closed
embedding.
17
Proof. This follows from [5], §3.
Remark 1.3.6. Note however that open immersions do not necessarily go to
open embeddings. For example, the topology on the idèles IK Ď AK is not the
subspace topology of AK .
Proposition 1.3.7. Let X{K be a proper variety, and let XS {OK,S be proper
as in Proposition 1.3.1 (1),(3). Then
ź
ź
ź
XpKv q.
XpKv q “
XS pOv q ˆ
X pAK q “
vPΩK
vPΩ8
K
vPΩfK
Proof. Immediate from Proposition 1.3.2 and Theorem 1.2.14.
Definition 1.3.8. Let X{K be a proper variety. Then we write
ź
ź
XpAK q‚ “
XpKv q ˆ
π0 pXpKv qq,
vPΩ8
K
v real
vPΩfK
where π0 pXpKv qq denotes the set of connected components of XpKv q (in the
real topology).
Remark 1.3.9. This is the notation occurring in Stoll’s paper [22]. In the paper
itself, the set XpAK q‚ is defined as
ź
ź
XpKv q ˆ
π0 pXpKv qq,
vPΩfK
vPΩ8
K
i.e. including the π0 of the complex places as well. This is changed to the
definition above in the errata.
Note that if X is connected, then so is XpKv q for any complex place v, by
Shafarevich [19], Theorem VII.2.2.1. Hence, in this case the two definitions
coincide.
18
2
Group schemes
In this chapter, we will cover some results about group schemes. Sections 1
and 2 are rather general, while the last three sections focus on more specialistic
results we will need later on, in Chapter 5.
2.1
Group schemes
Definition 2.1.1. Let S be a base scheme. Then a group scheme over S is a
group object in the monoidal category pSch{S, ˆS q. That is, it is a scheme G{S
together with S-morphisms
µ
G ˆS G ÝÑ G
η
S ÝÑ G
ι
G ÝÑ G
such that the diagrams
1ˆµ
G ˆS G ˆS G
G ˆS G
µ
µˆ1
µ
G ˆS G
G ˆS S
1ˆη
G ˆS G
(2.1)
G,
η ˆ1
S ˆS G
(2.2)
µ
π1
π2
G,
G
∆G
G ˆS G
1ˆι
G ˆS G
µ
S
η
(2.3)
G
commute.
In general, we will assume that all group schemes are flat.
Since it is in general hard to check whether a given object G with morphisms
µ, η and ι is actually a group scheme, we will often use the following criterion:
Proposition 2.1.2. Let X{S be a scheme. Then X is a group scheme over S
if and only if for every scheme T {S, the set
XpT q “ HomS pT, Xq
is a group, and for every morphism g : T Ñ T 1 of S-schemes, the natural map
XpT 1 q Ñ XpT q
is a group homomorphism.
20
Proof. This is Corollary A.4.7.
Corollary 2.1.3. Let G{S be a group scheme, and let S 1 {S be arbitrary. Then
G1 “ G ˆS S 1 is a group scheme over S 1 .
Proof. Let T {S 1 be a scheme. Then
G1 pT q “ HomS 1 pT, G ˆS S 1 q “ HomS pT, Gq “ GpT q,
where the structure map of T as S-scheme is given by the composition T Ñ
S 1 Ñ S. Hence, the result follows from the proposition above.
Corollary 2.1.4. If G1 , G2 are two group schemes over S, then G1 ˆS G2 is a
group scheme over S.
Proof. Let T be an S-scheme. Then
pG1 ˆS G2 qpT q “ HomS pT, G1 ˆS G2 q “ HomS pT, G1 q ˆ HomS pT, G2 q,
and the result follows from the proposition.
Remark 2.1.5. This result also follows from Corollary A.4.16.
Remark 2.1.6. One could also prove the above two results directly, by defining
multiplication, unit and inversion, and showing that they satisfy the necessary
relations. However, the proofs we give are easier and in some way more intuitive,
since in many cases it is more natural to think of a scheme in terms of its T points (for all schemes T {S).
We recall from scheme theory the following adjunction.
Lemma 2.1.7. The functor Sch Ñ Ringop given by X ÞÑ ΓpX, OX q is the left
adjoint of the functor Spec : Ringop Ñ Sch.
Proof. We need to show that, for any scheme X and for any ring R, there is an
isomorphism
„
HomSch pX, Spec Rq ÝÑ HomRing pR, ΓpX, OX qq,
natural in both X and R. The isomorphism is given by taking global sections
(cf. Hartshorne [10], Exercise II.2.4), and naturality is easy to check.
This gives already many examples of group schemes.
Example 2.1.8. If G “ Spec ZrXs, then for any scheme T there is an isomorphism
GpT q – HomRing pZrXs, ΓpT, OT qq – ΓpT, OT q,
where the first isomorphism is given by the lemma, and the second since ZrXs
represents the forgetful functor Ring Ñ Set (cf. Example A.1.4).
21
Hence, since ΓpT, OT q is a group and each ΓpT 1 , OT 1 q Ñ ΓpT, OT q is a group
homomorphism (for T Ñ T 1 a morphism of schemes), this shows that G is a
group scheme. It is denoted Ga , for the additive group.
Example 2.1.9. Similarly, if G “ Spec ZrX, X ´1 s, then there is a natural
isomorphism
GpT q “ ΓpT, OT qˆ ,
since ZrX, X ´1 s represents the group of units functor Ring Ñ Set. Hence, G is
a group scheme, called the multiplicative group. It is denoted Gm .
Example 2.1.10. If G “ Spec ZrtXij uni,j“1 , det´1 s, then there is a natural
isomorphism
GpT q “ GLn pΓpT, OT qq,
since ZrtXij uni,j“1 , det´1 s represents the functor GLn : Ring Ñ Set. Here, det
denotes the element
n
ÿ
ź
det “
sgnpσq
Xiσpiq .
i“1
σPSn
The group scheme G is called the general linear group scheme, and is denoted
GLn .
Example 2.1.11. If G “ Spec ZrXs{pX n ´ 1q, then there is an isomorphism
GpT q – tx P ΓpT, OT q : xn “ 1u.
Then G is called the group of n-th roots of unity, and is denoted µn .
š
Example 2.1.12. Let G be a finite group. Put X “ SpecpZG q “ gPG Spec Z.
Then
XpT q – Gπ0 pT q ,
since a morphism T Ñ X is uniquely determined by choosing a connected
component of X for each connected component of T . The thus obtained group
scheme is called the constant group scheme on the group G, and is denoted G
as well.
In particular, for G “ 0, we get the trivial group scheme.
Definition 2.1.13. Let K be a field. Then a group variety over K is a group
scheme that is a variety over K.
Example 2.1.14. We get group varieties
Ga,K , Gm,K , GLn,K , GK pG finiteq
over K, by extension of scalars. Note that µn,K is only a group variety over K
when charK - n. Indeed, if charK | n, then KrXs{pX n ´ 1q is not reduced, so
G “ µn,K is not a variety by our conventions.
Lemma 2.1.15. Let X be a variety over an algebraically closed field K. Then
there exists a dense open subset U Ď X that is nonsingular.
22
Proof. For X irreducible, this is Hartshorne [10], Cor. II.8.16.
For general X, let X1 , . . . , Xn be the irreducible components of X. Define
ď
V “ Xz pXi X Xj q
i‰j
as the (open) set of points that are in one component only. Note that V is
nonsingular at a point x P V if and only if X is nonsingular at x, since x is in
one irreducible component only.
The irreducibility of each Xi and the fact that Xi ­Ď Xj for i ‰ j force that
each V X Xi is nonempty. Hence, V is dense in X, since its closure contains
each Xi . Moreover, the irreducible components of V are the Vi “ V X Xi , and
each two have empty intersection. That is, V is the disjoint union of the Vi .
Now each Vi contains some (dense) open Ui that is nonsingular. Then clearly
the union U of these Ui is open and dense in V , and nonsingular. The result
follows since V is open and dense in X.
Lemma 2.1.16. Let G be a group variety over K. Then G is smooth.
Proof. Recall that G is smooth if and only if Ḡ is nonsingular. By the lemma
above, Ḡ has an open dense subset U which is nonsingular. There are translations
µ
1ˆa
τa : Ḡ ÝÑ Ḡ ˆK̄ Ḡ ÝÑ Ḡ
for various a P GpK̄q. Each translation is an isomorphism, and the translates
of U cover Ḡ. Hence, Ḡ is nonsingular.
Remark 2.1.17. Some authors allow for a more general definition of group
variety, in which reducedness is not assumed. Then the lemma above does not
hold for G “ µp,Fp . What we call a group variety is then called a smooth group
variety, cf. the lemma.
Definition 2.1.18. Let G be a group variety over K, let P : Spec K̄ Ñ G be a
K̄-point of G. Observe that there is an isomorphism
„
ψ : ΓK ÝÑ AutSpec K pSpec K̄qop
mapping σ : K̄ Ñ K̄ to its associated morphism Spec K̄ Ð Spec K̄ (in the other
direction).
Let σ P ΓK . Then we define σP as the K̄-point P ˝ ψpσq of G. That is, there
is a commutative diagram
Spec K̄
σP
G.
ψpσq
P
Spec K̄
23
Remark 2.1.19. This makes GpK̄q into a ΓK -module, as
pστ qP “ P ˝ ψpστ q “ P ˝ ψpτ q ˝ ψpσq “ σpτ P q
for all σ, τ P ΓK , P P GpK̄q. Moreover, this is a discrete ΓK -module, since any
K̄-point of G is actually defined over some finite extension L{K.
Finally, if L{K is some separable algebraic extension, then GpLq “ GpK̄qΓL .
Indeed, if P P GpK̄q and U is some affine open neighbourhood of (the image of)
P , then U is closed in some AnK . The K-algebra homomorphisms
φ : KrX1 , . . . , Xn s Ñ K̄
such that σ ˝ φ “ φ for all σ P ΓL are exactly those with image inside L, so P
is an L-point if and only if it is ΓL -stable.
We shall simply write G for the ΓK -module GpK̄q.
2.2
Abelian varieties
Definition 2.2.1. An abelian variety over K is a geometrically connected,
proper group variety A over K.
Example 2.2.2. As an uninteresting example, the trivial group scheme gives
an abelian variety of dimension 0. Since abelian varieties are connected and
reduced, and have an identity section Spec K Ñ A, this is the only dimension 0
abelian variety.
Example 2.2.3. Any elliptic curve is an abelian variety of dimension 1. Using
the theory of Jacobians, one can also show that these are the only abelian
varieties of dimension 1. See Remark 2.5.4
Remark 2.2.4. As we will see, the group law on an abelian variety is indeed
commutative, justifying the name. Note however that it is not true that every
commutative group variety over a field K is an abelian variety. For instance,
Ga,K is clearly commutative, but not proper (since the only proper morphisms
of affine schemes over K are finite morphisms, by Corollary 1.2.6).
Theorem 2.2.5. Let X, Y and Z be varieties over K, such that X is proper
and X ˆK Y is irreducible. Let α : X ˆK Y Ñ Z be a morphism of varieties
over K, and suppose there exist closed points x P X, y P Y and z P Z such that
αpX ˆK tyuq “ tzu “ αptxu ˆK Y q.
Then αpX ˆK Y q “ tzu.
Proof. Let Z0 be an affine open neighbourhood of z, and let V be the inverse
image of the closed set ZzZ0 in X ˆK Y . Let U “ pX ˆK Y qzV “ α´1 pZ0 q be
its complement.
24
Since X is proper, the second projection π2 : X ˆK Y Ñ Y is closed, so the
image W of V is closed in Y . Moreover,
π2´1 ptyuq “ X ˆK tyu Ď U,
as αpX ˆK tyuq “ tzu Ď Z0 . Hence, y is not in W “ π2 pV q.
Now for any closed point y 1 P Y zW , the fibre X ˆK ty 1 u is inside U , so
αpX ˆK ty 1 uq Ď Z0 .
Since y 1 is closed, it is finite over K, so X ˆK ty 1 u is proper over K. Then
ˇ
αˇXˆ ty1 u : X ˆK ty 1 u Ñ Z0
K
is a map from a proper variety into an affine variety, so this map has to be
constant (Theorem 1.2.7). Since αptxu ˆK ty 1 uq “ tzu, in fact this constant
value has to be z. That is, for all y 1 P Y zW , it holds that
αpX ˆK ty 1 uq “ tzu.
Since the closed points of Y zW are dense in it (by the Nullstellensatz), this
forces
αpX ˆK pY zW qq “ tzu.
Now X ˆK pY zW q is a nonempty open subset of an irreducible variety, hence
it is dense. This gives
αpX ˆK Y q “ tzu,
as α is continuous and tzu is closed.
Corollary 2.2.6. Let f : A Ñ B be a morphism of abelian varieties over K.
Then f can be written uniquely as f “ τb ˝ g, with τb a right translation by some
closed point b P B and g a homomorphism of abelian varieties.
Proof. Let b “ f p0q, and put g “ τ´b ˝ f . Then gp0q “ 0. Now define
α : A ˆK A ÝÑ B
pa1 , a2 q ÞÝÑ gpa1 ` a2 q ´ gpa1 q ´ gpa2 q.
That is, α is the composition
AˆA
pµ,ιπ1 ,ιπ2 q
ÝÑ
gˆgˆg
µˆ1
µ
A ˆ A ˆ A ÝÑ B ˆ B ˆ B ÝÑ B ˆ B ÝÑ B,
where we drop the subscript from ˆK to ease notation. One easily sees that
αpA ˆK t0uq “ t0u “ αpt0u ˆK Aq,
so α is zero by the theorem. Hence, g is a homomorphism of abelian varieties.
It is clear that this factorisation is unique.
Corollary 2.2.7. Let A be an abelian variety over K. Then A is commutative.
Proof. The inversion ι : A Ñ A is a morphism fixing 0. Hence, by the corollary
above, it is a homomorphism.
25
2.3
Selmer groups
Here and henceforth, A will denote an abelian variety over a number field K.
Definition 2.3.1. Let rns : A Ñ A be the multiplication by n map. Then we
write Arns for the kernel of rns. It is a group scheme by Corollary A.4.16.
Remark 2.3.2. Since rns is unramified, it is in fact étale. The base change of
rns along the identity section 0 Ñ A is Arns (this holds in any category with
finite products). Hence, Arns is étale over K. In fact, it is finite étale over K.
Lemma 2.3.3. There is a short exact sequence
0 Ñ ApKq{nApKq Ñ H 1 pK, Arnsq Ñ H 1 pK, Aqrns Ñ 0.
Proof. There is a short exact sequence of ΓK -modules
n
0 Ñ Arns Ñ A Ñ A Ñ 0.
Then the long exact sequence of Galois cohomology groups gives
n
n
ApKq Ñ ApKq Ñ H 1 pK, Arnsq Ñ H 1 pK, Aq Ñ H 1 pK, Aq,
hence the result.
Remark 2.3.4. This induces a commutative diagram
0
0
ApKq{nApKq
ź
ApKv q{nApKv q
vPΩK
H 1 pK, Arnsq
ź
H 1 pK, Aqrns
H 1 pKv , Arnsq
vPΩK
ź
H 1 pKv , Aqrns
vPΩK
with exact rows.
Definition 2.3.5. The n-Selmer group of A is the kernel
˜
¸
ź
pnq
1
1
Sel pK, Aq “ ker H pK, Arnsq Ñ
H pKv , Aqrns
vPΩK
of the diagonal of the right hand square of the diagram above.
Definition 2.3.6. The Tate–Shafarevich group of A is
˜
¸
ź
1
1
XpK, Aq “ ker H pK, Aq Ñ
H pKv , Aq .
vPΩK
Corollary 2.3.7. There is a short exact sequence
0 Ñ ApKq{nApKq Ñ Selpnq pK, Aq Ñ XpK, Aqrns Ñ 0.
26
0
0,
Proof. Restrict the short exact sequence of the lemma to the elements that map
to zero in the bottom right term
ź
H 1 pKv , Aqrns
vPΩK
of the commutative diagram above.
Remark 2.3.8. One can show that Selpnq pK, Aq is finite. The proof is essentially the same as that of the weak Mordell–Weil theorem. For an elliptic curve,
it is given in Theorem X.4.2(b) of Silverman [20].
In particular, XpK, Aqrns is finite. There is the following conjecture.
Conjecture 2.3.9. (Shafarevich–Tate) The group XpK, Aq is finite.
Remark 2.3.10. Since XpK, Aq lives inside H 1 pK, Aq, it is torsion. Therefore,
the conjecture breaks up into two statements:
• XpK, Aqrps “ 0 for almost all primes p,
• XpK, Aqtpu is finite for all primes p.
Only in particular cases are we able to compute XpK, Aqtpu, so we are still
quite far away from proving the conjecture.
Lemma 2.3.11. Let m | n. Then there is a commutative diagram
0
ApKq{nApKq
Selpnq pK, Aq
XpK, Aqrns
0
n
m
0
ApKq{mApKq
Selpmq pK, Aq
XpK, Aqrms
0
with exact rows.
Proof. There is a commutative diagram
0
Arns
A
n
A
0
n
m
0
Arms
A
m
A
0
with exact rows. The associated long exact sequence is
n
A
A
H 1 pK, Arnsq
H 1 pK, Aq
n
m
A
n
H 1 pK, Aq
m
H 1 pK, Aq.
n
m
m
A
H 1 pK, Armsq
H 1 pK, Aq
The result follows by restricting to the respective Selmer groups.
27
{ for the limit
Definition 2.3.12. Write ApKq
{ “ lim ApKq{nApKq.
ApKq
ÐÝ
n
Also, put
x
SelpK,
Aq “ lim
Selpnq pK, Aq,
ÐÝ
n
where the limit is taken over the maps above. Finally, if B is any abelian group,
put
T B “ lim
Brns
ÐÝ
n
for the (absolute) Tate module of B, where the limit is taken with respect to
the maps
n
m
Brms
Brns ÝÑ
for m | n.
Proposition 2.3.13. Let I be a directed set, and let
0 Ñ pAi q Ñ pBi q Ñ pCi q Ñ 0
be a short exact sequence of projective systems. Then there is an exact sequence
C.
B Ñ lim
A Ñ lim
0 Ñ lim
ÐÝ i
ÐÝ i
ÐÝ i
i
i
i
If moreover I is countable and the maps Ai Ñ Aj for j ď i are surjective, then
the sequence is exact on the right as well, i.e. lim Bi Ñ lim Ci is surjective.
ÐÝ
ÐÝ
Proof. Note that the limit of a projective system pMi q is given by
˜
¸
ź
ź
ψ
lim
Mi “ ker
Mj ÝÑ
Mi ,
ÐÝ
i
jďi
i
where ψ is the map given by
pmj qj,i ÞÝÑ pmj ´ m̄i q,
where m̄i denotes the image of mi in Mj under the natural map Mi Ñ Mj .
Then the limits of pAi q, pBi q and pCi q are the kernels of the vertical maps in
the diagram
ź
ź
ź
0
Aj
Bj
Cj
0
jďi
jďi
ψA
0
ź
i
Ai
jďi
ψB
ź
Bi
i
ψC
ź
Ci
0,
i
and the first statement follows from taking vertical kernels.
Now if I is countable, say I “ ti0 , . . .u, we inductively construct a subset J Ď I
of the form tj0 , . . .u such that ik ď jk and the map k ÞÑ jk gives an isomorphism
„
of ordered sets Zě0 ÝÑ J.
28
Namely, pick j0 “ i0 , and inductively let jk be such that jk´1 ď jk and ik ď jk .
Such jk exists since I is directed, and it is clear that J satisfies the promised
properties. We shall identify Zě0 with J via k ÞÑ jk .
Now J is cofinal since ik ď jk for all k P Zě0 , so
lim
Mi “ lim
Mk
ÐÝ
ÐÝ
iPI
kPZě0
for any projective system pMi q. Moreover, the order on J is linear, so
˜
¸
ź
ź
∆
lim
Mk “ ker
Mk ÝÑ
Mk ,
ÐÝ
k
k
k
where ∆ is the map given by
pmk qk ÞÝÑ pmk ´ m̄k`1 qk ,
where m̄k`1 denotes the image of mk`1 in Mk under the map Mk`1 Ñ Mk . We
now get a commutative diagram with exact rows
ź
ź
ź
Ak
Bk
Ck
0
0
k
k
k
∆A
ź
0
Ak
k
∆B
ź
∆C
ź
Bk
k
Ck
0.
k
If pak qk is given, set m0 “ a0 and inductively pick some mk`1 P Ak`1 such
that m̄k`1 “ mk ´ ak . We can do this since Ak`1 Ñ Ak is surjective. Then by
definition
∆A ppmk qk q “ pmk ´ m̄k`1 qk “ pak qk .
Since pak qk was arbitrary, this shows that ∆A is surjective. The result now
follows from the snake lemma.
Corollary 2.3.14. There is a short exact sequence
{ Ñ SelpK,
x
0 Ñ ApKq
Aq Ñ T XpK, Aq Ñ 0.
{ – SelpK,
x
If moreover XpK, Aqdiv “ 0, then ApKq
Aq.
Proof. The first statement follows from the proposition, as the maps
ApKq{nApKq ÝÑ ApKq{mApKq
for m | n are surjective.
The second statement follows as T B “ T pBdiv q for any abelian group B. Indeed,
if
ź
pbn qnPZą0 P T B Ď
Brns
nPZą0
then that means exactly that
mbmn “ bn
for all m, n P Zą0 . Hence, each bn is divisible, so pbn qn P T pBdiv q.
29
Lemma 2.3.15. Let B be a torsion abelian group such that Brns is finite for
all n P Zą0 . Then Bdiv is a divisible group, so in particular it is the maximal
divisible subgroup of B.
Proof. Let b P Bdiv be given, then there exist bn P B such that nbn “ b. What
we have to prove is that we can take bn to be in Bdiv .
Let d be the order of b. Then any bn with nbn “ b has order nd. Hence, the
subsets
ˇ
(
Cn “ cn P Brdns ˇ ncn “ b
of Brdns are all nonempty. Moreover, there are natural maps
Cmn ÝÑ Cn
cmn ÞÝÑ mcmn
for all m, n. Since Brdns is finite, so is Cn . Since a projective limit of finite
nonempty sets is nonempty, there exists an element pcn q P lim Cn Ď T B such
ÐÝ
that ncn “ b for all n. Saying that pcn q P lim Cn means that mcmn “ cn for all
ÐÝ
m, n, hence each cn is divisible. Hence, b is divisible in Bdiv as well.
Remark 2.3.16. The lemma is no longer valid if we drop the assumption that
Brns be finite. Indeed, let B be the quotient of the group
à
xn pZ{2nZq
n
by the subgroup generated by nxn ´ x1 . Then B is a torsion group, and x1 is
divisible by construction (and it has order 2). However, no other xn is divisible,
and in fact one can show that Bdiv “ t0, x1 u. This is clearly not a divisible
group.
What happens here is that we can find xn with nxn “ x1 , but we can not do
this in a compatible way, i.e. we do not have mxmn “ xn .
{ “ SelpK,
x
Corollary 2.3.17. We have ApKq
Aq if and only if XpK, Aqdiv “ 0.
Proof. One implication was already noted in Corollary 2.3.14. Conversely, if
{ “ SelpK,
x
ApKq
Aq, then T XpK, Aq “ 0. The proof of the lemma above shows
that if Bdiv ‰ 0, then T B ‰ 0. Since XpK, Aqrns is finite for each n, we can
apply this to B “ XpK, Aq to obtain the result.
Remark 2.3.18. The article of Stoll [22] uses the notation Bdiv to mean the
maximal divisible subgroup of an abelian group B, as opposed to the set of
divisible elements. By the lemma above, in the case of XpK, Aq there is no
difference.
Finally, we will compare ApAK q with ApAK q‚ (see section 1.3 for this notation).
Proposition 2.3.19. There is a short exact sequence
0 Ñ ApAK qdiv Ñ ApAK q Ñ ApAK q‚ Ñ 0.
30
Proof. There is a short exact sequence
ź
ApKv q0 Ñ ApAK q Ñ ApAK q‚ Ñ 0
0Ñ
vPΩ8
K
induced by the short exact sequences
0 Ñ ApKv q0 Ñ ApKv q Ñ π0 pApKv qq Ñ 0
0
for v P Ω8
K . Here, ApKv q denotes the connected component of the origin,
which is a (normal) subgroup with quotient π0 pApKv qq. Note that since A is
connected, the complex places give a trivial π0 , so they do not contribute to
ApAK q‚ (compare Remark 1.3.9).
It remains to prove that ApKv q0 “ ApKv qdiv for all infinite places v and
ApKv qdiv “ 0 for all finite places v. But ApAK q‚ is compact and totally disconnected, hence profinite. Hence, it contains no divisible elements, so
ApKv qdiv Ď ApKv q0
for all infinite places, and ApKv qdiv “ 0 for all finite places.
Finally, a standard result on real or complex Lie groups shows that for a compact
Lie group, the exponential
exp : LiepApKv qq Ñ ApKv q
is a surjection onto ApKv q0 . Hence, every element of ApKv q is divisible, so
ApKv qdiv “ ApKv q0 ,
which finishes the proof.
Corollary 2.3.20. Let n P Zą0 . Then
ApAK q{nApAK q “ ApAK q‚ {nApAK q‚ .
Proof. Take vertical cokernels in the diagram
0
ApAK qdiv
ApAK q
n
0
ApAK q‚
n
ApAK qdiv
ApAK q
0
n
ApAK q‚
0.
The result then follows since the left vertical map is surjective.
Corollary 2.3.21. There is a natural identification
{
ApA
K q – ApAK q‚ .
Proof. This follows since ApAK q‚ is its own profinite completion.
31
Lemma 2.3.22. There is a chain of maps
{ ãÑ SelpK,
x
ApKq ãÑ ApKq
Aq Ñ ApAK q‚ ,
inducing isomorphisms
„ {
„ x
ApKqtors Ñ ApKq
tors Ñ SelpK, Aqtors .
Proof. By the definition of the Selmer group, there are maps
ź
Selpnq pK, Aq Ñ
ApKv q{nApKv q.
v
{
x
These induce a map SelpK,
Aq Ñ ApA
K q by taking the limit, and the right hand
side is ApAK q‚ by the corollary above.
By the Mordell–Weil theorem, there is an isomorphism
ApKq – ∆ ˆ Zr
for some finite group ∆ and some r P Zě0 . In particular, ApKqdiv “ 0, so the
{ is injective. Injectivity of ApKq
{ Ñ SelpK,
x
map ApKq Ñ ApKq
Aq is Corollary
2.3.14.
Finally, the explicit description of ApKq also gives
{ – ∆ ˆ Ẑr ,
ApKq
{ . The identification
so ApKqtors “ ApKq
tors
{
x
ApKq
tors “ SelpK, Aqtors
follows from Corollary 2.3.14, taking into account that pT Bqtors “ 0 for any
abelian group B.
The main result of the next section (and indeed one of the main results of this
work) is that the map
x
SelpK,
Aq Ñ ApAK q‚
is also injective.
2.4
Adelic points of abelian varieties
This section is essentially section 3 of Stoll’s paper [22]. Its main ingredient is
a theorem of Serre (which we will not prove); see Theorem 2.4.2.
Lemma 2.4.1. Let A{K be an abelian variety. Then there is an isomorphism
AutpAtors q – GL2g pẐq
of groups.
32
Proof. Over the complex numbers, there is an isomorphism
ApCq – Cg {Λ,
for some lattice Λ Ď Cg . Since multiplication by n is defined over K, all ntorsion of A is defined over K̄. That is,
ApK̄qtors – ApCqtors “ pQ{Zq2g .
Since Q{Z is the colimit of
1
n Z{Z,
we get
HomppQ{Zq2g , pQ{Zq2g q “ lim
Hompp n1 Z{Zq2g , pQ{Zq2g q
ÐÝ
n
“ lim
Hompp n1 Z{Zq2g , p n1 Z{Zq2g q
ÐÝ
n
“ lim
M2g pZ{nZq “ M2g pẐq.
ÐÝ
n
Taking invertible elements gives the result.
Theorem 2.4.2. Let A{K be an abelian variety. Then the image of ΓK Ñ
GL2g pẐq contains a subgroup pẐˆ qd of d-th power scalars, for some d P Zą0 .
Proof. See [17].
Lemma 2.4.3. Let d P Zą0 be even, and let S “ pẐˆ qd Ď Zˆ . Let Ẑˆ –
AutpQ{Zq act on Q{Z in the canonical way. Then there exists D P Zą0 killing
H i pS, Q{Zq,
for i P t0, 1u.
Proof. For p prime, put νp “ mintvp pad ´ 1q | a P Zˆ
p u. Put
ź
D“
pνp ,
p
and note that this indeed a finite product, for instance since, for p ‰ 2,
νp ď vp p2d ´ 1q,
which is zero for almost all p. There is a decomposition
à
à
Q{Z –
pQ{Zqtpu “
Qp {Zp .
p
p
Since any isomorphism has to map the p-primary torsion into the p-primary
torsion, the restriction of the action of Ẑˆ to Qp {Zp is just given by the action of
ˆ
the subgroup Zˆ
p Ď Ẑ . One easily sees that in fact this is just the multiplication
ˆ
action of Zp on Qp {Zp .
We have a decomposition
pQ{ZqS “
à
ˆ d
pQp {Zp qpZp q .
p
33
By definition, each term of the right hand side is given by
ˆ d
pQp {Zp qpZp q “ tx P Qp {Zp | pad ´ 1qx “ 0 for all a P Zˆ
p u.
ˆ d
Hence, if ap is an element such that vp padp ´ 1q is minimal, any x P pQp {Zp qpZp q
is killed by adp ´ 1, hence also by pνp . Hence, pQ{ZqS is killed by D.
Now the logarithm induces isomorphisms
„
Zˆ
2 ÝÑ t˘1u ˆ Z2 ,
„
pp oddq.
Zˆ
p ÝÑ Z{pp ´ 1qZ ˆ Zp
Since d is even, this gives isomorphisms (including for p “ 2)
„
d
pZˆ
p q ÝÑ dpZ{pp ´ 1qZ ˆ Zp q.
d
In particular, pZˆ
p q has a topological generator α, corresponding to the element
d ¨ p1, 1q P dpZ{pp ´ 1qZ ˆ Zp q.
d
Since α generates pZˆ
p q topologically, any continuous 1-cocycle
d
a : pZˆ
p q Ñ Qp {Zp
is uniquely determined by its value at α. Moreover, it comes from the coboundary σ ÞÑ σb ´ b if and only if aα “ pα ´ 1qb. This gives an injection
d
H 1 ppZˆ
p q , Qp , Zp q ãÑ
Qp {Zp
.
pα ´ 1qQp {Zp
Since α ´ 1 ‰ 0 and since Qp {Zp is divisible (by elements of Zp , even), the right
hand side is 0. Hence,
d
H 1 ppZˆ
p q , Qp {Zp q “ 0.
ś
Also, the action of q‰p Zˆ
q on Qp {Zp is trivial, so
d
¸pZˆ
pq
˜
H1
ź
d
pZˆ
q q , Qp {Zp
d
¸pZˆ
pq
˜
ź
“ Homcont
q‰p
d
pZˆ
q q , Qp {Zp
q‰p
˜
“ Homcont
¸
ź
d
d
pZˆ
pq
pZˆ
q q , pQp {Zp q
.
q‰p
ˆ d
The latter is killed by D since pQp {Zp qpZp q is. The inf-res sequence gives
d
¸pZˆ
pq
˜
d
1
1
0 Ñ H 1 ppZˆ
p q , Qp {Zp q Ñ H pS, Qp {Zp q Ñ H
ź
d
pZˆ
q q , Qp {Zp
.
q‰p
The left term is zero, and the right term is killed by D. Hence, the middle term
is killed by D as well. Taking the sum over all primes gives the result.
34
Definition 2.4.4. Let A{K be an abelian variety. Put
Kn “ KpArnsq
for the field obtained by adjoining (all coordinates of) the n-torsion to K. Also
put
8
ď
K8 “
Kn .
n“1
Remark 2.4.5. Note that Kn {K is a finite extension. Moreover, if nP “ 0,
then nσpP q “ σpnP q “ 0 as well, for σ P ΓK . Hence, Kn {K is Galois. It follows
that also K8 {K is Galois.
Proposition 2.4.6. Let A{K be an abelian variety. Then there exists m P Zą0
killing all H 1 pKn {K, Arnsq.
Proof. Note that K8 is defined to be the fixed field of the kernel of ΓK Ñ
GL2g pẐq. Hence, the image of this morphism is isomorphic to G “ GalpK8 {Kq.
By Theorem 2.4.2, it contains S “ pẐˆ qd for some d P Zą0 . By making S smaller
if necessary, we can assume that d is even. Then by the lemma above, there
exists D P Zą0 killing
H i pS, Ators q “ pH i pS, Q{Zqq2g
for i P t0, 1u.
Since S is central in GL2g pẐq, it is normal in G. Then the inf-res sequence gives
0 Ñ H 1 pG{S, AStors q Ñ H 1 pG, Ators q Ñ H 1 pS, Ators q.
The first term is killed by D since AStors is, and the third term is also killed by
D. Hence, the middle term is killed by D2 .
Now the short exact sequence
0 Ñ Arns Ñ Ators Ñ Ators Ñ 0
of G-modules (note that all torsion is defined over K8 ) gives a long exact
sequence
. . . Ñ ApKqtors Ñ H 1 pK8 {K, Arnsq Ñ H 1 pK8 {K, Ators q Ñ . . . .
The third term is killed by D2 , and the first term is finite. Hence, the middle
term is killed by
m “ #ApKqtors ¨ D2 .
Finally, the inflation map gives an injection
H 1 pKn {K, Arnsq ãÑ H 1 pK8 {K, Arnsq,
which gives the result.
35
Corollary 2.4.7. The kernel of
Selpnq pK, Aq Ñ Selpnq pKn , Aq
is killed by m.
Proof. There is an inf-res sequence
0 Ñ H 1 pKn {K, Arnsq Ñ H 1 pK, Arnsq Ñ H 1 pKn , Arnsq.
(2.4)
By definition, the Selmer groups live inside the second and third term, and the
map Selpnq pK, Aq Ñ Selpnq pKn , Aq is the one induced by (2.4). Hence, the kernel
is killed by m, since H 1 pKn {K, Arnsq is.
Lemma 2.4.8. Let Q P Selpnq pK, Aq, and let α be the image of Q under the
map
Selpnq pK, Aq Ñ Selpnq pKn , Aq Ď H 1 pKn , Arnsq “ Homcont pΓKn , Arnsq.
Let L be the fixed field of the kernel of α. Let v be a place of K that splits
completely in Kn . Then v splits completely in L if and only if the image of Q
in ApKv q{nApKv q is zero.
Proof. Let w be a place of Kn above v. Then v splits completely in L iff w does,
and the latter is equivalent to
ˇ
α ˇΓ
“ 0.
Kn,w
We have a commutative diagram
Selpnq pK, Aq
Selpnq pKn , Aq
HompΓKn , Arnsq
ApKv q{nApKv q
ApKn,w q{nApKn,w q
HompΓKn,w , Arnsq.
The result follows since both horizontal arrows of the right hand square are
injections and the bottom arrow of the left hand square is an isomorphism.
Lemma 2.4.9. Let Q P Selpnq pK, Aq, and let d be the order of mQ. Then the
density of places v of K such that v splits completely in Kn and the image of Q
in ApKv q{nApKv q is trivial is at most drK1n :Ks .
Proof. Let α : ΓKn Ñ Arns and L be as in the lemma above. Let e be the order
of α.
By Corollary 2.4.7, the kernel of
Selpnq pK, Aq Ñ Selpnq pKn , Aq
is killed by m. Since eα “ 0, it follows that eQ is in this kernel, so meQ “ 0.
Since d is the order of mQ, this forces d | e.
36
On the other hand, one can easily see that the order of α is the exponent of its
image. That is,
e “ exppGalpL{Kn qq.
In particular, it divides rL : Kn s. Hence, d | e | rL : Kn s, so
rL : Ks “ rL : Kn srKn : Ks ě d ¨ rKn : Ks.
The result now follows from the lemma above and Chebotarev’s Density Theorem.
Lemma 2.4.10. Let I be a directed set, and let pBi q be some projective system.
Put B “ lim Bi . Let b1 , . . . , br P B have infinite order, and let d P Zą0 be given.
ÐÝ
Then there exists i P I such that the images of b1 , . . . , br all have order at least
d in Bi .
Proof. The elements pd ´ 1q!bk P B are nonzero, so there exist ik P I such that
pd ´ 1q!pbk qik P Bik is nonzero.
Since I is directed, there exists i P I with i ě ik for all k P t1, . . . , ru. Then
pd ´ 1q!pbk qi ‰ 0 P Bi .
Hence, each pbk qi has at least order d.
Theorem 2.4.11. Let Z Ď A be a finite subscheme of A such that ZpKq “
x
ZpK̄q. Let P P SelpK,
Aq be such that the image Pv of P in ApKv q is inside
ZpKv q “ ZpKq for a set of finite places v of density 1. Then P P ZpKq.
Proof. Let d ą #ZpKq. Write
x
P ´ ZpKq “ tQ1 , . . . , Qr u Ď SelpK,
Aq.
Note that the assumption on P is that there is a set of finite places v of density
1 such that one of the Qi maps to 0 in ApKv q.
Now suppose that all the Qi have infinite order. Then so do the mQi , so by the
lemma above, there exists n P Zą0 such that the image of mQi in Selpnq pK, Aq
has order di ě d for all i P t1, . . . , ru. By Lemma 2.4.9, the set of places of
K that split completely in Kn such that the image of at least one of the Qi in
ApKv q{nApKv q is trivial is at most
r
ÿ
1
r
1
ď
ă
.
d
rK
:
Ks
drK
:
Ks
rK
: Ks
i
n
n
n
i“1
Hence, there is a set of places of positive density that split completely, but for
which the images of all the Qi in ApKv q{nApKv q are nonzero. This contradicts
the assumption on P , so at least one of the Qi must have finite order. Hence
x
P P ZpKq ` SelpK,
Aqtors “ ZpKq ` ApKqtors Ď ApKq.
37
Now pick a finite place such that Pv P ZpKv q “ ZpKq. Since the map
ApKq Ñ ApKv q
is injective and P lands inside ZpKv q “ ZpKq, in fact P itself is in ZpKq.
Theorem 2.4.12. Let A{K be an abelian variety, and let S be a set of places
of K of density 1. Then the map
x
SelpK,
Aq Ñ ApASK q‚
is injective.
Proof. We can without loss of generality remove the infinite places from S. Then
apply the proposition above to the finite subscheme Z “ t0u.
Corollary 2.4.13. The map
{ Ñ ApAK q‚
ApKq
{ and the topological closure ApKq of
induces an identification between ApKq
ApKq in ApAK q‚ .
Proof. The map is injective by the above, and it is continuous since it was
defined via the quotients
ApKq{nApKq Ñ ApAK q{nApAK q.
{ is compact and ApAK q‚ is Hausdorff, so the
It is a closed map since ApKq
{ is just the subspace topology of the closed subset ApKq
{ Ď
topology on ApKq
{
ApAK q‚ . The result follows since ApKq is dense in ApKq.
Corollary 2.4.14. Let L{K be finite. Then the map
x
x
SelpK,
Aq Ñ SelpL,
Aq
is injective.
Proof. Let S “ ΩfK be the set of finite places. Then we have a commutative
diagram
x
SelpK,
Aq
ApAfK q‚
ApAfL q‚ .
x
SelpL,
Aq
The two horizontal maps are injective by the theorem, and the right vertical
map is injective since ApAfK q‚ is just ApAfK q. Hence, the left vertical map is
injective as well.
38
Remark 2.4.15. This last corollary can also be proven directly from the definitions. Together with the theorem, it allows us to think of all the arrows in
the diagram
ZpKq
ApKq
{
ApKq
x
SelpK,
Aq
ApASK q‚
ZpLq
ApLq
z
ApLq
x
SelpL,
Aq
ApATL q‚
as inclusions, whenever S Ď ΩfK is a set of finite places of density 1 and T is
the set of places of L above S.
For more flexibility, we remove from Theorem 2.4.11 the restriction that all
points of Z are defined over K.
x
Theorem 2.4.16. Let Z Ď A be a finite subscheme. Let P P SelpK,
Aq be such
that the image Pv P ApKv q is inside ZpKv q for a set S of finite places v of K
of density 1. Then P P ZpKq.
Proof. Since Z is a finite scheme, there is a finite Galois extension L{K such that
x
ZpLq “ ZpK̄q. Theorem 2.4.11 then implies that the image of P in SelpL,
Aq is
inside ZpLq.
But the image of P in ApATL q‚ is GalpL{Kq-stable since it comes from ApASK q‚
(where T is the set of places of L above S). Hence,
P P ZpLqGalpL{Kq “ ZpKq.
Remark 2.4.17. The proof uses that ZpLqGalpL{Kq “ ZpKq. I do not know
x
whether the analogous statement for SelpK,
Aq is true as well, i.e. whether the
S
x
x
Aq.
intersection of SelpL,
Aq and ApAK q‚ inside ApATL q‚ is SelpK,
2.5
Jacobians
We will not prove the existence of Jacobians, but we will state the definitions
and main properties, as well as some results we will need later on.
Theorem 2.5.1. Let C be a curve of genus g over a field K. Then there exists
an abelian variety J of dimension g together with a map
Pic0 pC ˆK Lq Ñ JpLq
for all L{K finite separable (functorial in L) that is an isomorphism whenever
CpLq ‰ ∅. Moreover, J is unique up to a unique isomorphism, and is called
the Jacobian of C.
Proof. See e.g. [14], Theorem III.1.6.
39
Theorem 2.5.2. Let C be a curve over K, and let L{K be a finite separable extension such that CpLq ‰ ∅. Then any point P gives rise to a closed
immersion
f P : C ˆK L ÝÑ J ˆK L,
which on M -points (for M {L finite separable) is given by
CpM q ÝÑ Pic0 pC ˆK M q – JpM q
Q ÞÝÑ rQ ´ P s.
Proof. See [14], Proposition III.2.3.
Remark 2.5.3. Note that the map f P only depends on P up to translation.
That is,
1
f P “ τrP ´P 1 s ˝ f P ,
for P, P 1 P CpLq.
Remark 2.5.4. For C “ E an abelian variety of dimension 1, one sees that E
is its own Jacobian. Since the dimension of J is the genus of C, this shows that
E has genus 1, hence is an elliptic curve.
We will now turn to Jacobians over the real numbers. In what follows, C will
be a curve over R such that CpRq ‰ ∅.
Lemma 2.5.5. Let x1 , x2 P CpRq be distinct points. Then there exists a function f P RpCq with no real poles, such that f px1 q ‰ f px2 q.
Proof. Since CpCq is a complex manifold, it is also a real manifold of dimension
2. Hence, as CpRq Ď CpCq has real dimension 1, there exist infinitely many
points P P CpCq that are not in CpRq. Fix such a P , and consider the divisor
D “ npP ` P q ´ x1 , for some n, where P is the complex conjugate of P . Note
that D is defined over R.
If 2n ´ 1 ą g ` 1, then Riemann–Roch shows that
h0 pC, L pDqq “ deg D “ 2n ´ 1,
and
h0 pC, L pD ´ x2 qq “ degpD ´ x2 q “ 2n ´ 2.
Hence, there exists a function f P RpCq such that div f ` D is effective, but
div f ` D ´ x2 is not. Hence, div f ` D contains no terms x2 , so div f contains
no terms x2 .
Then f is an element of RpCq, and f has only poles at the non-real points P
and P . Moreover, it has a zero at x1 , and neither a zero nor a pole at x2 .
Corollary 2.5.6. The set of functions f P RpCq with no real poles is dense in
MappCpRq, Rq, with respect to the topology of uniform convergence.
Proof. This is the Stone–Weierstrass theorem, applied to the compact space
CpRq.
40
Proposition 2.5.7. Let C1 , . . . , Cr be the connected components of CpRq. Then
there exists a function f P RpCq with no real zeroes or poles such that f is
negative on C1 and positive on all other Ci .
Proof. We can create a sequence pfn q of functions uniformly converging to the
function that is ´1 on C1 and 1 on the other Ci . Hence, eventually fn will be
negative on all of C1 and positive on all of Ci . Note that f then automatically
has no real zeroes.
Proposition 2.5.8. Suppose that P, Q P CpRq are two real points such that
rP ´ Qs is divisible by 2. Then P and Q lie in the same connected component
of CpRq.
Proof. Let C1 , . . . , Cr be the connected components of CpRq, in such a way that
P P C1 . By the proposition above, there exists f P RpCq such that f has neither
zeroes nor poles on CpRq and f is negative on C1 and positive on all other Ci .
Since all zeroes and poles of f are non-real, they come in pairs, and div f is of
the form
ÿ
ni pPi ` Pi q.
i
Now if rP ´ Qs is divisible, there exists a divisor M and an element g P CpCq
such that P ´ Q “ 2M ` div g. Hence,
f pP q
“ f p2M q ¨ f pdiv gq.
f pQq
Note that f p2M q “ f pM q2 , which is always nonnegative. Moreover, a standard
result shows that f pdiv gq “ gpdiv f q. Since div f is of the form
ÿ
ni pPi ` Pi q,
i
this gives
gpdiv f q “
ź
ź
pf pPi qf pPi qqni ,
pf pPi qf pPi qqni “
i
i
which is nonnegative as well. Hence,
f pP q
“ f pM q2 gpdiv f q ě 0.
f pQq
Since P P C1 , we have f pP q ă 0. Hence, f pQq is negative as well, so Q P C1 .
Corollary 2.5.9. Let C be a curve over R, such that CpRq ‰ ∅. Then the map
π0 pCpRqq Ñ π0 pJpRqq
induced by the embedding of C into its Jacobian J is injective.
Proof. We saw in the proof of Proposition 2.3.19 that JpRq0 “ JpRqdiv , i.e. the
identity component of the Jacobian is the subgroup of divisible elements. Now if
P, Q P CpRq map to the same component of JpRq, then rP ´Qs is in the identity
component, hence it is divisible. Hence, P and Q lie in the same component of
CpRq.
41
Corollary 2.5.10. Let K be a number field, and let C be a curve over K with
CpKq ‰ ∅. Then the map
CpAK q‚ Ñ JpAK q‚
induced by the embedding of C into its Jacobian J is injective.
Proof. At the finite places and the complex places, there is nothing to prove.
At the real places, it follows from the corollary above.
42
3
Torsors
In this chapter, we will look closely at the first cohomology of representable
sheaves of abelian groups. We will give an ad hoc definition of Ȟ 1 for sheaves
of (not necessarily commutative) groups, and show that its elements correspond
to certain geometrical objects.
3.1
First cohomology groups
Lemma 3.1.1. Let G : B Ñ A be a functor with an exact left adjoint F . Then
G preserves injectives.
Proof. Let I P ob B be injective. Let 0 Ñ A Ñ B be an injection in A . Then
0 Ñ FA Ñ FB
is exact in B. Hence, the map
BpF B, Iq Ñ BpF A, Iq
is surjective. But this is A pB, GIq Ñ A pA, GIq, by the adjunction.
In order to compute the first cohomology, we want to use Čech cohomology. In
order to do this, we need some comparison results between Čech cohomology
and sheaf cohomology.
Proposition 3.1.2. Let I be an injective presheaf, and let U “ tUi Ñ U uiPI
be a covering of some U P ob C . Then Ȟ i pU , I q “ 0 for all i ą 0.
Proof. This is Lemma III.2.4 of [15].
Theorem 3.1.3. The functors Ȟ i pU , ´q : PShpC q Ñ PShpC q are the right
derived functors of Ȟ 0 pU , ´q.
Proof. Note that for a short exact sequence of presheaves
0 Ñ F Ñ G Ñ H Ñ 0,
we get a commutative diagram
0
0
0
F pU q
ś
0
G pU q
ś
0
H pU q
ś
0
iPI
iPI
iPI
0
F pUi q
ś
G pUi q
ś
H pUi q
ś
pi,jqPI 2
pi,jqPI 2
pi,jqPI 2
0
F pUi ˆU Uj q
...
G pUi ˆU Uj q
...
H pUi ˆU Uj q
...,
0
44
with exact columns. That is, we have an exact sequence
0 Ñ Č ‚ pU , F q Ñ Č ‚ pU , G q Ñ Č ‚ pU , H q Ñ 0
of chain complexes.
This gives a long exact cohomology sequence
0 Ñ Ȟ 0 pU , F q Ñ Ȟ 0 pU , G q Ñ Ȟ 0 pU , H q Ñ Ȟ 1 pU , F q Ñ . . . .
The result now follows from the definition of right derived functor by a routine
computation, since any injective sheaf is acyclic by the proposition above.
Proposition 3.1.4. Let F be a sheaf. Then there is an isomorphism
Hˇ 1 pF q – H 1 pF q.
Proof. Note that the inclusion ShpC q Ñ PShpC q preserves injectives by Lemma
3.1.1, since it has an exact left adjoint by Theorem B.3.6 and Proposition B.3.17.
Now let F Ñ I be a monomorphism into an injective sheaf I . Let G be the
presheaf cokernel, then G ` is the sheaf cokernel (Corollary B.3.10). Moreover,
G is separated by Lemma B.3.14. The long exact sequence of H i gives an exact
sequence
0 Ñ F Ñ I Ñ G ` Ñ H 1 pF q Ñ 0
(3.1)
in PShpC q, since I is injective. On the other hand, since G is separated, we
have Hˇ 0 pG q “ G ` , so the short exact sequence of presheaves
0ÑF ÑI ÑG Ñ0
gives the long exact Hˇ i -sequence
0 Ñ F Ñ I Ñ G ` Ñ Hˇ 1 pF q Ñ 0,
since I is also injective as presheaf. Comparing with (3.1) gives the result.
3.2
Nonabelian cohomology
We give an ad hoc definition of Ȟ 1 pU, G q when G is a sheaf of (not necessarily
abelian) groups.
Definition 3.2.1. Let G be a sheaf of groups on a site C , and let U “ tUi Ñ
U uiPI be a covering of some U P ob C . Then a 1-cocycle for U with values in
G is a family
ź
pgij qpi,jq P
G pUij q
pi,jqPI 2
satisfying
´ ˇ
gij ˇU
ijk
¯ ´ ˇ
¨ gjk ˇU
ijk
¯
ˇ
“ gik ˇU
ijk
,
for all pi, j, kq P I 3 , where the product is the group law of G pUijk q. We will
usually write g for the cocycle pgij qpi,jq .
45
Definition 3.2.2. Let g, h be two
ś 1-cocycles. Then g is cohomologous to h if
there exists a family b “ pbi qi P iPI G pUi q such that
´ ˇ ¯
´ ˇ ¯´1
hij “ bi ˇUij ¨ gij ¨ bj ˇUij
,
for all pi, jq P I 2 . This is clearly an equivalence relation, and the set of cohomology classes is denoted
Ȟ 1 pU , G q.
It is called the first cohomology of G with respect to U .
Remark 3.2.3. If G is a sheaf of abelian groups, then Ȟ 1 pU , G q was already
defined, namely as the cohomology of the complex
Č 0 pU , G q Ñ Č 1 pU , G q Ñ Č 2 pU , G q.
The map d1 : Č 1 pU , G q Ñ Č 2 pU , G q is defined by
´´ ˇ
¯ ´ ˇ
¯ ´ ˇ
pgij qpi,jq ÞÑ gjk ˇU
´ gik ˇU
` gij ˇU
ijk
ijk
ijk
¯¯
,
pi,j,kq
hence a 1-cocycle is exactly an element of ker d1 . The map d0 is given by
´ ˇ ¯ ´ ˇ ¯
pbi qi ÞÑ bj ˇUij ´ bi ˇUij ,
hence two 1-cocycles g, h are cohomologous if and only if their difference is in
im d0 . Hence, the definition of Ȟ 1 pU , G q given here is the same as the one given
in section B.2.
Remark 3.2.4. Just like in the abelian case, if V is a refinement of U , there
is a natural morphism
Ȟ 1 pU , G q Ñ Ȟ 1 pV , G q.
We define
Ȟ 1 pU, G q “ colim
Ȟ 1 pU , G q.
ÝÑ
U PJU
We have already seen in Remark B.2.14 that JU is a directed set, so the colimit
is just a direct limit.
Definition 3.2.5. A sequence
1ÑF ÑG ÑH Ñ1
of sheaves of groups is exact if for every U P ob C , the sequence
1 Ñ F pU q Ñ G pU q Ñ H pU q
is exact, and for every h P H pU q, there exists a covering U “ tUi Ñ U u of U
such that the restrictions h|Ui come from elements gi P G pUi q.
Remark 3.2.6. For the case where F , G and H are all sheaves of abelian
groups, this notion corresponds to the notion of exactness in the abelian category
ShpC q, by Corollary B.3.7 and Corollary B.3.13.
46
In general, analogously to Theorem B.3.6 and Corollary B.3.7, one can see that
limits in the category of sheaves of groups are just pointwise. This justifies
the first part of the definition of exactness. In general one would hope that
Corollary B.3.13 generalises to the statement that regular epimorphisms in the
category of sheaves of groups are exactly the G Ñ H satisfying the second part
of the definition of exactness. We do not prove this, and we will use the ad hoc
notion of surjectivity instead.
Proposition 3.2.7. Let 1 Ñ F Ñ G Ñ H Ñ 1 be a short exact sequence, and
let U P ob C . Then there is a long exact sequence of pointed sets
1 Ñ F pU q Ñ G pU q Ñ H pU q Ñ Ȟ 1 pU, F q Ñ Ȟ 1 pU, G q Ñ Ȟ 1 pU, H q.
If moreover F is abelian and F Ñ G lands inside the centre (pointwise), then
this sequence can be extended to
. . . Ñ Ȟ 1 pU, F q Ñ Ȟ 1 pU, G q Ñ Ȟ 1 pU, H q Ñ Ȟ 2 pU, F q.
Proof. See [7], sections III.3 and IV.3.
3.3
Torsors
Definition 3.3.1. Let X be a scheme, and let G be a group scheme over X.
Then a sheaf torsor for G on Xét (or Xfppf ) is a sheaf of sets S together with a
right G-action such that there exists a covering tUi Ñ Xu such that each S |Uét
(resp. S |Ufppf ) is isomorphic to G ˆX U with the canonical G-action.
We will later turn to the case where S can be represented by some X-scheme
S. However, we will firstly characterise sheaf torsors.
Definition 3.3.2. Let S be a sheaf torsor for G. Let U “ tUi Ñ Xu be a
covering that trivialises S . Then in particular S pUi q is nonempty for all i, so
we can pick si P S pUi q. Since the action of G ˆX U on S pUi q is isomorphic to
G ˆX U , there exists a unique gij P GpUij q such that
´ ˇ ¯
ˇ
si ˇUij gij “ sj ˇUij .
Then
´ ˇ
si ˇU
¯
ijk
´ ˇ
gij gjk “ sk “ si ˇU
ijk
¯
gik .
Since the action of G on S pUijk q is simply transitive, this forces
gij gjk “ gik ,
so g “ pgij q is a 1-cocycle. (We have omitted the restrictions to ease notation.)
Lemma 3.3.3. The cohomology class of g does not depend on the choice of U
or of the si . Moreover, it depends on S only up to isomorphism (of sheaves
with G-action).
47
Proof. If we choose other s1i , then there exist unique bi P GpUi q with s1i bi “ si .
Hence, (omitting restrictions to ease notation)
1
s1i gij
bj “ s1j bj “ sj “ si gij “ s1i bi gij ,
1
1
so gij
“ bi gij b´1
j , and g and g are cohomologous.
Independence of the choice of U follows from independence of the choice of
the si , taking a common refinement and restricting the chosen si . The last
statement is clear.
Definition 3.3.4. The association of the cocycle g is denoted S ÞÑ gpS q.
Definition 3.3.5. Conversely, let g P Ȟ 1 pX, Gq be a 1-cocycle; say that g P
Ȟ 1 pU , Gq for a covering U “ tUi Ñ Xu. Let Fn be the presheaf defined by
ź
Fn pV q “
GpUi0 ¨¨¨in ˆX V q,
pi1 ,...,in qPI n
for n P t0, 1u, and note that it is in fact a sheaf. Let d : F0 Ñ F1 be the
morphism given on an arbitrary V by
ź
ź
GpUi ˆX V q ÝÑ
GpUij ˆX V q
pi,jqPI 2
iPI
phi qi ÞÝÑ ph´1
i hj qi,j .
Now, g is by definition a global section of F1 , so it defines elements
´ ˇ
¯
ˇ
P F1 pV q.
g ˇV “ gij ˇU ˆ V
ij
X
i,jPI 2
Then define S as the presheaf inverse image of g. That is, for any V , we have
ˇ
ˇ (
S pV q “ s P F0 pV q ˇ dpsq “ g ˇV .
Remark 3.3.6. Note that S is in fact a subsheaf of F0 : if s P F0 pV q is given,
and tVi Ñ V u is some covering such that s|Vi P S pVi q for all i, then
´ ˇ ¯
ˇ
ˇ
dpsqˇV “ d sˇV “ g ˇV
i
i
i
for all i, hence by the sheaf condition of F1 , we must have dpsq “ g|V , so
s P S pV q.
Definition 3.3.7. We equip S with a right G-action in the following way: for
any V , we define
S pV q ˆ GpV q ÝÑ S pV q
ˆ´
ˇ
ppsi qiPI , hq ÞÝÑ
hˇU
i ˆX V
˙
¯´1
si
,
iPI
where we think of an element s P S pV q as the element psi qiPI P F0 pV q. Note
that if psi q P S pV q, then (once again omitting restrictions)
``
˘˘ `
˘
`
˘
d h´1 si i “ ph´1 si q´1 h´1 sj i,j “ s´1
i sj i,j “ d ppsi qi q .
48
Hence, ph´1 si qi is in S pV q as well. Clearly, this gives a right GpV q-action to
each S pV q, and the maps S pV q Ñ S pV 1 q are G-invariant, so it indeed gives
a G-action on S .
Lemma 3.3.8. The sheaf S with the given right G-action is a torsor. Up to
isomorphism, it depends neither on the choice of representative for the cocycle
we started with, nor on the covering U trivialising g.
Proof. For each n P I, we have
´ ˇ
¯´ ˇ
¯´1
ˇ
gij ˇUijn “ gin ˇUijn gjn ˇUijn
.
´1
Hence, setting spnq “ psi qi “ pgin
qi P F0 pUn q, the above identity reads:
´
¯
ˇ
g ˇUn “ d spnq .
pnq
That is, S |Un has a global section spnq . But then the morphism of sheaves
G|Un Ñ S |Un given on V Ñ Un by
GpV q ÝÑ S pV q
ˇ
h ÞÝÑ h´1 spnq ˇV
gives an isomorphism of sheaves of sets with right G-actions, showing that Un
trivialises S . Since tUn Ñ Xu is a covering, this shows that S is a torsor.
Now if g 1 is a cohomologous cocycle, then there exists pbi qi P F0 pXq such that
1
gij
“ bi gij b´1
j .
Define the maps
f0 : F0 pV q ÝÑ F0 pV q
phi qi ÞÝÑ phi b´1
i qi
and
f1 : F1 pV q ÝÑ F1 pV q
paij qi,j ÞÝÑ pbi aij b´1
j qi,j .
This gives a commutative diagram of morphisms of sheaves
F0
F1
f1
d
„
f0
d
„
F0
F1 .
Since f1 pgq “ g 1 , this diagram induces an isomorphism S – S 1 of the inverse
images of g and g 1 in F0 . Since f0 is defined by multiplication on the right, it
commutes with the action of G on S and S 1 , which is given on the left. Hence,
S – S 1 as G-sheaves, hence as torsors.
49
Hence, S does not depend on the cocycle representing our class. If we chose
a different covering U 1 , then restricting our cocycles to a common refinement
shows that S also does not depend on U .
Definition 3.3.9. The association of the sheaf torsor S is denoted g ÞÑ Sg .
Theorem 3.3.10. The maps S Ñ gpS q and g ÞÑ Sg give a bijection between
the set of isomorphism classes of sheaf torsors on Xét (or Xfppf ) and Ȟ 1 pXét , Gq
(resp. Ȟ 1 pXfppf , Gq).
Proof. If g is a 1-cocycle, say g P Ȟ 1 pU , Gq, then the lemma above shows that
U trivialises g, and there are sections spnq P Sg pUn q, defined by
pnq
si
´1
“ gin
.
The cocycle condition on g asserts that (omitting restrictions)
´1
´1
´1
gnm
gin
“ gim
,
which by the definition of the G-action on S (Definition 3.3.7) gives
´
¯
´
¯
pmq
´1 pnq
spnq gnm “ pgnm q si
“ si
“ spmq .
i
i
Hence, g is the cocycle gpSg q associated to Sg , by Definition 3.3.2.
Conversely, let S be a sheaf torsor. We fix U “ tUi Ñ XuiPI trivialising S ,
and we fix sections si P S pUi q. Then gij is defined by
si gij “ sj .
„
Moreover, the isomorphism of sheaves G|Ui ÝÑ S |Ui given on V Ñ Ui by
ψ : GpV q Ñ S pV q
g ÞÑ si g ´1
induces an isomorphism
„
ψ : F0 ÝÑ S0 ,
where S0 pV q “ i S pUi ˆX V q. By the definition of the right action on F0 ,
for any V Ñ X we have
ś
ψppgi qi hq “ ψpph´1 gi qi q “ psi gi´1 hqi ,
whenever pgi qi P F0 pV q and h P GpV q. Hence, the right action on S0 given by
pti qi h :“ pti hqi
makes ψ into a G-invariant map. Now let t “ pti qi “ psi h´1
i qi P S0 pV q be the
element corresponding to some h “ phi qi P F0 under the isomorphism ψ.
Then dphq “ g if and only if h´1
i hj “ gij , i.e. ti |Uij “ tj |Uij . Hence, under this
identification, the inverse image sheaf of g corresponds to the equaliser of
ź
ź
S pUi ˆX V q ÝÑ
S pUij ˆX V q.
ÝÑ
iPI
pi,jqPI 2
50
But this equaliser is just S pV q, since S is a sheaf. Hence, SgpS q is isomorphic
to S as sheaf. Since ψ is G-invariant, the actions agree as well, so SgpS q is the
same torsor as S .
We will now turn to sheaf torsors that are representable.
Definition 3.3.11. Let X be a scheme, and G a group scheme over X. Then
a torsor for G on Xét (resp. Xfppf ) is a scheme S over X, together with a right
G-action on S such that the sheaf represented by S becomes a sheaf torsor.
Remark 3.3.12. That is, a torsor is a scheme S over X with a right G-action
such that there exists a covering tUi Ñ Xu in Xét (resp. Xfppf ) such that S|Ui
is isomorphic to G|Ui , with its canonical G-action.
Definition 3.3.13. The set of G-torsors on Xét (resp. Xfppf ) up to isomorphism
is denoted
PHSpG{Xét q
(resp. PHSpG{Xfppf q).
It is short for principal homogeneous spaces, which is another word for torsors.
Corollary 3.3.14. There is an injection
PHSpG{Xét q Ñ Ȟ 1 pXét , Gq.
Proof. Clear from the theorem.
Proposition 3.3.15. Let S be an X-scheme with a right G-action. Then the
following are equivalent:
(1) S is a G-torsor on Xfppf ,
(2) S is faithfully flat and locally of finite type over X, and the morphism
pπ1 ,mq
S ˆX G ÝÑ S ˆX S
is an isomorphism (where π1 : S ˆX G Ñ S is the first projection, and
m : S ˆ G Ñ S is the action).
Proof. It is clear that in the second case, the one-object covering tS Ñ Xu
trivialises S, hence S is a torsor.
Conversely, suppose that G is a torsor.
Then there is a covering tUi Ñ Xu
š
trivialising S. Hence, also U “
U
trivialises
S. Note that U Ñ X is
i
i
faithfully flat and locally of finite type. Now G Ñ X is flat (by our assumptions
on group schemes), and in fact faithfully flat since η is a section. The morphism
ˇ
ˇ
pS ˆX Gqˇ ÝÑ pS ˆX Sqˇ
U
U
is an isomorphism. Then by descent theory (see EGA 4 [8], Prop. 2.7.1), S ˆX
G Ñ S ˆX S is an isomorphism. Since S becomes isomorphic to G after the
faithfully flat base change along S Ñ X, another application of descent theory
shows that S is faithfully flat over X.
51
Corollary 3.3.16. Suppose G is smooth over X. Then so is any G-torsor S.
Proof. After the faithfully flat base change along S Ñ X, S becomes isomorphic
to G. Hence the result follows from descent theory.
Corollary 3.3.17. Suppose G is smooth over X. Then any G-torsor S for the
fppf topology is actually a torsor for the étale topology.
Proof. We have to show that there exists an étale covering tUi Ñ Xu trivialising S. We have a smooth covering S Ñ X trivialising S. By EGA 4 [8],
Cor. 17.16.3(ii), there exists a surjective étale morphism S 1 Ñ X and an Xmorphism S 1 Ñ S. That is, tS 1 Ñ Xu is a refinement of tS Ñ Xu, and it is an
étale covering. It trivialises S since tS Ñ Xu does.
Corollary 3.3.18. Suppose G is smooth over X. Let S be an X-scheme with
a right G-action. Then the following are equivalent:
(1) S is a G-torsor on Xét ,
(2) S is faithfully flat and locally of finite type over X, and the morphism
pπ1 ,mq
S ˆX G ÝÑ S ˆX S
is an isomorphism (where π1 : S ˆX G Ñ S is the first projection, and
m : S ˆ G Ñ S is the action).
Proof. This is a reformulation of the above.
We will use without proof the following theorem.
Theorem 3.3.19. Assume that we are in one of the following situations:
(1) G is affine over X,
(2) G is smooth and separated over X, and dim X ď 1;
(3) G is smooth and proper over X, has geometrically connected fibres, and G
is regular.
Then the inclusion PHSpG{Xfppf q Ñ Ȟ 1 pXfppf , Gq is an isomorphism.
Proof. This is Theorem 4.3 and Corollary 4.7 of [15].
Corollary 3.3.20. If G is smooth and satisfies one of the conditions of the
theorem, then
PHSpG{Xét q “ Ȟ 1 pXét , Gq “ Ȟ 1 pXfppf , Gq “ PHSpG{Xfppf q.
Proof. By the theorem, any sheaf torsor S for Xfppf is representable by some
S. Since G is smooth, S is in fact a torsor over Xét , so S is a sheaf torsor on
Xét .
52
Remark 3.3.21. If G is commutative and quasi-projective, then Theorem 3.9
of [15] proves that the canonical maps
H i pXét , Gq Ñ H i pXfppf , Gq
are isomorphisms, which for i “ 1 gives part of the corollary (since Ȟ 1 “ H 1 ).
3.4
Descent data
Definition 3.4.1. Let A Ñ B be a faithfully flat ring homomorphism. Then a
descent datum for B{A is a B-module N with an isomorphism
„
φ : N bA B ÝÑ B bA N
of B bA B-modules, such that the diagram
φ2
N bA B bA B
B bA B bA N
φ3
φ1
B bA N bA B
commutes, where φi is obtained by tensoring φ with 1B on the i-th coordinate.
Example 3.4.2. Let M be an A-module, and let N “ B bA M . Then the
canonical descent datum pN, canq is given by the isomorphism
can : pB bA M q bA B ÝÑ B bA pB bA M q
pb b mq b c ÞÝÑ b b pc b mq.
One easily verifies that this is indeed a descent datum.
Definition 3.4.3. Let pN 1 , φ1 q, pN 2 , φ2 q be descent data. Then a morphism of
descent data is a B-linear map ψ : N 1 Ñ N 2 making commutative the diagram
N 1 bA B
φ1
B bA N 1
ψb1
N 2 bA B
1bψ
φ2
B bA N 2 .
(We used superscript in pN i , φi q since φ1 already has a different meaning.)
Example 3.4.4. Clearly, if M1 Ñ M2 is a morphism of A-modules, then it
induces a morphism pB bA M1 , canq Ñ pB bA M2 , canq of descent data. This
makes the canonical descent datum into a functor.
Definition 3.4.5. Let pN, φq be a descent datum. Then define N φ “ ker α,
where α is the map
α : N ÝÑ B bA N
n ÞÝÑ 1 b n ´ φpn b 1q.
53
It is an A-module, and there is a canonical A-linear map
fφ : N φ bA B ÝÑ N
n b b ÞÝÑ bn.
Proposition 3.4.6. Let pN, φq be a descent datum. Then fφ is an isomorphism.
Proof. Write α0 for the map n ÞÑ 1 b n, and α1 for n ÞÑ φpn b 1q, so that
α “ α0 ´ α1 . Now consider the diagram
α0 b 1
N bA B
B bA N bA B
α1 b 1
φ
φ1
(3.2)
d10 b 1
B bA N
B bA B bA N,
d11 b 1
where d1i is as in Lemma B.7.2. If n P N and b P B are given, then
`
˘
pφ1 ˝ pα0 b 1qq pn b bq “ φ1 p1 b n b bq “ 1 b φpn b bq “ pd10 b 1q ˝ φ pn b bq.
ř
Moreover, if we write φpn b bq “ i bi b ni for certain bi P B, ni P N , then the
definition of φ2 gives
ÿ
` 1
˘
pd1 b 1q ˝ φ pn b bq “
bi b 1 b ni “ φ2 pn b 1 b bq.
i
On the other hand, we have
pφ1 ˝ pα1 b 1qq pn b bq “ φ1 pφpn b 1q b bq “ φ1 pφ3 pn b 1 b bqq,
so the descent datum assumption on φ forces
φ1 ˝ pα1 b 1q “ pd11 b 1q ˝ φ.
Hence, the squares for the top and bottom arrows of the horizontal pairs in (3.2)
commute. Since φ and φ1 are isomorphisms, this implies that the equalisers of
the pairs of horizontal arrows are isomorphic.
But since A Ñ B is faithfully flat, the equaliser of the top pair is just N φ bA B,
by definition of N φ . On the other hand, the equaliser of the bottom pair is N ,
by Lemma B.7.2. Hence, N φ bA B – N .
φ
Finally, the map N φ bA B Ñ N bA B ÝÑ B bA N is given by
n b b ÞÝÑ φpn b bq “ p1 b bqφpn b 1q “ p1 b bqp1 b nq “ p1 b bnq,
which is the image of fφ pnbbq in BbA N . Hence, the isomorphism N φ bA B – N
given above is given by fφ .
Theorem 3.4.7. The canonical descent datum functor gives an equivalence between the category of A-modules and the category of descent data.
54
Proof. Let M be an A-module, and let N “ BbA M . Then one sees immediately
from the definitions that the sequence
α
0 Ñ M Ñ N ÝÑ B bA N
is isomorphic to the one from Lemma B.7.2. Hence, it is exact, so N can – M .
Conversely, if pN, φq is a descent datum, then the proposition above shows that
fφ is an isomorphism. Moreover, for all n P N φ , b, c P B it holds that
φpbn b cq “ φppb b cqpn b 1qq “ pb b cqφpn b 1q “ pb b cqp1 b nq “ b b nc.
Hence, the diagram
pN φ bA Bq bA B
can
B bA pN φ bA Bq
fφ b 1
1 b fφ
N bA B
φ
B bA N
commutes, so fφ is an isomorphism of descent data.
3.5
Hilbert’s theorem 90
We will prove a generalisation of Hilbert’s theorem 90 for étale cohomology,
giving a concrete description of Ȟ 1 pXét , GLn q. Note that since GLn is smooth
and affine over X, the above shows that this is the same as Ȟ 1 pXfppf , GLn q, so
will compute the latter instead.
Proposition 3.5.1. Let X “ Spec A be affine. Then Ȟ 1 pXfppf , GLn q is the set
of locally free A-modules of rank n, up to isomorphism.
Proof. If g P Ȟ 1 pXfppf , GLn q, then g is trivialised by some covering U “ tUi Ñ
Xu. By refining, we can assume that all the Ui are affine. Since each Ui Ñ X is
flat and locally of finite type, it is open, and since X is compact we only need
finitely many Ui to cover X.
š
Now define U “ i Ui . Since g is trivialised by U , it is also trivialised by the
one-object covering U Ñ X, since this is a refinement of U . Now U is affine
since each Ui is; say U “ Spec B.
Then g is a cocycle in Ȟ 1 ptU Ñ Xu, GLn q. Setting I for the index set t˚u, we
get
´ ˇ
¯ ´ ˇ
¯
ˇ
gij ˇU
¨ gjk ˇU
“ gik ˇU ,
ijk
ijk
ijk
for all i, j, k P t˚u. Although i, j and k will always be equal to each other, it
is still useful to keep separate indices, to clarify which restriction maps we are
talking about.
Now gij is an element of GLn pUij q “ GLn pB bA Bq. We will view it as an
isomorphism
„
gij : B bA B n ÝÑ B n bA B.
55
We will write N “ B n . Then gij |Uijk is an isomorphism B bA N bA B Ñ
N bA B bA B, and likewise for gjk and gik . We get a commutative diagram
B bA B bA N
gik
gjk
N bA B bA B,
gij
B bA N bA B
´1
hence φ “ g˚˚
makes N into a descent datum. Conversely, starting with this
descent datum, it is clear that we can recover g.
By Theorem 3.4.7, descent data of this form correspond to A-modules M . For
such an M , we have M bA B – B n , so standard descent theory shows that M
is locally free of rank n (see EGA 4 [8], Prop. 2.5.2).
Corollary 3.5.2. Let X “ Spec A be affine. Then the canonical map
Ȟ 1 pXZar , GLn q Ñ Ȟ 1 pXfppf , GLn q
is an isomorphism.
Proof. It is a standard result that Ȟ 1 pXZar , GLn q characterises locally free Amodules of rank n. Moreover, the proof above essentially shows that any fppflocally free A-module of rank n is already trivialised by a Zariski covering. This
proves the result.
Theorem 3.5.3. (Hilbert’s theorem 90) Let X be a scheme. Then the canonical
map
Ȟ 1 pXZar , GLn q Ñ Ȟ 1 pXfppf , GLn q
is an isomorphism.
Proof. Let g be trivialised by some tUi Ñ Xu. The images Xi of the Ui are open,
and by refining we can assume that the Xi are affine. Then g|Xi is trivialised
by the one object cover tUi Ñ Xi u, and by the corollary above this shows that
it is trivialised by some Zariski cover. But then g itself is trivialised by a Zariski
cover.
We state a number of immediate corollaries, each of which can be referred to as
Hilbert’s theorem 90.
Corollary 3.5.4. Let X be a scheme. Then the canonical maps
PicpXq “ H 1 pXZar , Gm q Ñ H 1 pXét , Gm q Ñ H 1 pXfppf , Gm q
are isomorphisms.
Proof. The second isomorphism follows from Remark 3.3.21, and the first from
the theorem, setting n “ 1, and observing that Ȟ 1 “ H 1 by Proposition 3.1.4.
56
Corollary 3.5.5. Let K be a field. Then
H 1 pK, GLn q “ 0.
Proof. Follows since étale cohomology is Galois cohomology by Corollary B.8.7,
using the theorem above. Here, H 1 denotes the ad hoc definition given in [18]
of non-abelian first cohomology (one checks that it still corresponds to the ad
hoc definition of étale Ȟ 1 ).
Corollary 3.5.6. (Classical Hilbert’s theorem 90) Let K be a field. Then
H 1 pK, K̄ ˆ q “ 0.
Proof. Follows from either of the corollaries above.
57
4
Brauer groups
In this chapter, we will treat the Azumaya Brauer group and the cohomological
Brauer group of a scheme. In the case of Spec K, they are the classical Brauer
group of K, which we will assume familiar. Note however that for a general
scheme, the two different Brauer groups need not coincide.
4.1
Azumaya algebras
Definition 4.1.1. Let X be a scheme. An OX -algebra A is called an Azumaya
algebra over X if it is a locally free OX -module of finite rank, and if moreover
the canonical map
A bOX Aop ÝÑ End OX pAq
given on any affine (Zariski) open U Ď X by
ApU q bOX pU q ApU qop ÝÑ EndOX pU q pApU qq
a b b ÞÝÑ px ÞÑ axbq
is an isomorphism.
Remark 4.1.2. If X “ Spec R is affine, then an Azumaya algebra is just an
R-algebra A that is a projective module of finite type such that the map
A bR Aop ÝÑ EndR pAq
a b b ÞÝÑ px ÞÑ axbq
is an isomorphism. In this case, we also call A an Azumaya algebra over R.
Note that the isomorphism pAbR Aop q˜– ÃbOX Ãop is automatic (cf. Hartshorne
[10], Prop. II.5.2(b)), but for the identification
pEndR pAqq˜– End OX pÃq
we use that A is coherent and X noetherian.
Lemma 4.1.3. Let A be an Azumaya algebra over a ring R. Then the centre
of A is R.
Proof. Note that R Ñ A is injective, since R Ñ A bR Aop “ EndR pAq is, as A
is projective. Also, it is clear that the image of R lands inside the centre ZpAq
of A. Let C be the R-module ZpAq{R.
Firstly, suppose that R is local, with maximal ideal m and residue field k. By
Nakayama’s lemma, any lift of a k-basis of A bR k generates A, hence is a basis
since A is free. Hence, we can pick a basis a1 , . . . , an of A such that a1 “ 1. Let
φ : A Ñ A be defined by
"
1 i “ 1,
φpai q “
0 i ‰ 1.
58
Now if a P ZpAq, then a b 1 is central in A bR Aop , so px ÞÑ axq is central in
EndR pAq. In particular, it commutes with φ, so
a “ a ¨ φp1q “ φpa ¨ 1q.
Since the image of φ is in R, this gives a P R, so ZpAq “ R.
Now turn to the general case. By exactness of localisation, any prime p Ď R
gives a short exact sequence
0 Ñ Rp Ñ ZpAqp Ñ Cp Ñ 0.
It is clear that the subset ZpAqp Ď Ap is actually inside ZpAp q. Hence, Rp Ď
ZpAqp Ď ZpAp q, and by the local case they are all equal. Hence, Cp “ 0, and
we are done since p is arbitrary.
Lemma 4.1.4. Let R be a local ring, and A an Azumaya algebra over R. Then
the ideals I Ď A correspond bijectively to the ideals J Ď R via
I ÞÑ I X R,
JA Ðß J.
Proof. Note that if φ P EndR pAq, then φ is some sum of endomorphisms of the
form x ÞÑ axb. Hence, if I Ď A is an ideal, then φpxq P I whenever x P I.
Hence,
φpIq Ď I.
Now let a1 , . . . , an be a basis for A with a1 “ 1, as above. Let φ1 , . . . , φn be the
elements of EndR pAq defined by
"
1 i “ j,
φi paj q “ δij “
0 i ‰ j,
where δij P R is viewed as an element of A. Note that φi pAq Ď R for all i.
If I Ď A is some ideal, and a P I, then
a“
n
ÿ
φi paqai .
i“1
Since φi pIq Ď I, each term φi paq is in I X R. Hence, a P pI X RqA.
Conversely, if J Ď R is some ideal, and a P JA X R, then we can write
a“
n
ÿ
ri ai
i“1
for certain ri P J. As a P R we must have ri “ 0 for i ą 1, so a P J.
Hence, the maps I ÞÑ I X R and J ÞÑ JA are each others inverse.
Lemma 4.1.5. Let K be a field, and let A be a finite K-algebra. Then A is an
Azumaya algebra over X “ Spec K if and only if it is a central simple algebra
over K.
59
Proof. If A is a central simple algebra, then so is Aop , and a standard result
then shows that also A bK Aop is a central simple algebra. Therefore, the map
A bK Aop ÝÑ EndK pAq
is injective, so by dimension reasons it is an isomorphism. Since A is clearly
free of finite rank, this shows that A is an Azumaya algebra.
Conversely, if A is an Azumaya algebra, then A is central by Lemma 4.1.3, and
simple by Lemma 4.1.4.
Lemma 4.1.6. Let R be a local ring with maximal ideal m and residue field k.
Let A be an R-algebra that is a free module of finite rank. Then A is an Azumaya
algebra over R if and only if A bR k is a central simple algebra over k.
Proof. We write f and fk respectively for the morphisms
A bR Aop ÝÑ EndR pAq
pA bR kq bk pA bR kqop ÝÑ Endk pA bR kq.
There is a commutative diagram
f b1
EndR pAq bR k
„
„
pA bR Aop q bR k
op
pA bR kq bk pA bR kq
fk
Endk pA bR kq.
Therefore, if f is an isomorphism, so is fk .
Conversely, if fk is an isomorphism, then so is f b 1. But since A is free of finite
rank, so are M “ A bR Aop and N “ EndR pAq. Then both the kernel K and
the cokernel C of f are finitely generated, since R is noetherian.
By right exactness of the tensor product, we have C bR k “ 0. By Nakayama’s
lemma, this forces C “ 0, so f is surjective. But then we get a long exact
Tor-sequence:
TorR
1 pN, kq Ñ K bR k Ñ M bR k Ñ N bR k Ñ 0.
Since N is free, the first term vanishes. Since f b1 is an isomorphism, this forces
K bR k “ 0, hence K “ 0 by Nakayama’s lemma. That is, f is an isomorphism,
and the result now follows from the previous lemma.
Lemma 4.1.7. Let A be an OX -algebra that is a locally free OX -module of finite
rank. Then A is an Azumaya algebra over X if and only if Ax is an Azumaya
algebra over OX,x for all x P X.
Proof. Let x P X be a point. Since A is coherent, there is a natural isomorphism
pA bOX Aop qx – Ax bOX,x Aop
x . Moreover, since A is locally free, there is also
a natural isomorphism
pEnd OX pAqqx – EndOX,x pAx q.
60
Hence, the map
A bOX Aop ÝÑ End OX pAq
is an isomorphism if and only if for each x P X the map
Ax bOX,x Aop
x ÝÑ EndOX,x pAx q
is.
Proposition 4.1.8. Let A be an OX -algebra that is locally free of finite rank as
OX -module. Then the following are equivalent:
(1) A is an Azumaya algebra;
(2) Ax is an Azumaya algebra over OX,x for every x P X;
(3) A bOX kpxq is a central simple algebra over kpxq for every x P X.
Proof. Clear from Lemma 4.1.5, 4.1.6, and 4.1.7.
Corollary 4.1.9. Let F be a locally free OX -module of finite rank n. Then
EndOX pF q is an Azumaya algebra over X.
Proof. The module A “ End OX pF q is locally free of finite rank since F is,
and it is an Azumaya algebra since A bOx kpxq – Mn pkpxqq is a central simple
algebra over kpxq, for each x P X.
Corollary 4.1.10. If A and B are Azumaya algebras over X, then so is A bOX
B.
Proof. It is locally free of finite rank since A and B are, and the result now
follows from part (3) of the proposition, using the analogous result on central
simple algebras.
Remark 4.1.11. We could also prove this directly, by constructing a morphism
End OX pAq bOX End OX pBq ÝÑ End OX pA bOX Bq
φ b ψ ÞÝÑ φ b ψ,
where the first φ b ψ indicates the formal element of the tensor product, and
the second indicates the map
φ b ψ : A bOX B ÝÑ A bOX B
a b b ÞÝÑ φpaq b ψpbq.
Locally, one shows this to be an isomorphism by choosing bases for A and B.
Definition 4.1.12. Let A and B be two Azumaya algebras over X. Then A
and B are called similar if there exist locally free OX -modules E , F of finite
rank such that
A bOX End OX pE q – B bOX End OX pF q.
This is clearly an equivalence relation, and the equivalence class of A is denoted
rAs.
61
Definition 4.1.13. The Azumaya-Brauer group BrA pXq of a scheme X is the
set of equivalence classes of Azumaya algebras on X. Given two Azumaya
algebras A, B over X, we define rAs ¨ rBs to be rA bOX Bs. This is well-defined
since
End OX pE q bOX End OX pF q – End OX pE bOX F q,
for any two locally free OX -modules of finite rank E , F (similar to the preceding
remark).
Remark 4.1.14. The multiplication on BrA pXq indeed makes it into a group,
with unit element OX . The inverse of an element A is given by Aop , by the very
definition of an Azumaya algebra!
Remark 4.1.15. If X “ Spec k is the spectrum of a field k and E is a locally
free Ox -module of rank n, then E – k̃ n , and
End OX pE q – pMn pkqq˜.
Hence, Lemma 4.1.5 gives BrA pXq “ Brpkq.
Definition 4.1.16. Let f : X Ñ Y be a morphism of schemes. Then an Azumaya algebra A on Y gives rise to the Azumaya algebra f ˚ A on Y . This induces
a map
f ˚ : BrA pY q Ñ BrA pXq,
making BrA into a functor Schop Ñ Ab.
4.2
The Skolem–Noether theorem
The aim of this section is to prove a generalisation of the Skolem–Noether
theorem for Azumaya algebras over schemes. The treatment is largely based
on [12].
Definition 4.2.1. Let R be a ring, and let A be an Azumaya algebra over
R. Let α, β be R-algebra automorphisms of A. Then we write α Aβ for the
A-bimodule A whose left action is given by α and whose right action is given
by β.
If α “ 1, we will just write Aβ for α Aβ . In particular, we write A1 for the usual
A-bimodule structure on A.
Given an A-bimodule M , we will denote by M A the R-submodule
M A “ tm P M | am “ ma for all a P Au.
For any R-module M , we will equip A bR M with the A-bimodule structure
given by
xpa b mqy :“ pxayq b m,
for x, y P A, a P A and m P M .
62
Proposition 4.2.2. Let α be an R-algebra automorphism of A. Then the natural map
ψ : A bR pAα qA ÝÑ Aα
a b b ÞÝÑ ab
is an A-bimodule isomorphism.
Proof. This follows from III.5.1 of [12]. The proof uses a descent argument.
Definition 4.2.3. If α is an R-algebra automorphism of A, then we write Iα
for the R-module pAα qA , as above.
Remark 4.2.4. Since A1 bR Iα – Aα , this forces Iα to be projective and
faithfully flat, since both A1 and Aα are. By a dimension argument, it must be
a line bundle, so it gives an element of PicpRq.
Lemma 4.2.5. The map
AutR– alg pAq Ñ PicpRq
α ÞÑ Iα
is a group homomorphism.
Proof. If α, β P AutR– alg pAq are given, then the map
ψ : Aα bA Aβ ÝÑ Aαβ
(4.1)
a b b ÞÝÑ aαpbq
is pA, Aq-linear, since
ψpxpa b bqyq “ ψpxa b bβpyqq “ xaαpbqαpβpyqq
for all a P Aα , b P Aβ and x, y P A. It is an isomorphism since the element a b b
equals the element aαpbq b 1, for all a P Aα , b P Aβ . Similarly, the map
Iα bR Iβ ÝÑ Aαβ
(4.2)
a b b ÞÝÑ aαpbq
is pA, Aq-linear. The image is inside Iαβ since xaαpbq “ aαpxbq “ aαpbqαpβpxqq
for all a P Aα , b P Aβ , x P A. Combining the proposition above (for α, β and
αβ) with the isomorphism (4.1), we find that the map
pA1 bR Iα q bA pA1 bR Iβ q Ñ A1 bR Iαβ
induced by (4.2) is an isomorphism. Since A is faithfully flat, this shows that
(4.2) itself is an isomorphism.
Theorem 4.2.6. (Rosenberg–Zelinsky sequence) Let A be an Azumaya algebra.
Then the sequence
0 Ñ Rˆ Ñ Aˆ Ñ AutR– alg pAq Ñ PicpRq
is exact.
63
Proof. The map Rˆ Ñ Aˆ is the obvious one, and the map Aˆ Ñ AutR– alg pAq
is given by
u ÞÝÑ px ÞÑ uxu´1 q.
Injectivity of Rˆ Ñ Aˆ and exactness at Aˆ follow since the centre of A is R.
It remains to show that Iα – R (as R-modules) if and only if α is inner.
Suppose α is inner, say αpxq “ uxu´1 for all x P A. Then Iα “ ZpAqu´1 Ď A,
which is isomorphic to R since ZpAq “ R and right multiplication by u´1 is
R-linear.
Conversely, suppose Iα – R. Then the proposition above gives an isomorphism
„
ψ : A1 ÝÑ Aα .
Let u “ ψp1q´1 . By pA, Aq-linearity of ψ, we have ψpaxbq “ aψpxqαpbq for all
a, b, x P A. Setting x “ 1 and a “ b´1 , this gives
u´1 “ b´1 u´1 αpbq,
or αpbq “ ubu´1 , for all b P A. Hence, α is inner.
Corollary 4.2.7. Let A be an Azumaya algebra over a ring R with trivial Picard
group. Then every automorphism of A is inner.
Proof. The map Aˆ Ñ AutR– alg pAq is surjective.
Corollary 4.2.8. (Skolem–Noether for local rings) Let A be an Azumaya algebra over a local ring R. Then every automorphism of A is inner.
Proof. Over a local ring, every projective module is free, hence PicpRq “ 0.
Remark 4.2.9. We will see later that, for A “ Mn pRq, the exact sequence from
the theorem is part of a (non-abelian) long exact cohomology sequence.
Theorem 4.2.10. (Skolem–Noether for schemes)
Let X be a scheme, and let A be an Azumaya algebra over X. Let α be an
OX -algebra automorphism of A. Then there exists a Zariski covering tUi u of X
and elements ui P ApUi q such that
ˇ
αˇVi pvq “ ui vu´1
i
for all Vi Ď Ui open and all v P ApVi q. That is, α is Zariski-locally given by an
inner automorphism.
Proof. Let x P X be a point. Then Ax is an Azumaya algebra over OX,x , so by
the corollary above, αx is given by a ÞÑ uau´1 for some u P Aˆ
x.
Let V be an open neighbourhood of x on which u is defined, and let a P ApV q.
Then αpaq ´ uau´1 vanishes at x, hence in some open neighbourhood Ua of x.
Taking the intersection of these Ua for some finite set of generators tai u, we find
that α is given by a ÞÑ uau´1 on some open neighbourhood of x. Since x is
arbitrary, this gives the result.
64
Definition 4.2.11. Let PGLn be the sheaf of sets on Xfppf represented by
Spec S0 , where S0 is the degree 0 part of the graded ring
S “ ZrtXij uni,j“1 , det´1 s,
where all the Xij have degree 1, and det is the element
det “
ÿ
sgnpσq
n
ź
Xiσpiq .
i“1
σPSn
Then PGLn is called the projective general linear group scheme. We will prove
that it is indeed a group scheme.
Lemma 4.2.12. Let R be a ring such that Pic R “ 0. Then
PGLn pRq “ GLn pRq{Rˆ .
Proof. We know that giving a morphism from an arbitrary scheme X into Pm
m`1
is the same as giving a surjection of OX -modules OX
Ñ L , where L is a
m`1
line bundle. Moreover, two such maps OX Ñ L , OX Ñ L 1 give rise to the
same morphism X Ñ Pm if and only if there exists an isomorphism φ : L Ñ L 1
making commutative the diagram
L
m`1
OX
φ
L 1.
Now since R has trivial Picard group, we find that
HomSch pSpec R, Pm q “ tpa0 , . . . , an q P Rm`1 | pa0 q ` . . . ` pan q “ Ru{ „,
where two n`1-tuples pa0 , . . . , an q, pb0 , . . . , bn q are equivalent if and only if there
exists λ P Rˆ such that ai “ λbi for all i P t0, . . . , nu. We write ra0 : . . . : an s
for the equivalence class of pa0 , . . . , an q.
Now let S be the graded ring from the definition of PGLn . Then Spec S0 is a
2
standard open inside Proj S – Pn ´1 . Hence,
ˇ
#
+
n
ˇ ÿ
ˇ
n
HomSch pSpec R, Proj Sq “ paij qi,j“1 ˇ
pa q “ R { „,
ˇ i,j“1 ij
and
#
HomSch pSpec R, Spec S0 q “
paij qni,j“1
ˇ
+
n
ˇ ÿ
ˇ
paij q “ R, detpaij q P Rˆ { „ .
ˇ
ˇ i,j“1
řn
But the condition detpaij q P Rˆ forces i,j“1 paij q “ R, hence the right hand
side is GLn pRq{Rˆ . By definition, the left hand side is PGLn pRq, which gives
the result.
65
Corollary 4.2.13. Let R be a ring such that Pic R “ 0 (e.g. R is a local ring).
Then
PGLn pRq “ AutR– alg pMn pRqq.
Proof. Immediate from Lemma 4.2.12 and Theorem 4.2.6.
Proposition 4.2.14. Let X be a scheme. Then
PGLn pXq “ AutOX – alg pMn pOX qq.
Proof. Let F be the Zariski presheaf
U ÞÝÑ AutOU – alg pMn pOU qq
on X. It is a subpresheaf of the sheaf End OX pMn pOX qq of local OU -module homomorphisms. For a local OU -module homomorphism, being an automorphism
is a local condition. Similarly for being an OU -algebra morphism. Hence, F is
a sheaf of groups for the Zariski topology.
Let U Ď X be open, and let φ : U Ñ Spec S0 be a morphism. The discussion in
m`1
Lemma 4.2.12 shows that φ is given by a surjection OU
Ñ L for some line
bundle L on U . If we let tUi u be a Zariski covering of U trivialising L , then
φi “ φ|Ui is given by some matrix Ai P GLn pUi q.
Moreover, for any i, j, there exists λij P ΓpUi X Uj , Oˆ q such that Ai “ λij Aj .
Hence, the inner automorphisms of Mn pOq on Ui and Uj defined by Ai and
Aj respectively coincide on Ui X Uj . Hence, since F is a sheaf, this gives a
well-defined element of F pU q.
This clearly defines a morphism of sheaves
PGLn Ñ F ,
which is an isomorphism since it is so locally, cf. Corollary 4.2.8 and 4.2.13.
Theorem 4.2.15. The projective general linear group PGLn gives a sheaf of
groups on the Zariski, étale and fppf sites. Moreover, there is a short exact
sequence
1 Ñ Gm Ñ GLn Ñ PGLn Ñ 1
on any of these sites.
Proof. The first statement is immediate from the proposition above. The sequence is clearly exact at Gm and at GLn . Exactness on the right follows from
Skolem–Noether for schemes and the definition of an exact sequence of sheaves
of groups (Definition 3.2.5).
Remark 4.2.16. The Rosenberg–Zelinsky sequence (Theorem 4.2.6) for Mn is
now just a nonabelian long exact cohomology sequence, cf. Proposition 3.2.7.
66
4.3
Brauer groups of Henselian rings
In this section, R will be a local ring with maximal ideal m and residue field k.
Definition 4.3.1. A local ring R is called Henselian if, given a monic polynomial f P RrXs and two coprime monic polynomials g0 , h0 P krXs such that
f¯ “ g0 h0 , there exist g, h P RrXs such that ḡ “ g0 and h̄ “ h0 and f “ gh.
Example 4.3.2. By (the classical version of) Hensel’s lemma, any complete
DVR is Henselian. This is the motivation for the term.
Theorem 4.3.3. Let x be the closed point in X “ Spec R. Then the following
are equivalent:
(1) R is Henselian,
(2) any finite R-algebra is isomorphic to a product of local rings,
(3) any étale map f : Y Ñ X such that Y has a point y with f pyq “ x and
kpyq “ kpxq admits a section s : X Ñ Y .
Proof. See [15], Theorem I.4.2.
Corollary 4.3.4. If R is Henselian and R1 is a finite local R-algebra, then R1
is Henselian.
Proof. This follows from p1q ô p2q of the theorem.
Remark 4.3.5. Note that if R1 is finite and local, then the morphism R Ñ R1
is automatically a local ring homomorphism, by the going-up theorem.
Lemma 4.3.6. Let R be Henselian, and let B, C be finite étale local R-algebras.
Then the map
HomR– alg pB, Cq ÝÑ Homk– alg pB bR k, C bR kq
is injective.
Proof. Put X “ Spec C, Y “ Spec B and S “ Spec R. Let x, y, s be the unique
closed points. By the going-up theorem, the fibre above s in X (resp. Y ) is txu
(resp. tyu). In particular, if f : X Ñ Y is an S-morphism, then f pxq “ y. Note
also that B bR k is finite étale over k, and local since B is. Hence, it is a field,
so it is the residue field of B.
Now if f, g : B Ñ C are two R-algebra homomorphisms inducing the same map
B bR k Ñ C bR k, then Corollary 1.1.13 asserts that f “ g.
Lemma 4.3.7. Let R be Henselian, and let B, C be finite étale local R-algebras.
Then the map
HomR– alg pB, Cq ÝÑ Homk– alg pB bR k, C bR kq
is surjective.
67
Proof. Let g : B bR k Ñ C bR k be an R-algebra homomorphism. This defines
a surjective homomorphism
ψ : C bR B Ñ C bR k
c b b ÞÑ c̄gpb̄q.
This corresponds to a kpxq-point z of X ˆS Y , and since the kernel of C Ñ
C bR B Ñ C bR k is the maximal ideal of C, the image of z under the morphism
f : X ˆS Y Ñ X
is x. Now by Corollary 4.3.4, C is Henselian, and by Theorem 4.3.3 (3), we get
a section s : X Ñ Y of f . That is, we get a map
C bR B Ñ C
such that the composition C Ñ C bR B Ñ C is the identity. The composition
B Ñ C bR B Ñ C now gives a map inducing the map B bR k Ñ C bR k we
started with.
Lemma 4.3.8. Let R be Henselian, and let l be a local étale k-algebra. Then
there exists a local étale R-algebra B such that B bR k “ l.
Proof. Since l is finite étale and local, it must be a finite separable field extension
of k. Hence, by the theorem of the primitive element, there exists a separable
monic irreducible polynomial f0 P krXs such that
l “ krXs{pf0 q.
Set B “ RrXs{pf q for any monic lift f P RrXs of f0 . Then f01 pXq is invertible in
l, hence f 1 pXq is not in m, hence invertible in R. This shows that f is separable
as well, so B is étale over R. Clearly B is finite over R and local (since f is
irreducible), and B bR k “ l.
Proposition 4.3.9. If R is Henselian, then the functor B ÞÑ B bR k gives an
equivalence between the category of finite étale R-algebras and the category of
finite étale k-algebras.
Proof. By Theorem 4.3.3 (2) (and since ´ bR k commutes with finite products),
we only need to consider local R-algebras. But on the categories of local finite
étale algebras, we have seen that the functor ´ bR k is full (Lemma 4.3.7),
faithful (Lemma 4.3.6) and essentially surjective (Lemma 4.3.8).
Theorem 4.3.10. If R is Henselian, then the map
BrA pRq ÝÑ BrA pkq
is injective.
Proof. Let A be an Azumaya algebra over R, and let φ be an isomorphism
„
A bR k ÝÑ Mn pkq.
68
Let ε P A bR k correspond to the matrix e1,1 with coefficient 1 on the upper left
entry and 0 elsewhere. Note that ε is an idempotent.
Let a P A be any lift of ε. Then Rras is a finite commutative R-algebra,
so by Theorem 4.3.3 (2), it is a product of local R-algebras. It must be a
product of exactly two local R-algebras, since Rras bR k “ krεs is isomorphic to
krXs{pX 2 ´ Xq “ k ˆ k. Then Rras – B1 ˆ B2 , and one of the elements p1, 0q,
p0, 1q gives a nontrivial idempotent e in Rras mapping to ε in krεs.
„
There is an isomorphism of R-modules A ÝÑ Ae ‘ Ap1 ´ eq. Hence, Ae is a
finitely generated projective R-module, hence free of finite rank since R is local.
Now consider the R-algebra homomorphism
ψ : A Ñ EndR pAeq
b ÞÑ pc ÞÑ bcq.
Let I be the kernel of ψ. Then I XR “ 0 since Ae is a free module. Hence, I “ 0
by Lemma 4.1.4. Similarly, the map ψ̄ : A bR k Ñ Endk ppA bR kqεq induced by
ψ is injective as well. By dimension reasons, ψ̄ is an isomorphism.
Now write C for the cokernel of ψ. Since ψ̄ is an isomorphism and ψ is injective, by right exactness of the tensor product we get C bR k “ 0. Hence,
by Nakayama’s lemma, C “ 0. Hence, ψ is an isomorphism, so A is a matrix
algebra.
Corollary 4.3.11. If R is Henselian and k is either finite or separably closed,
then BrA pRq “ 0.
Proof. In this case, BrA pkq “ Brpkq is zero.
Corollary 4.3.12. If R is Henselian and A is an Azumaya algebra over R,
then there exists a finite étale local R-algebra R1 such that
A bR R1 – Mn pR1 q.
Proof. If R “ k is a field, this says that there is a finite separable field extension
l{k splitting A, cf. [6], Proposition 2.2.5. The general result now follows from
Proposition 4.3.9 and Theorem 4.3.10.
4.4
Cohomological Brauer group
Definition 4.4.1. Let x P X be a point, and write x̄ for Spec kpxq. Then we
define
OX,x̄ “ colim
ΓpU, OU q,
ÝÑ
U
where the limit is taken over all étale maps U Ñ X with a factorisation x̄ Ñ
U Ñ X. It is a Henselian local ring whose residue field is kpxq “ kpx̄q (see [15],
section I.4).
69
Proposition 4.4.2. Let X be a scheme, and let A be an OX -algebra that is
of finite type as OX -module. Then A is an Azumaya algebra if and only if
there exists an étale covering tUi Ñ Xu such that each A|Ui is isomorphic to
Mni pOUi q.
Proof. Let A be an Azumaya algebra. If x is a point, then OX,x̄ is a Henselian
local ring with separably closed residue field, hence
A bOX OX,x̄ – Mn pOX,x̄ q
for some n, by Corollary 4.3.11. But this isomorphism is already defined over
some U Ñ X through which x̄ Ñ X factors. In particular, U Ñ X is étale,
A|U – Mn pOU q, and the image of U in X contains x. Since x was arbitrary,
this proves the assertion.
Conversely, suppose there exists an étale
š covering tUi Ñ Xu such that each A|Ui
is isomorphic to Mni pOUi q. Let U “ i Ui . Then A|U is locally free, hence by
descent theory the same goes for A (since U Ñ X is faithfully flat).
Moreover, if x P X is given, then there exists some y P Ui over x, and the
field extension kpxq Ñ kpyq is finite and separable. The assumption forces that
pA|U qy – Mni pOU,y q, which implies that
pA bOX kpxqq bkpxq kpyq – Mni pkpyqq.
Hence, A bOX kpxq is a central simple algebra over kpxq, so A is an Azumaya
algebra by Proposition 4.1.8.
Lemma 4.4.3. The set of isomorphism classes of Azumaya algebras of rank n2
is isomorphic to Ȟ 1 pXét , PGLn q.
Proof. By the proposition above, an Azumaya algebra of rank n2 is an étale
twist of Mn pOU q. By Proposition 4.2.14, we have
AutOU – alg pMn q – PGLn pU q
for all U P obpÉt{Xq. Hence, an Azumaya algebra defines an element g P
Ȟ 1 pXét , PGLn q (compare section 3.3).
Conversely, a 1-cocycle for PGLn “ AutOX – alg pMn q defines in particular a 1cocycle on AutOX pMn q “ GLn2 . By Hilbert 90, this corresponds to a locally
free module A of rank n.
Moreover, the construction from Definition 3.4.5 shows that A “ N φ is now
the equaliser of two OX -algebra homomorphisms, hence is itself an OX -algebra.
Any covering tUi Ñ Xu that trivialises A as OX -module also trivialises it as
OX -algebra, hence A is an Azumaya algebra.
Proposition 4.4.4. Let X be connected. Then there is a canonical injective
homomorphism
BrA pXq Ñ Ȟ 2 pXét , Gm q.
70
Proof. By Theorem 4.2.15, we have a short exact sequence
1 Ñ Gm Ñ GLn Ñ PGLn Ñ 1
of sheaves on Xét . Since Gm lands in the centre of GLn , Proposition 3.2.7 gives
an exact sequence of pointed sets
δ
n
Ȟ 2 pXét , Gm q.
. . . Ñ Ȟ 1 pXét , GLn q Ñ Ȟ 1 pXét , PGLn q ÝÑ
A direct computation shows that Ȟ 1 pXét , GLn q Ñ Ȟ 1 pXét , PGLn q maps the
locally free OX -module E to the Azumaya algebra End OX pE q. Moreover, a
further computation shows that
δn`m pA bOX Bq “ δn pAqδm pBq
for any Azumaya algebras A, B of ranks n2 and m2 respectively (see [15], Theorem IV.2.5 for the explicit definition of δn ).
Now since X is connected, the rank of an Azumaya algebra A is constant, so
we have A P Ȟ 1 pXét , PGLn q for some n. We define the image of A to be δn pAq.
The above shows that
δn pAq “ δn`m pA bOX End OX pE qq
for any locally free OX -module E of rank m. Hence, if rAs P BrA pXq, then the
image of rAs under the map above does not depend on the Azumaya algebra A
representing rAs. Moreover, the above show that
BrA pXq Ñ Ȟ 2 pXét , Gm q
is a homomorphism, and that the inverse image of 1 is just the trivial class
rOX s.
Corollary 4.4.5. Let X be a compact scheme. Then there is a canonical injective homomorphism
BrA pXq Ñ Ȟ 2 pXét , Gm q.
š
Proof. Let X “ i Xi be the decomposition into connected components. Note
that there are only finitely many. Now both BrA pXq and Ȟ 2 pXét , Gm q break
up into a direct product over the connected components, so the result follows
from the above.
Definition 4.4.6. The (cohomological) Brauer group BrpXq of a scheme X is
the group H 2 pXét , Gm q.
Remark 4.4.7. Some authors write BrpXq for the Brauer group in terms of
Azumaya algebras, and Br1 pXq for the cohomological one. However, since we
are mostly interested in the latter, we have introduced the notation above (BrA
and Br, respectively).
We will use without proof the following theorem.
71
Theorem 4.4.8. Let X be a compact scheme such that every finite subset of X
is contained in an affine open set. Then there are canonical isomorphisms
„
Ȟ i pXét , ´q ÝÑ H i pXét , ´q.
Proof. See [15], Theorem III.2.17. The proof given there uses an article of
M. Artin [1].
Theorem 4.4.9. Let X be a scheme satisfying the assumptions of the theorem
above. Then there is an injection
BrA pXq Ñ BrpXq.
Proof. By the theorem above, we have BrpXq “ Ȟ 2 pXét , Gm q. The result then
follows from the corollary preceding it.
Remark 4.4.10. The theorem remains valid if the assumptions from Theorem
4.4.8 are dropped, but a different proof is required. A sketch of the proof is
given in [15], Theorem IV.2.5.
Remark 4.4.11. Observe that Br X is functorial in X. Indeed, for any smooth
group scheme G over S and any morphism f : X Ñ Y of S-schemes, we have a
restriction map
H i pYét , Gq Ñ H i pXét , Gq.
If X Ñ Y is étale, it is given by the restriction
H i pGqpYét q Ñ H i pGqpXét q,
where H i is the derived functor of the inclusion ShpYét q Ñ PShpYét q.
For a general morphism X Ñ Y , we have to go to the big étale site, and we get
the same result (we do not include the details of this argument).
We need two more propositions about Brauer groups, which we will state without proof.
Proposition 4.4.12. Let R be a Henselian local ring. Then the map
BrA pXq Ñ BrpXq
is an isomorphism.
Proof. See [15], Corollary IV.2.12.
Corollary 4.4.13. Let R be a Henselian local ring, and suppose that its residue
field k is either finite or separably closed. Then
BrpXq “ 0.
Proof. Immediate from the proposition above and Corollary 4.3.11.
72
Proposition 4.4.14. If X is a smooth variety over K, then every element
α P Br X arises Zariski-locally from an Azumaya algebra.
Proof. See [15], Proposition IV.2.15.
73
5
Obstructions for the existence of rational points
In this chapter, K will be a number field, and AK the ring of adèles. We will
study the set of K-rational points on a K-variety X.
5.1
Descent obstructions
In this section, X will be a K-variety, G an affine group variety that is étale
over K, and f : S Ñ X a torsor under G. We need the following construction.
Proposition 5.1.1. Let ξ P H 1 pK, Gq, and let S0 Ñ X be the torsor corresponding to the image of ξ in Ȟ 1 pX, Gq. Then there exist a group variety Gξ
and a torsor fξ : Sξ Ñ X (depending on S) over Gξ , satisfying the following
properties:
(1) Gξ is locally on pSpec Kqfppf isomorphic to G;
(2) The map
Ȟ 1 pX, Gq ÝÑ Ȟ 1 pX, Gξ q
S ÞÝÑ Sξ
is a bijection, mapping S0 to the trivial torsor;
(3) If G is commutative, then Gξ “ G, and the map S ÞÑ Sξ is given by
S ÞÑ S ´ S0 ,
where ` is the addition on Ȟ 1 pX, Gq induced by the addition on G.
(4) Sξ is stable under base change: if Y Ñ X is a morphism of K-varieties,
then
pS ˆX Y qξ “ Sξ ˆX Y.
Proof. See [21], Lemma 2.2.3 and the examples following.
Definition 5.1.2. The group Gξ is called the inner form of G, and Sξ is called
the twist of S (with respect to ξ).
Definition 5.1.3. The torsor S defines a map
θS : Xpkq ÝÑ H 1 pK, Gq
mapping x : Spec K Ñ X to the element of H 1 pK, Gq corresponding to the
torsor S ˆX K Ñ K induced by x : Spec K Ñ X.
Lemma 5.1.4. Let ξ P H 1 pK, Gq, and let x : Spec K Ñ X be a K-point of X.
Then θS pxq “ ξ if and only if Sξ ˆX K has a K-point.
Proof. Let T “ S ˆX K. Then the torsor Tξ has a K-point if and only if it is
the trivial torsor. By Proposition 5.1.1 (2), this is equivalent to T “ ξ. But the
class corresponding to T is θS pxq.
74
Corollary 5.1.5. There is a decomposition
ž
fξ pSξ pKqq .
XpKq “
ξPH 1 pK,Gq
Proof. An element x P XpKq is in fξ pSξ pKqq if and only if the fibre Sξ ˆX K
has a K-rational point. Hence, the result follows from the lemma, since each
x P XpKq satisfies θS pxq “ ξ for exactly one ξ.
Definition 5.1.6. Let f : S Ñ X be a torsor under an étale affine K-variety G.
Then we put
ď
(5.1)
XpAK qf‚ “
fξ pSξ pAK q‚ q .
ξPH 1 pK,Gq
We write
XpAK qf-cov
,
‚
XpAK qf-sol
,
‚
XpAK qf-ab
‚
for the intersections over XpAK qf‚ , where f runs over all torsors under finite,
finite soluble, finite abelian K-group varieties, respectively.
Remark 5.1.7. One can also introduce the above notation with XpAK q instead
of XpAK q‚ . Note that XpAK qf‚ is the image of XpAK qf under the natural
surjection XpAK q Ñ XpAK q‚ . Indeed, if prxv sq P XpAK qf‚ , then there exist ξ
and pryv sq P Sξ pAK q‚ such that
fξ pryv sq “ rxv s.
Since Sξ pCq ‰ ∅ when X ‰ ∅, we can pick yv P Sξ pKv q for each complex place,
giving an element pyv q P Sξ pAK q mapping to pryv sq. Then fξ pyv q “ xv for all
finite places, and fξ pyv q is in the connected component rxv s for all real places.
Hence, pfξ pyv qq is an element of XpAK qf mapping to prxv sq in XpAK q‚ .
Note that we do not necessarily have fξ pyv q “ xv for real v, but they lie in the
same connected component, which is good enough. In other words, the inverse
image of XpAK qf‚ in XpAK q might a priori be larger than XpAK qf .
Lemma 5.1.8. If ψ : X 1 Ñ X is a morphism of K-varieties, then
ψpX 1 pAK qf-cov
q Ď XpAK qf-cov
,
‚
‚
and similarly for the soluble and abelian versions.
Proof. We will prove the first statement; the other two follow similarly. Let
pxv q P X 1 pAK qf-cov
. Let G be an étale affine K-group, and f : S Ñ X a G-torsor.
‚
Then f 1 : S ˆX X 1 Ñ X 1 is a G-torsor on X 1 , so there exists ξ P H 1 pK, Gq and
pzv q P pSξ ˆX X 1 qpAK q‚ with fξ1 ppzv qq “ pxv q. Then
`
`
˘˘
` `
˘˘
fξ p1 ˆ ψq pzv q “ ψ fξ1 pzv q “ ψppxv qq,
hence ψppxv qq P XpAK qf‚ . Since f was arbitrary, the result follows.
We also need the functoriality with respect to K.
75
Proposition 5.1.9. Let L{K be a finite field extension. Then the image of
XpAK qf-cov
in XpAL q‚ is contained in XpAL qf-cov
, and similarly for the soluble
‚
‚
and abelian versions.
Proof. See [22], Proposition 5.16.
We will now compare XpKq to XpAK qf-cov
. We will use the following theorem.
‚
Theorem 5.1.10. Suppose X is proper. Then there are only finitely many
twists Sξ such that Sξ pAK q‚ ‰ ∅.
Proof. This is Proposition 5.3.2 of [21].
Lemma 5.1.11. Let X be a proper variety. Then XpKq Ď XpAK qf-cov
, where
‚
XpKq denotes the topological closure of XpKq in XpAK q‚ .
Proof. Let G be a finite group scheme over K, let S Ñ X be a G-torsor, and let
ξ P H 1 pK, Gq. Then Gξ is finite over K by descent theory, since it is locally on
pSpec Kqfppf isomorphic to G. Hence, Gξ ˆK X Ñ X is finite, so fξ : Sξ Ñ X
is finite since Sξ is fppf-locally isomorphic to Gξ .
In particular, Sξ is a proper variety. Hence,
ź
Sξ pAK q “
Sξ pKv q,
vPΩK
which is compact since each Sξ pKv q is compact. Hence, Sξ pAK q‚ is compact
as well. Since XpAK q‚ is Hausdorff, the image of the compact space Sξ pAK q‚
is closed. By the theorem above, the union in (5.1) is finite, hence XpAK qf‚ is
is closed.
closed. Taking the intersection over all f , we find that XpAK qf-cov
‚
The result now follows since XpKq Ď XpAK qf-cov
,
by
Corollary
5.1.5.
‚
Corollary 5.1.12. Let X be a proper variety. Then we have a chain of inclusions
Ď XpAK qf-sol
Ď XpAK qf-ab
Ď XpAK q‚ .
XpKq Ď XpKq Ď XpAK qf-cov
‚
‚
‚
Proof. The second inclusion follows from the lemma, and the others are obvious.
Definition 5.1.13. Let X be a proper variety. Then X is good with respect
to all coverings (soluble coverings, abelian coverings) if XpKq “ XpAK qf-cov
‚
(resp. XpAK qf-sol
, XpAK qf-ab
‚
‚ ).
In the first case, we also say that X is good. In the third case, we also say that
X is very good.
Definition 5.1.14. Let X be a proper variety. Then X is excellent with respect
to all coverings (soluble coverings, abelian coverings) if XpKq “ XpAK qf-cov
‚
(resp. XpAK qf-sol
, XpAK qf-ab
‚
‚ ).
76
5.2
The Brauer–Manin obstruction
We will use without proof the following addendum to Proposition 1.3.1:
Theorem 5.2.1. Let X{K be a variety, and let XS {OK,S be a model. Let G be
a smooth group scheme over XS . Then the canonical maps
H n ppXT qét , Gq ÝÑ H n pXét , Gq
colim
ÝÑ
T
are isomorphisms for all n P Zě0 .
Proof. This is SGA 4 [2], Corollary VII.5.9.
Definition 5.2.2. If xv : Spec Kv Ñ X is a Kv -point of X, then we denote the
restriction map
Br X Ñ Br Kv
by α ÞÑ αpxv q. If α comes from an Azumaya algebra A, then
αpxv q “ A bOX Kv .
Proposition 5.2.3. Let X{K be a variety, let α P H 2 pXét , Gm q, and let
ź
x “ pxv qv P XpAK q Ď
XpKv q
v
be an AK -point of X. Then αpxv q “ 0 P H 2 pKv , Gm q for almost all v.
Proof. Let XS be a model over OK,S , cf. Proposition 1.3.1 (1). By enlarging S
if necessary, we can assume that xv P XpOv q for v R S. By Theorem 5.2.1, we
can assume that α comes from an element of BrpXS q.
Now for each v R S, the map xv : Spec Kv Ñ XS factors through Spec Ov .
Hence, the map
Br XS Ñ Br Kv
factors through Br Ov . But Br Ov “ 0 by Corollary 4.4.13. Since α P Br X is in
the image of Br XS Ñ Br X, this shows that αpxv q “ 0 for all v R S.
Definition 5.2.4. Let x “ pxv qv P XpAK q be an AK -point. Then we write
à
αpxq “ pαpxv qqv P
Br Kv .
vPΩK
It is indeed an element of the direct sum by the proposition above.
Recall from global class field theory the following theorem:
Theorem 5.2.5. (Brauer–Hasse–Noether) Let K be a number field. Then there
is a short exact sequence
0 ÝÑ Br K ÝÑ
à
ř
invv
Br Kv ÝÑ Q{Z ÝÑ 0,
vPΩK
77
where invv : Br Kv Ñ Q{Z is an injective map that is an isomorphism for all
finite places v.
This inspires the following definition.
Definition 5.2.6. Define a pairing
x´, ´y : Br X ˆ XpAK q Ñ Q{Z
pα, xq ÞÑ inv αpxq “
ÿ
invv pαpxv qq.
v
It is called the Brauer–Manin pairing.
Theorem 5.2.7. Let x P XpAK q. Then for x to come from a K-rational point,
it is necessary that xα, xy “ 0 for all α P Br X.
Proof. If x comes from a K-rational point, then αpxq P
element of Br K. Hence, it maps to 0 in Q{Z.
À
Br Kv comes from an
Definition 5.2.8. Let α P Br X. Then we denote by XpAK qα the set
ˇ
(
XpAK qα “ x P XpAK q ˇ xα, xy “ 0 .
Similarly, if B Ď Br X is a subset, then we put
ˇ
(
XpAK qB “ x P XpAK q ˇ xα, xy “ 0 for all α P B .
For B “ Br X, this set is called the Brauer–Manin obstruction.
Corollary 5.2.9. We have
XpKq Ď XpAK qBr X .
Proof. This is a reformulation of the theorem above.
Corollary 5.2.10. If XpAK qBr X “ ∅, then XpKq “ ∅.
Proof. Immediate from the preceding corollary.
Remark 5.2.11. If X is such that XpAK q ‰ ∅ but XpAK qBr X “ ∅, then
X has points everywhere locally, but not globally. Thus the Brauer–Manin
obstruction is an obstruction to the Hasse principle. The first counterexamples
to the Hasse principle can all be explained by the Brauer–Manin obstruction,
but today examples are known now where the failure of the Hasse principle is
not explained by the Brauer–Manin obstruction.
Finally, we will adjust the Brauer–Manin obstruction to the set XpAK q‚ instead
of XpAK q (see Definition 1.3.8 for this notation).
Proposition 5.2.12. Let X{K be a smooth variety, and let α P Br X. Then
the map
α : XpKv q ÝÑ Br Kv
xv ÞÝÑ αpxv q
78
is locally constant (for the topology on XpKv q induced by the topology on Kv ).
Proof. The question is local. Hence, by Proposition 4.4.14, we can assume that
α is given by an Azumaya algebra A. Moreover, we can assume that X is
connected, so that A is of pure rank n. Then A P Ȟ 1 pXét , PGLn q corresponds
to a PGLn -torsor f : S Ñ X.
If xv : Kv Ñ X is a Kv -point, then A maps to zero under Ȟ 1 pXét , PGLn q Ñ
Ȟ 1 pKv , PGLn q if and only if S ˆX Kv is the trivial torsor, i.e. S ˆX Kv has a
Kv -rational point. Hence, a Kv -point of X satisfies αpxv q “ 0 if and only if xv
is the image of a Kv -point of Y . That is:
α´1 p0q “ f pSpKv qq,
where α denotes the map XpKv q Ñ Br Kv . But f : S Ñ X is étale and surjective, hence
f : SpKv q Ñ XpKv q
is an open map (see e.g. the discussion before Theorem 4.5 of [5]). Hence,
f pSpKv qq is open, so α´1 p0q is open. By translation, each α´1 pcq is open for
c P Br Kv , hence α is locally constant.
Corollary 5.2.13. The Brauer–Manin pairing factors through Br X ˆXpAK q‚ .
Proof. If α P Br X is given and v is a real place, then the value αpxv q depends
only on the connected component on which xv P XpKv q lies.
Definition 5.2.14. This defines a modified Brauer–Manin pairing
x´, ´y : Br X ˆ XpAK q‚ Ñ Q{Z.
If α P Br X, then we put
ˇ
(
ˇ
XpAK qα
‚ “ x P XpAK q‚ xα, xy “ 0 .
If B Ď Br X, then we write
ˇ
(
ˇ
XpAK qB
‚ “ x P XpAK q‚ xα, xy “ 0 for all α P B .
Proposition 5.2.15. Let f : S Ñ X be a torsor on X under PGLn , and let
A P Ȟ 1 pXét , PGLn q be the corresponding Azumaya algebra. Then
XpAK qf‚ “ XpAK qA
‚.
(5.2)
Proof. Both sides of (5.2) are the images of the respective sets in XpAK q under
the natural surjection XpAK q Ñ XpAK q‚ . For the left hand side, this is Remark
5.1.7, and for the right hand side, this is by definition. Hence, it suffices to prove
the result for XpAK q instead of XpAK q‚ .
If θv : XpKv q Ñ H 1 pKv , PGLn q denotes the map associating to a Kv -point xv
the pullback S ˆX Kv , then an argument similar to Lemma 5.1.4 shows that
θv pxv q “ ξ if and only if S ˆX Kv has a Kv -rational point.
79
Hence, if pxv q is an AK -point of X, then θv pxv q “ ξ for all v if and only if
S ˆX AK has an AK -point. That is,
ˇ
(
fξ pXpAK qq “ pxv q P XpAK q ˇ θv pxv q “ ξ for all v .
f
Then
ś 1 XpAK q is the set of pxv q P XpAK q such that the element pθv pxv qq P
H pKv , PGLn q lies in the image of the diagonal map
ź
H 1 pK, PGLn q Ñ
H 1 pKv , PGLn q.
ś
On the other hand, XpAK qA is the set of points pxv q whose image in
Br Kv
lands inside the image of Br K. The result follows since we have a commutative
diagram
ś 1
H 1 pK, PGLn q
H pKv , PGLn q
„
„
v
ś
BrpKqrns
BrpKv qrns.
v
Corollary 5.2.16. We have
č
A pXq
XpAK qBr
“
‚
XpAK qf‚ .
nPZą0
f PȞ 1 pX,PGLn q
In particular, the Brauer–Manin obstruction is a special case of the descent
obstruction.
Proof. Follows by taking intersections over all elements of H 1 pX, PGLn q for
various n.
5.3
Obstructions on abelian varieties
We will firstly study the finite abelian obstruction on abelian varieties, and
then use the embedding of a curve into its Jacobian to deduce results about
obstructions on curves.
Definition 5.3.1. For each n P Zą0 , give the multiplication by n map A Ñ A
the structure of a torsor under Arns via
µ
A ˆA Arns ÝÑ A.
That is, on each U P obpXfppf q, it is given by
ApU q ˆ ArnspU q ÝÑ ApU q
pa, bq ÞÝÑ a ` b.
Lemma 5.3.2. We have
č
x
ApAK qrns
‚ “ SelpK, Aq.
n
80
θn ś
1
Proof. We have a map ApAK q‚ Ñ
v H pKv , Arnsq given by pxv q ÞÑ pθv pxv qq.
By a computation, one checks that this is the same map as the left arrow of the
bottom row of the diagram
0
ApKq{nApKq
0
ApAK q‚ {nApAK q‚
H 1 pK, Arnsq
ź
H 1 pKv , Arnsq
vPΩK
H 1 pK, Aqrns
ź
H 1 pKv , Aqrns
0
0
vPΩK
of Remark 2.3.4. As in the proof of Proposition 5.2.15, we find that ApAK q‚
is the set
ˇ
˜
¸+
#
ˇ
ź
ˇ
1
1
H pKv , Arnsq
. (5.3)
pxv q P ApAK q ˇ θn pxv q P im H pK, Arnsq Ñ
ˇ
v
rns
By definition of the Selmer group, if θn pxv q is in the image of H 1 pK, Arnsq,
it is in fact in the image of Selpnq pK, Aq, since it comes from an element of
ApAK q‚ {nApAK q‚ . Now write φn for the map
φn : Selpnq pK, Aq ÝÑ ApAK q‚ {nApAK q‚ .
pnq
pK, Aq are all nonempty. Since the
Then the sets Cn “ φ´1
n pθn pxv qq Ď Sel
Selmer group is finite, so is Cn . The projective limit of finite nonempty sets is
x
nonempty, hence pxv q is in the image of the injection φ : SelpK,
Aq Ñ ApAK q‚
induced by the φn (it is an injection by Theorem 2.4.12). This gives
č
x
ApAK qrns
‚ Ď SelpK, Aq,
n
and the other inclusion is obvious from (5.3).
Proposition 5.3.3. Let A be an abelian variety. Then
x
ApAK qf-cov
“ ApAK qf-ab
“ SelpK,
Aq.
‚
‚
Proof. We only sketch the proof, since it involves terminology and results beyond
the scope of this thesis. A standard result on abelian varieties shows that the
étale fundamental group of AK̄ is the projective limit
lim
ApK̄qrns.
ÐÝ
n
Hence, the multiplication by n maps rns : A Ñ A form a cofinal set inside the
family of all finite torsors. In particular, both ApAK qf-cov
and ApAK qf-ab
can be
‚
‚
computed as
č
ApAK qrns
‚ ,
n
by Lemma 5.7 and 5.8 of [22]. The result follows from the computation above.
Corollary 5.3.4. Let A be an abelian variety over K. Then A is very good if
and only if XpK, Aqdiv “ 0, and A is excellent with respect to abelian coverings
if and only if ApKq is finite and XpK, Aqdiv “ 0.
81
{ by Corollary 2.4.13, hence ApKq “ SelpK,
x
Proof. We have ApKq “ ApKq
Aq if
and only if XpK, Aqdiv “ 0 by Corollary 2.3.17. Moreover, by Mordell–Weil,
we have
ApKq “ ∆ ˆ Zr
for some finite group ∆ and some r P Zě0 , so
ApKq “ ∆ ˆ Ẑr .
In particular, ApKq “ ApKq if and only if ApKq is finite.
5.4
Obstructions on curves
Recall that for us, curves are smooth, projective, and geometrically connected
K-varieties of dimension 1.
The results in this section are basically section 8 of [22].
Theorem 5.4.1. Let C{K be a curve of genus at least 1, and let Z Ď C
and the image of
be a finite subscheme. Then the intersection of CpAK qf-ab
‚
ZpAK q‚ Ñ CpAK q‚ is ZpKq.
Proof. It clearly contains ZpKq. Conversely, let pxv q P CpAK qf-ab
be in the
‚
image of ZpAK q‚ . Let L{K be a finite Galois extension over which there is a
morphism
ψ : CL Ñ JL ,
where J “ JacpCq is the Jacobian of C.
Let Ξ be the image of ZL in JL , and note that it is still finite (over L). Let pyv q
be the image of pxv q in CpAL q‚ . By Proposition 5.1.9, we have pyv q P CpAL qf-ab
‚ .
x
by Lemma 5.1.8, and this set equals SelpL,
Aq by
Now ψppyv qq is in JpAL qf-ab
‚
Proposition 5.3.3. Since pxv q comes from an element of ZpAK q‚ , it is clear that
ψppyv qq comes from an element of ΞpAL q‚ . By Theorem 2.4.16, this forces
ψppyv qq P ΞpLq “ ψpZpLqq.
By Corollary 2.5.10, the map
ψ : CpAL q‚ Ñ JpAL q‚
is injective, hence we conclude that pyv q P ZpLq. Then in fact it must be in the
image of ZpKq in CpAL q‚ , since it is GalpL{Kq-invariant. Note that this does
not yet force that pxv q is in ZpKq, since the map CpAK q‚ Ñ CpAL q‚ is not in
general injective.
Now if ZpKq “ ∅, then the image of ZpKq in CpAL q‚ is also empty, which
shows that pxv q cannot exist, so we are done. If ZpKq ‰ ∅, then in particular
CpKq is nonempty, so ψ is already defined over K. Hence, following the above
with L “ K gives pxv q P ZpKq, which completes the proof.
82
Theorem 5.4.2. Let φ : C Ñ X be a non-constant morphism into some Kvariety X. If X is excellent with respect to all coverings (soluble coverings,
abelian coverings), then so is C.
Proof. We will prove the result for CpAK qf-cov
. The other two statements follow
‚
similarly.
If C has genus 0, then C is a Severi–Brauer variety of dimension 1, hence it
satisfies the Hasse principle. That is, CpAK q‚ “ ∅ if and only if CpKq “ ∅.
Hence, if CpKq “ ∅, the result is immediate. If CpKq ‰ ∅, then C – P1 , and
CpKq is dense in CpAK q‚ by weak approximation.
Hence, XpKq contains φpCpAK qq‚ . Pick a point x0 P CpKq and a point x1 P
CpK̄q with a different image in XpK̄q, and let L be a finite Galois extension
such that x1 P CpLq. Then L{K is completely split at a set of finite primes of
positive density. Now let
pxv q P CpAK q‚
be such that xv “ x1 is v is a finite place that is completely split (this makes
sense since K Ñ Kv factors through L), and xv “ x0 otherwise. Then ψpxv q is
not the same for all v, hence φppxv qq cannot be in XpKq. Hence,
,
XpKq Ĺ φpCpAK q‚ q Ď XpKq Ď XpAK qf-cov
‚
contradicting the assumption on X. This completes the proof for genus 0.
. Then by Lemma
Now suppose C has genus at least 1. Let pxv q P CpAK qf-cov
‚
5.1.8, we have φppxv qq P XpAK qf-cov
“
XpKq.
Let
Z
be
the inverse image
‚
scheme of P “ φppxv qq. Then Z is quasi-finite over K since φ is non-constant,
so in particular Z is finite. Moreover, pxv q is in the image of ZpAK q‚ Ñ CpAK q‚
since ppxv qq P φ´1 pP q. Hence, by the theorem above, pxv q is in ZpKq. In
particular, it is in CpKq, which completes the proof.
Corollary 5.4.3. Let C Ñ A be a non-constant map of C into an abelian
variety A{K, such that ApKq is finite and XpK, Aqdiv “ 0. Then C is excellent
with respect to abelian coverings.
Proof. This follows from the theorem, since A is excellent with respect to abelian
coverings, by Corollary 5.3.4.
Finally, we will state without proof some consequences. We firstly need a comparison result:
Theorem 5.4.4. Let X be a smooth, projective and geometrically connected
variety. Then
X
XpAK qBr
Ď XpAK qf-ab
‚
‚ .
Proof. This is a consequence of Theorem 6.1.1 of [21]. It is also included as
Theorem 7.1 in [22].
83
In Stoll’s article [loc. cit.], he even proves the following:
C
Theorem 5.4.5. If C is a curve, then CpAK qf-ab
“ CpAK qBr
.
‚
‚
Proof. This is Corollary 7.3 of [22].
Corollary 5.4.6. If C Ñ A is a non-constant map of C into an abelian variety
A{K such that ApKq is finite and XpK, Aqdiv “ 0, then the Brauer–Manin
obstruction is the only obstruction for the existence of rational points on C.
C
Proof. This means that CpKq “ ∅ if and only if CpAK qBr
“ ∅. It is immedi‚
ate from the above. Note that we only need Theorem 5.4.4, and not the slightly
stronger Theorem 5.4.5.
This leads to the following conjecture.
Conjecture 5.4.7. If C is a curve, then C is very good. In other words, CpKq
is dense in CpAK qf-ab
‚ .
Remark 5.4.8. We can make the following observations:
• For genus 0, the proof of Theorem 5.4.2 shows that C is very good.
• For elliptic curves E, we have seen in Corollary 5.3.4 that E is very good
if and only if XpK, Eqdiv “ 0. The Tate–Shafarevich conjecture predicts
that in fact XpK, Aq is finite.
• By Faltings’ theorem, for curves of genus at least 2, the finiteness of CpKq
implies that C is very good if and only if it is excellent with respect to
abelian coverings.
• We have proven the result when C maps nontrivially into an abelian variety of algebraic rank 0 whose Tate–Shafarevich group contains no nonzero
divisible elements.
C
The conjecture would imply that CpKq “ ∅ if and only if CpAK qBr
“ ∅,
‚
i.e. the Brauer–Manin obstruction is the only obstruction for the existence of
rational points on a curve C.
84
A
Category theory
We will recall the basic notions of category theory. We will assume familiar the
notions of category, functor and natural transformation.
Recall that a category C is locally small if for any two objects A, B P ob C ,
the collection C pA, Bq of morphisms A Ñ B is a set. If C is locally small and
moreover the collection ob C of objects is a set, then C is called small. The
occasional remark aside, we ignore set-theoretic issues.
A.1
Representable functors
Lemma A.1.1. Let C be a locally small category, and let A P ob C . Then the
association B ÞÑ C pA, Bq defines a functor C pA, ´q : C Ñ Set.
Proof. Given a morphism f : B Ñ C, we get a natural map
C pA, Bq Ñ C pA, Cq
g ÞÑ f ˝ g.
This clearly makes C pA, ´q into a functor.
Definition A.1.2. Let C be a category, and let F : C Ñ Set be a functor. Then
„
F is representable if there exists a natural isomorphism C pA, ´q Ñ F for some
object A P ob C .
Example A.1.3. The forgetful functor F : Ab Ñ Set is representable: for each
abelian group B, the underlying set F B corresponds bijectively to HompZ, Bq.
Example A.1.4. The forgetful functor F : Ring Ñ Set is representable: for
each ring R, the underlying set F R corresponds bijectively to HompZrXs, Rq.
Example A.1.5. Let F : Topop Ñ Set be the functor associating to a topological space pX, T q the set T of open sets of X, and to a continuous map f : X Ñ Y
the map f ´1 : T pY q Ñ T pXq mapping an open set to its inverse image.
Then F is representable: for each topological space pX, T q, the open sets of X
correspond bijectively with the continuous maps X Ñ A, where A “ t0, 1u with
topology t∅, t1u, t0, 1uu. That is, Topop pA, ´q “ Topp´, Aq – F .
Lemma A.1.6. Let C be a locally small category. Then the association A ÞÑ
C pA, ´q defines a functor Y : C op Ñ rC , Sets.
Proof. If f : A Ñ B is a morphism in C , then it is easy to check that the maps
C pB, Cq Ñ C pA, Cq
g ÞÑ g ˝ f
for all C P ob C form a natural transformation. This makes Y into a functor.
86
Definition A.1.7. The functor Y : C op Ñ rC , Sets defined above is called the
Yoneda embedding.
Theorem A.1.8. (The Yoneda Lemma) Let C be a locally small category. Let
A P ob C be given, and let F : C Ñ Set be a functor. Then there is an isomorphism
„
ψ : NatpC pA, ´q, F q ÝÑ F A,
α ÞÑ αA p1A q.
Moreover, this isomorphism is natural in both A and F .
Proof. Let x P F A be given. Then for B P ob C , define the map
θpxqB : C pA, Bq Ñ F B
f ÞÑ F f pxq.
Then a straightforward argument shows that θpxq is a natural transformation
C pA, ´q Ñ F . Moreover, it is clear that
ψpθpxqq “ θpxqA p1A q “ F p1A qpxq “ 1F A pxq “ x.
Conversely, given a natural transformation α : C pA, ´q Ñ F . Then by naturality of α, for each morphism f : A Ñ B, we have a commutative diagram
C pA, Aq
αA
FA
f ˝´
C pA, Bq
Ff
αB
F B.
Hence,
θpψpαqqB pf q “ F f pψpαqq “ pF f ˝ αA qp1A q “ αB pf ˝ 1A q “ αB pf q.
Hence, as f : A Ñ B was arbitrary, θpψpαqq “ α.
Hence, θ is the inverse of ψ. Naturality in F and A is an easy check.
Corollary A.1.9. The Yoneda embedding is full and faithful.
Proof. Set F “ C pB, ´q, then we have a natural isomorphism
NatpC pA, ´q, C pB, ´qq – C pB, Aq.
It is straightforward to check that it is given by Y g Ðß g.
Definition A.1.10. Let F : C Ñ Set be a functor. Then a pair pA, xq consisting of an object A P ob C and an element x P F A is said to represent F if
θpxq : C pA, ´q Ñ F (as above) is a natural isomorphism.
Note that a functor F : C Ñ Set is representable if and only if there exists a
pair pA, xq representing F . We will show what it means for the examples given
above.
87
Example A.1.11. The forgetful functor F : Ab Ñ Set is represented by the
pair pZ, 1q, since for any abelian group B the map θp1qB : HompZ, Bq Ñ B is
defined by g ÞÑ gp1q. Note that we could also have chosen pZ, ´1q, since the
map g ÞÑ gp´1q also defines an isomorphism HompZ, Bq Ñ B.
Example A.1.12. The forgetful functor F : Ring Ñ Set is represented by
pZrXs, Xq, since for any ring R the map HompZrXs, Rq Ñ R given by g ÞÑ gpXq
is an isomorphism.
Here, we could have equally well chosen pZrXs, φpXqq for any automorphism φ
of ZrXs (since then Y φ is an automorphism of RingpZrXs, ´q). For example,
we could have chosen pZrXs, aX ` bq for a P t˘1u, b P Z.
Example A.1.13. The functor F : Topop Ñ Set described above is represented
„
by pA, t1uq, since the isomorphism ToppX, Aq Ñ T pXq is given by g ÞÑ g ´1 t1u.
Note that in the first two examples, there are several possible choices for the
pair pA, xq. However, it is almost unique, in the following sense.
Corollary A.1.14. (of the Yoneda Lemma) Let pA, xq and pB, yq be two pairs
representing a functor F : C Ñ Set. Then there exists a unique isomorphism
„
f : A Ñ B such that F f pxq “ y.
Proof. The elements x P F A and y P F B correspond to isomorphisms
„
a : C pA, ´q Ñ F
„
b : C pB, ´q Ñ F.
„
Hence, there is a unique isomorphism h : C pB, ´q Ñ C pA, ´q such that a˝h “ b
(namely, h “ a´1 ˝b). But by the previous corollary, such an isomorphism comes
from a unique isomorphism f : A Ñ B. It is straightforward to check that the
condition ah “ b is equivalent to the condition F f pxq “ y.
„
Remark A.1.15. Some authors say that the pair pA, aq (where a : C pA, ´q Ñ F
is the natural isomorphism corresponding to x) represents F , instead of the pair
pA, xq. By the Yoneda lemma, this distinction is purely a matter of taste. We
have included this definition because we believe it gives some intuition behind
the Yoneda lemma; see the examples given above.
For reference purposes, we will state the dual of the Yoneda lemma.
Theorem A.1.16. (Contravariant Yoneda Lemma) Let C be a locally small
category. Let A P ob C be given, and let F : C op Ñ Set be a functor. Then there
is an isomorphism
„
ψ : NatpC p´, Aq, F q ÝÑ F A,
α ÞÑ αA p1A q.
Moreover, this isomorphism is natural in both A and F .
Proof. This follows from replacing C by C op in the Yoneda lemma.
88
A.2
Limits
Limits and colimits will always be assumed to have a small index category,
unless otherwise stated.
Definition A.2.1. Let C be a category, and let J be a small category. Then
a diagram of shape J in C is a functor D : J Ñ C .
Example A.2.2. If J is the category ‚ Ñ ‚ consisting of two objects with
exactly one non-identity morphism, then a diagram of shape J is a pair of
objects A, B P ob C together with a morphism f : A Ñ B.
Definition A.2.3. Let D : J Ñ C be a diagram of shape J . Then a cone
over D is a pair pA, taj ujPob J q, where A is an object of C , and
aj : A Ñ Dpjq
is a morphism in C , for all j P ob J , such that for each morphism α : j Ñ j 1 in
J the diagram
A
aj 1
aj
Dpjq
Dpαq
Dpj 1 q
commutes.
Definition A.2.4. Let D : J Ñ C be a diagram, and let pA, taj ujPob J q and
pB, tbj ujPob J q be two cones over D. Then a morphism of cones f : pA, taj uq Ñ
pB, tbj uq is a morphism f : A Ñ B such that for each j P ob J the diagram
f
A
aj
B
bj
Dpjq
commutes.
Clearly, the above definitions define a category of cones over D.
Remark A.2.5. If you will, the category of cones over D is just the comma
category p∆ Ó Dq, where ∆ : C Ñ rJ , C s is the diagonal embedding mapping
A P ob C to the constant functor j ÞÑ A.
Definition A.2.6. Let D : J Ñ C be a diagram, and let pL, tλj uq be a cone
over D. Then pL, tλj uq is called a limit of D if for every other cone pA, taj uq
there exists a unique morphism of cones f : pA, taj uq Ñ pL, tλj uq.
Remark A.2.7. That is, pL, tλj uq is terminal in p∆ Ó Dq.
Lemma A.2.8. The limit, if it exists, is unique up to unique isomorphism.
89
Proof. We noted that it is a terminal object in a certain category. Hence, it
suffices to prove that those are unique up to unique isomorphism.
Let D be a category, and suppose A, B P ob D are both terminal objects. Then
there is a unique morphism f : A Ñ B and a unique morphism g : B Ñ A.
Moreover, the composition gf is the unique morphism A Ñ A, hence has to
be the identity. Similarly, f g “ 1B . Hence, f is an isomorphism. It is clearly
unique.
Remark A.2.9. We often drop the maps tλj u from the notation, and simply
call L a limit for D. By the lemma, we can even say that L is the limit of D.
We will denote this by
L “ lim Dpjq.
jPob J
Remark A.2.10. One defines cocones under a diagram in a dual way. Then a
colimit is an initial object in pD Ó ∆q, and it is denoted by colim Dpjq.
jPob J
Example A.2.11. Let J be a discrete category, i.e. the only morphisms in J
are the identity morphisms. Then a diagram D : J Ñ C is just an pob J qindexed set of objects Dj P ob C . Then the limit of D (if it exists) is called the
product of the Dj , and it is denoted by
ź
L“
Dpjq.
jPob J
Example A.2.12. In Set, the product is just the Cartesian product. Indeed, if
Xj for j P J are sets, then every set of functions taj : A Ñ Xj u gives a unique
function
ź
a: A Ñ
Dpjq
jPJ
x ÞÑ paj pxqqjPJ ,
suchś
that πj a “ aj , where the product is one in the usual sense, with projections
πj : jPJ Dpjq Ñ Dpjq. Hence, it is also a product in our sense.
The reader is invited to check that products in categories like Gp, Ab, Ring,
Top, Sch{S in the way they are usually defined are indeed products in our sense.
Example A.2.13. More generally, in Set, a limit of an arbitrary diagram
D : J Ñ Set is given by
$
ˇ
,
ˇ
&
.
ź
ˇ
L :“ pxj qjPob J P
Dpjq ˇˇ Dpαqpxj q “ xj 1 for all α : j Ñ j 1 .
%
ˇ
jPob J
We end this section with the following useful characterisation of a limit.
Lemma A.2.14. Let D : J Ñ C be a diagram. Then an object A P ob C is
the limit of D if and only if for every B P ob C there is a natural isomorphism
C pB, Aq – lim C pB, Dpjqq,
jPob J
90
where the limit on the right hand side is one in Set.
Proof. This follows from the description of limits in Set above: the set
lim C pB, Dpjqq
jPob J
is the set of series pfj : B Ñ DpjqqjPob J of morphisms in C such that
Dpαq ˝ fj “ C pB, Dpαqqpfj q “ fj 1
for all α : j Ñ j 1 in J . This is exactly the condition on pB, tfj uq to be a cone
over D. The result follows since A is the limit if and only if the existence of a
morphism f : B Ñ A is equivalent to B being a cone.
A.3
Functors on limits
Definition A.3.1. Let J be a small category, and let F : C Ñ D be a functor.
(1) We say that F preserves limits of shape J if for any diagram D : J Ñ C
and any limit cone pL, tλj uq of D, the cone pF L, tF λj uq is a limit for F D.
(2) We say that F reflects limits of shape J if for any diagram D : J Ñ C
and any cone pL, tλj uq such that pF L, tF λj uq is a limit of F D, the cone
pL, tλj uq is a limit of D.
(3) We say that F creates limits of shape J if for any diagram D : J Ñ
C and any limit pM, tµj uq of F D, there exists a cone pL, tλj uq over D
such that pF L, tF λj uq is isomorphic to pM, tµj uq, and moreover any such
pL, λj uq is a limit of D.
Remark A.3.2. Dually, we define when a functor preserves, reflects or creates
colimits in the obvious way.
Example A.3.3. The forgetful functor F : Ab Ñ Set preserves products. Indeed,śif tAj ujPJ is a set of groups, then the underlying set of the product
A “ jPJ Aj is just the product set of the Aj . In fact, F preserves all limits.
Example A.3.4. However, the forgetful functor F : Ab Ñ Set does not preserve
coproducts. Indeed, if abelian groups tAj ujPJ are given, then the coproduct in
Ab is the direct sum
à
A“
Aj .
jPJ
However, the underlying set of A is not the disjoint union of the Aj , which
would be the coproduct in Set.
Lemma A.3.5. Let F : C Ñ D be a functor that creates limits of shape J .
Then it also reflects them. If moreover D has limits of shape J , then F preserves them.
91
Proof. The first statement is immediate from the definition of creating limits.
Now suppose D has limits of shape J , and let D : J Ñ C be a diagram of
shape J .
Suppose pL, tλj uq is a limit for D. Then pF L, tF λj uq is a cone over F D. Since
D has limits of shape J , we can form the limit pM, tµj uq of F D. Since F
creates limits, there exists a cone pL1 , tλ1j uq over D such that pF L1 , tF λ1j uq is
isomorphic to pM, tµj uq, and moreover L1 is a limit of D.
Hence, pL, tλj uq is isomorphic to pL1 , tλ1j uq, since limits are unique up to unique
isomorphism. But this clearly forces
pF L, tF λj uq – pF L1 , tF λ1j uq – pM, tµj uq,
so pF L, tF λj uq is also a limit of F D. Hence, F preserves limits.
We will now prove some results about limits in functor categories.
Definition A.3.6. Let C be a category. Then the discrete category on C is
ι
the subcategory C disc Ñ C with the same objects as C , but only identity
morphisms.
Lemma A.3.7. Let J be a small category, and D a category that has limits
of shape J . Then the functor rC , Ds Ñ rC disc , Ds creates limits of shape J .
Proof. Note that limits in rC disc , Ds are just pointwise limits. Hence rC disc , Ds
has and evA : rC disc , Ds Ñ D preserves all limits of shape J , for all A P
ob C disc .
Let D : J Ñ rC , Ds be a diagram, and suppose pL, tλpjquq is a limit cone of
ιD. Let f : A Ñ B be any morphism in C . Then LpBq is a limit of evB D.
For each morphism α : j Ñ j 1 in J , naturality of Dpαq gives a commutative
diagram
LpAq
λpj 1 qA
λpjqA
DpjqpAq
DpαqA
Dpj 1 qpAq
Dpj 1 qpf q
Dpjqpf q
DpjqpBq
DpαqB
Dpj 1 qpBq.
This shows that pLpAq, tDpjqpf q ˝ λpjqA uq is a cone over evB D. Hence, there
is a unique morphism Lpf q : LpAq Ñ LpBq making commutative the prism
LpBq
Lpf q
LpAq
DpjqpBq
DpjqpAq
Dpj 1 qpAq
92
Dpj 1 qpBq.
This shows that L can be extended to a functor C Ñ D.
Finally suppose any cone pM, tµpjquq over D mapping to pL, tλpjquq is given.
Let pC, tγpjquq be any cone over D. Since ιM is a limit for ιD, for each A P ob C
there is a unique morphism F pAq : CpAq Ñ M pAq such that µpjqA ˝ F pAq “
γpjqA for every j P ob J .
Now let f : A Ñ B be any morphism in C . We have a diagram
CpAq
F pAq
Cpf q
CpBq
M pAq
M pf q
F pBq
M pBq.
It is easily seen that the two cones
pCpAq, tµpjqB ˝ M pf q ˝ F pAquq,
pCpAq, tµpjqB ˝ F pBq ˝ Cpf quq
over evB D are the same cone. Hence, by the universal property of M pBq, we
have M pf q ˝ F pAq “ F pBq ˝ Cpf q. Hence, the diagram above commutes, so F
is in fact a natural transformation C Ñ M . Hence, pM, tµpjquq is a limit cone
for D.
Remark A.3.8. For D “ Set, the lemma above can also be proved with the
Yoneda lemma. However, getting the dual statement in a similar way would
require some isomorphism
NatpF, GA q – F pAq
for some functor GA depending on A. This can not be done canonically (for
instance because the left hand side is contravariant in F , whereas the right hand
side is covariant). I am not aware of any way to circumvent this. Since we want
to know both the lemma above and its dual (even for D “ Set), we have chosen
for the argument given above.
Corollary A.3.9. Suppose D is complete. Let A P ob C . Then the category
rC , Ds has and the evaluation functor evA : rC , Ds Ñ D preserves all limits.
Proof. The functor ι : rC , Ds Ñ rC disc , Ds creates limits. In particular, rC , Ds
has all limits, and ι preserves them. Hence, evA also preserves limits, since
limits in rC disc , Ds are pointwise.
Corollary A.3.10. Suppose D is cocomplete. Let A P ob C . Then the category
rC , Ds has and the evaluation functor evA : rC , Ds Ñ D preserves all colimits.
Proof. This follows dually.
This gives the following addendum to the Yoneda lemma.
93
Lemma A.3.11. Let C be locally small. Then the (covariant) Yoneda functor
Y : C Ñ rC op , Sets preserves and reflects limits.
Proof. Let J be a small category, and let D : J Ñ C be a diagram of shape
J . By Lemma A.2.14, an object A P ob C is a limit for D if and only if C pB, Aq
is the limit for the functor
J Ñ Set
j ÞÑ C pB, Dpjqq,
for each B P ob C . But this functor is none other than evB Y D. Hence, the
lemma above asserts that A is the limit for D if and only if Y A is the limit for
Y D, which is exactly what we needed to prove.
A.4
Groups in categories
Definition A.4.1. Let C be a category with finite products (in particular, it
has a terminal object T ). Then a group object in C is an object G P ob C
together with morphisms
µ
G ˆ G ÝÑ G
η
T ÝÑ G
ι
G ÝÑ G
such that the diagrams
1ˆµ
GˆGˆG
GˆG
µ
µˆ1
µ
GˆG
GˆT
1ˆη
GˆG
(A.1)
G,
η ˆ1
T ˆG
(A.2)
µ
π1
π2
G,
G
∆G
GˆG
1ˆι
GˆG
µ
η
T
(A.3)
G
commute. If moreover the diagram
GˆG
pπ2 , π1 q
µ
GˆG
µ
G,
commutes, then G is commutative (or abelian).
94
(A.4)
Remark A.4.2. The morphisms µ, η and ι are the analogues of multiplication,
the unit element and inversion, respectively. We will use these names to refer to
these morphisms. Diagram (A.1) is associativity, diagram (A.2) is the neutral
property of the unit element and diagram (A.3) is invertibility. Diagram (A.4)
is commutativity.
Note that we have only asserted that ι is a right inverse; the similar diagram
showing that ι is also a left inverse is a formal consequence of these three diagrams, in the same way that for groups any one-sided inverse is automatically
two-sided.
Example A.4.3. If C “ Set, then a group object is just a group, in the ordinary
sense. Moreover, G is abelian if and only if it is abelian in the ordinary sense.
Example A.4.4. If C “ Top, then a group object is a group G such that the
multiplication and inversion are continuous. That is, G is a topological group.
Note that the unit T Ñ G is automatically continuous, since the terminal object
in Top is the discrete space t˚u.
Also in this example, G is abelian if and only if it is abelian in the ordinary
sense.
Proposition A.4.5. Let C be a category with finite products. Let Y : C Ñ
rC op , Sets be the (covariant) Yoneda embedding. Then G P ob C is a group
object in C if and only if Y G is a group object in rC op , Sets.
Proof. By Lemma A.3.11, Y preserves limits. Since moreover Y is full and
faithful, giving a multiplication µ : G ˆ G Ñ G is equivalent to giving a multiplication
µ : Y pG ˆ Gq “ Y G ˆ Y G ÝÑ Y G
in rC op , Sets, and similarly for η and ι. Moreover, since Y is faithful, the
diagrams (A.1), (A.2) and (A.3) hold for G in C if and only if they hold for Y G
in rC op , Sets.
Lemma A.4.6. Let C be a category. Then a functor G : C op Ñ Set is a
group object if and only if each GA is a group object in Set, and moreover
each Gf : GB Ñ GA induced by f : A Ñ B is a group homomorphism.
Proof. Note that products are just pointwise. Hence, giving multiplication, unit
and inversion morphisms in rC op , Sets is equivalent to giving them on each GA,
such that µ, η and ι are natural. Clearly, they satisfy the diagrams in rC op , Sets
if and only if they satisfy the diagrams in Set for each A P ob C .
Also, note that naturality of µ is commutativity of the diagram
GB ˆ GB
µB
Gf ˆGf
GB
Gf
GA ˆ GA
95
µA
GA,
for any morphism f : A Ñ B in C . This is equivalent to each GB Ñ GA being
a group homomorphism.
Hence, if G is a group, then each GA is a group and each Gf : GB Ñ GA is a
group homomorphism, for f : A Ñ B. Conversely, if each GA is a group and
each Gf is a group homomorphism, then we only need to show naturality of η
and ι. But this is just the statement that group homomorphisms preserve the
identity and commute with inversion.
Corollary A.4.7. Let C be a category with finite products. Then G is a group
object in C if and only if C pA, Gq is a group for each A P ob C and the map
C pB, Gq Ñ C pA, Gq is a group homomorphism for each f : A Ñ B.
Proof. This is immediate from the proposition and the lemma above.
Definition A.4.8. Let C be a category (not necessarily having finite products).
Then a group object in C is an object G such that C pA, Gq is a group for all
A P ob C and the map C pB, Gq Ñ C pA, Gq is a group homomorphism for every
f : A Ñ B.
Definition A.4.9. Let C be a category with finite products, and let G, H
be group objects in C . Then a morphism f : G Ñ H of internal groups is a
morphism f : G Ñ H in C such that the diagram
µG
GˆG
f ˆf
H ˆH
G
f
µH
H,
commutes.
Lemma A.4.10. Let C be a category, and let G, H : C op Ñ Set be internal
group objects in rC op , Sets. Then a natural transformation f : G Ñ H is a
morphism of internal groups if and only if each fA : GA Ñ HA is a group
homomorphism.
Proof. This is obvious.
Corollary A.4.11. Let C be a category. Then the category GppC q of internal
group objects in rC op , Sets is the category rC op , Gps.
Proof. By Lemma A.4.6, an internal group object in rC op , Sets is exactly an object of rC op , Gps. By Lemma A.4.10, the notion of morphism in both categories
coincides as well.
Corollary A.4.12. Let C be a category. Then the category AbpC q of internal
abelian group objects in rC op , Sets is the category rC op , Abs.
Proof. This follows from the observation that an internal group object G in
rC op , Sets is abelian if and only if GA is abelian for each A P ob C .
96
Corollary A.4.13. Let C be a category with finite products, and let Y : C Ñ
rC op , Sets be the Yoneda embedding. Then the category of internal (abelian)
group objects in C is equivalent to the subcategory of rC op , Gps (resp. rC op , Abs)
of functors F for which the composition with the forgetful functor Gp Ñ Set
(resp. Ab Ñ Set) is representable.
Proof. This is clear from the above.
Lemma A.4.14. Let J be a small category, and let C be a category with limits
of shape J . Let G : J Ñ C be a diagram such that Gj is an internal group
object for each j P ob J . Then lim Gj is a group object in a canonical way.
Proof. By Lemma A.2.14, for each A P ob C it holds that
C pA, lim Gj q “ lim C pA, Gj q.
j
j
But in fact, the forgetful functor Gp Ñ Set creates limits. That is, we can
uniquely put a group structure on the limit making it the limit in Gp. It is
clear that this makes Y plim Gj q into a group object in rC op , Gps, which is by
definition representable.
Corollary A.4.15. Let C be a category with limits of shape J . Then the
category of internal group objects has and the forgetful functor GppC q Ñ C
preserves and reflects limits of shape J .
Proof. This is clear from the lemma.
Corollary A.4.16. Let C be a category with finite limits. Then the category
of group objects in C has and the forgetful functor GppC q Ñ C preserves and
reflects kernels and finite products.
Proof. Special case of the corollary above.
ś
In other words: if Gi is a finite set of group objects (i P t1, . . . , nu), then
Gi
is a group object as well. If f : G Ñ H is a morphism of group objects, then its
kernel (the equaliser of f and G Ñ T Ñ H) is a group object as well.
Definition A.4.17. Let C be a category with finite limits, and G a group
object in C . Then a left action of G on an object S is a morphism
m: G ˆ S Ñ S
such that the diagrams
GˆGˆS
1ˆm
GˆS
m
µˆ1
m
GˆS
97
S,
T ˆS
η ˆ1
GˆS
m
π1
S
commute.
Proposition A.4.18. Let C be a category with finite products, and let G be a
group object in C . Then giving a left action of G on S is equivalent to giving a
left action of Y G on Y S.
Proof. Similar to Proposition A.4.5.
Lemma A.4.19. Let C be a category, and G a group object in rC op , Sets. Then
giving a left action of G on S : C op Ñ Set is equivalent to giving a left action of
GA on SA for every A P ob C such that all the maps Gf : GB Ñ GA induced
by f : A Ñ B are G-invariant maps.
Proof. Similar to Lemma A.4.6.
Corollary A.4.20. Let C be a category with finite products, and let G be a group
object in C . Then giving an action of G on S is equivalent to giving an action of
C pA, Gq on C pA, Sq for every A P ob C , such that the map C pB, Sq Ñ C pA, Sq
is G-invariant for all f : A Ñ B.
Proof. Clear from the proposition and the lemma.
Remark A.4.21. One can define a morphism of G-actions in the obvious way,
and prove the analogous results. Also, one can define the notion of right action
and their morphisms, and show that the same properties hold.
98
B
Étale cohomology
This chapter will treat étale cohomology and fppf cohomology. The treatment
is based on [15], [2] and [11].
B.1
Sites and sheaves
A Grothendieck topology is a generalisation of a topological space. We will
study three such topologies in more detail, namely the Zariski topology, the
étale topology and the fppf topology.
Remark B.1.1. Recall that, in the Zariski topology, we define presheaves on
the scheme X as functors ToppXqop Ñ Set, where the category ToppXq has as
objects the open sets U Ď X and as morphisms the inclusions U Ď V . Then a
sheaf is defined to be a presheaf F with the extra condition that for every open
set U P ob ToppXq and for every open covering tUi uiPI of U , the diagram
ź
ź
F pU q ÝÑ
F pUi X Uj q
F pUi q ÝÑ
ÝÑ
pi,jqPI 2
iPI
is an equaliser diagram.
The idea of the étale topology is to replace the category ToppXq by a larger
category of which the “open sets” are no longer solely actual open sets, but
rather étale morphisms U Ñ X for (abstract) schemes U . In order to be able
to talk about coverings, we introduce the following.
Definition B.1.2. A covering family (or simply covering) of an object U in a
f
i
category C is a family tUi ÝÑ
U uiPI of morphisms to U .
In our applications (where C “ Sch{X), we will usually require the maps fi to
be jointly surjective, i.e. the union of their images should equal U .
Definition B.1.3. A Grothendieck pretopology on a category C is a collection
fi
CovpC q of coverings tUi ÝÑ
U uiPI for objects U P ob C , subject to the following
conditions:
(0) If U0 Ñ U occurs in some covering of U , and V Ñ U is any morphism in
C , then the fibred product U0 ˆU V exists in C ;
(1) If tUi Ñ U u is a covering of U and V Ñ U is arbitrary, then tUi ˆU V Ñ V u
is a covering of V ;
(2) If tUi Ñ U uiPI is a covering of U , and for each Ui we have a covering
tUij Ñ Ui ujPJi , then the family of composites tUij Ñ Ui Ñ U uiPI,jPJi is
a covering of U ;
(3) If V Ñ U is an isomorphism, then tV Ñ U u is a one-object covering.
The collection of all coverings is denoted CovpC q.
Note that condition (0) is only included to assure condition (1) makes sense.
100
Definition B.1.4. A site is a category C together with a Grothendieck pretopology CovpC q.
Definition B.1.5. A D-presheaf on a category C is a functor F : C op Ñ D.
We write PShD pC q for the category of D-presheaves on C .
Remark B.1.6. We will be mostly interested in the case D “ Ab. However,
at certain points we will need the case D “ Set or D “ Gp, so we will develop
a slightly more general theory.
Definition B.1.7. A presheaf of abelian groups on a category C is a functor
F : C op Ñ Ab. Equivalently, it is an internal abelian group object in PShSet pC q;
see Corollary A.4.12. We simply write PShpC q for the category of presheaves
of abelian groups on C .
Definition B.1.8. Let D be a category with products. Let F be a D-presheaf
on a site pC , CovpC qq. Then F is a sheaf (with respect to the Grothendieck
pretopology CovpC q) if for any covering tUi Ñ U uiPI , the diagram
ź
ź
F pU q ÝÑ
F pUi ˆU Uj q
F pUi q ÝÑ
ÝÑ
pi,jqPI 2
iPI
is an equaliser diagram. We write ShD pC , CovpC qq for the full subcategory of
PShD pC q of sheaves. If no confusion about the chosen Grothendieck pretopology
is possible, we will simply write ShD pC q. In the case D “ Ab, we will drop the
subscript, and simply write ShpC q.
Remark B.1.9. The pair of parallel arrows in the diagram are given as follows.
The projections Ui ˆU Uj Ñ Ui and Uj ˆU Ui Ñ Ui induce maps
ź
πi
F pUk q ÝÑ
Ñ F pUi q Ñ F pUi ˆU Uj q,
kPI
ź
π
i
F pUk q ÝÑ
Ñ
F pUi q Ñ F pUj ˆU Ui q.
kPI
The parallel arrows above are given by the respective products over all pi, jq P I 2
of these two maps.
Example B.1.10. The (small) Zariski site is the category ToppXq of Zariski
open sets of X, together with the Grothendieck pretopology given by open
coverings tUi Ď U uiPI of U P ob ToppXq.
Fibred products of open subsets are just intersections (this holds both in the
category Sch{X and in ToppXq). Hence, it is easy to check that the Zariski site
is indeed a site. Note that, in this case, the definition of sheaves coincides with
the usual one.
Remark B.1.11. Note that the definition of sheaf depends heavily on the given
Grothendieck pretopology. We will see examples of presheaves which are a sheaf
for the Zariski topology, but not for the étale topology.
We recall the following results from category theory.
101
Lemma B.1.12. Suppose D is complete. Let A P ob C . Then the category
rC , Ds has and the evaluation functor evA : rC , Ds Ñ D preserves all limits.
Proof. This is Corollary A.3.9.
Corollary B.1.13. Suppose D is cocomplete. Let A P ob C . Then the category
rC , Ds has and the evaluation functor evA : rC , Ds Ñ D preserves all colimits.
Proof. This follows dually.
Corollary B.1.14. The category PShpC q is both complete and cocomplete, and
limits and colimits are pointwise.
Proof. This follows since Ab is both complete and cocomplete.
Corollary B.1.15. Let α : F Ñ G be a morphism in PShpC q. Then α is monic
(resp. epic) if and only if αA is injective (resp. surjective) for every A P ob C .
Proof. It is easy to see that any morphism f : B Ñ C in some category D is
monic if and only if the diagram
B
1
B
f
1
B
f
C
is a pullback square. By the above, F is the pullback of α along α if and
only if F pAq is the pullback of αA along αA for each A P ob C . The result now
follows since monomorphisms in Ab are exactly injective maps. The result about
epimorphisms follows dually, since epimorphisms in Ab are exactly surjective
maps.
Definition B.1.16. If a morphism of presheaves α : F Ñ G is a monomorphism
(resp. an epimorphism), we will say it is injective (resp. surjective).
By the corollary above, a morphism α : F Ñ G is injective (resp. surjective) if
and only if the same holds for each αA : F pAq Ñ GpAq.
Corollary B.1.17. The category PShpC q is balanced.
Proof. Let α : F Ñ G be both monic and epic. Then each αA is both injective
and surjective, hence an isomorphism. This forces α to be an isomorphism.
Remark B.1.18. The results of Corollary B.1.14 through B.1.17 also hold for
the category of presheaves of sets, since Set is complete, cocomplete and balanced.
102
B.2
Čech cohomology
In the section above, we have seen some of the basic properties of the category
of presheaves on a category C . In this section and the next, we will show that
a certain adjunction (“sheafification”) gives similar results about the category
of sheaves on a site. On the way, we develop another useful tool, namely Čech
cohomology.
Definition B.2.1. Let C be a site, and let F be a presheaf of abelian groups
on C . Let U “ tUi Ñ U uiPI be a covering of some U P ob C . Then we write
Ui0 ¨¨¨ip “ Ui0 ˆU . . . ˆU Uip
whenever pi0 , . . . , ip q P I p`1 . Define
Č p pU , F q “
ź
`
˘
F Ui0 ¨¨¨ip ,
pi0 ,...,ip qPI p`1
for all p P Zě0 . For each j P t0, . . . , pu, there is a natural map
resp
j
F pUi0 ¨¨¨ij´1 ij`1 ¨¨¨ip q ÝÑ F pUi0 ¨¨¨ip q
defined by the projection
Ui0 ˆU . . . ˆU Uip Ñ
Ñ Ui0 ˆU . . . ˆU Uij´1 ˆU Uij`1 ˆU . . . ˆU Uip .
This defines homomorphisms
dp´1
: Č p´1 pU , F q Ñ Č p pU , F q
j
`
˘
psi qiPI p ÞÑ respj psi0 ¨¨¨ij´1 ij`1 ¨¨¨ip q pi
satisfying the relations
0 ,¨¨¨ ,ip qPI
p`1
,
dpk dp´1
“ dpj dp´1
j
k´1
whenever 0 ď j ă k ď p ` 1. Hence, they define a cosimplicial abelian group
ÝÑ Č 2 pU , F q ¨ ¨ ¨ .
1
Č 0 pU , F q ÝÑ
ÝÑ Č pU , F q ÝÑ
ÝÑ
Definition B.2.2. The Čech complex associated to F with respect to U is the
complex
d0
d1
0 ÝÑ Č 0 pU , F q ÝÑ Č 1 pU , F q ÝÑ . . . .
associated to the cosimplicial abelian group above. That is, its arrows are given
by
p
ÿ
dp´1 “
p´1qj dp´1
.
j
j“0
It is a complex since it comes from a cosimplicial abelian group. That is,
p p´1
d d
p`1
p
ÿ ÿ
“
p´1qk`j dpk dp´1
j
k“0 j“0
ÿ
“
p´1qk`j dpj dp´1
k´1 `
0ďjăkďp`1
ÿ
0ďkďjďp
103
p´1qk`j dpk djp´1 .
Setting k 1 “ k ´ 1 in the first sum gives
ÿ
1
p´1qk `1`j dpj dp´1
`
dp dp´1 “
k
0ďjďk1 ďp
ÿ
,
p´1qk`j dpk dp´1
j
0ďkďjďp
which is zero since the two sums are equal exactly up to a factor ´1.
Definition B.2.3. Let C be a site, F a presheaf of abelian groups on C and
U a covering of U P ob C . Then the Čech cohomology of F with respect to U
is the cohomology
`
˘
Ȟ i pU , F q “ H i Č ‚ pU , F q
of the Čech complex.
We now want to compare the Čech cohomology with respect to different coverings. We firstly need a way to compare to coverings of U .
gj
f
i
Definition B.2.4. Let U “ tUi ÝÑ
U uiPI and V “ tVj ÝÑ U ujPJ be two
coverings of U P ob C . Then V is a refinement of U if there exists a map
α : J Ñ I and for each j P J a morphism ηj : Vj Ñ Uαpjq such that the diagram
ηj
Vj
gj
Uαpjq
fαpjq
U
commutes. The pair pα, tηj uq is called a refining morphism from V to U .
Definition B.2.5. Let pα, tηj ujPJ q be a refining morphism from V to U as
above. Then the ηj induce maps
ηj0 ¨¨¨jp : Vj0 ¨¨¨jp Ñ Uαpj0 q¨¨¨αpjp q
for any pj0 , . . . , jp q P J p`1 . This is turn defines a morphism
p
ψpα,tη
: Č p pU , F q Ñ Č p pV , F q
j uq
given by
`
si0 ¨¨¨ip
˘
pi0 ,...,ip
qPI p`1
¯
´
ÞÑ resηj0 ¨¨¨jp psαpj0 q¨¨¨αpjp q q
pj0 ,...,jp qPJ p`1
.
Remark B.2.6. Note that for all pj0 , . . . , jp q P J p`1 , k P t0, . . . , pu, the diagram
Vj0 ¨¨¨jp
Vj0 ¨¨¨jk´1 jk`1 ¨¨¨jp
ηj0 ¨¨¨jk´1 jk`1 ¨¨¨jp
ηj0 ¨¨¨jp
Uαpj0 q¨¨¨αpjp q
Uαpj0 q¨¨¨αpjk´1 qαpjk`1 q¨¨¨αpjp q
p
commutes. Hence, the maps ψpα,tη
commute with all the dpk , in the sense that
j uq
p
p´1
ψpα,tη
˝ dpk “ dpk ˝ ψpα,tη
.
j uq
j uq
104
It follows that the ψpα,tηj uq commute with the coboundary maps dp , hence they
define morphisms
ρppα,tηj uq : Ȟ p pU , F q Ñ Ȟ p pV , F q.
Lemma B.2.7. Let U , V be two coverings of U P ob C . Then any two refining
morphisms pα, tηj uq, pβ, tθj uq define the same map on Čech cohomology.
Proof. We will show that the maps on the Čech complex are chain homotopic.
For each k P t0, . . . , pu, the maps ηj and θj define a morphism
ηj0 ¨¨¨jk ˆ θjk ¨¨¨jp : Vj0 ¨¨¨jp ÝÑ Uαpj0 q¨¨¨αpjk qβpjk q¨¨¨βpjp q .
This induces maps
χp`1
: Č p`1 pU , F q ÝÑ Č p pV , F q
k
defined by
´
¯
psi qiPI p`2 ÞÝÑ resηj0 ¨¨¨jk ˆθjk ¨¨¨jp psαpj0 q¨¨¨αpjk qβpjk q¨¨¨βpjp q q
pj0 ,...,jp qPJ p`1
.
p
We note that χp`1
˝ dp0 is none other than ψpβ,tθ
, since the diagram
0
j uq
Vj0 ¨¨¨jp
ηj0 ˆ θj0 ¨¨¨jp
Uαpj0 qβpj0 q¨¨¨βpjp q
θj0 ¨¨¨jp
Uβpj0 q¨¨¨βpjp q
p
˝ dpp`1 is just ψpα,tη
commutes. Similarly, χp`1
. By similar arguments, we
p
j uq
find the following relations:
p
χp`1
˝ dpl “ dp´1
k
l´1 ˝ χk ,
χp`1
˝ dpl
k
χp`1
˝ dpk`1
k
We now put χp`1 “
“
“
dp´1
˝ χpk´1 ,
l
p
χp`1
k`1 ˝ dk`1 ,
řp
k p`1
k“0 p´1q χk .
χp`1 dp ` dp´1 χp “
0 ď k ă l ´ 1 ď p.
(B.1)
0 ď l ă k ď p,
(B.2)
0 ă k ă p ´ 1.
(B.3)
Then we get:
p p`1
p p´1
ÿ
ÿ
ÿ
ÿ
p
p´1qk`l χp`1
d
`
p´1qk`l dp´1
χpk .
k
l
l
k“0 l“0
Now the first double sum splits into
¨
˛
ÿ ÿ
ÿ
ÿ ‹
ÿ
˚ ÿ
˚
‹ p´1qk`l χp`1 dp .
`
`
`
`
`
k
l
˝
‚
k“l“0
lăk
l“k
k‰0
l“k`1
k‰p
ląk`1
ląk
105
(B.5)
k“p
l“p`1
The second double sum in (B.4) splits into
˜
¸
ÿ ÿ
`
p´1qk`l dp´1
χpk .
l
lďk
(B.4)
l“0 k“0
(B.6)
p
Now the first term of (B.5) sum gives ψpβ,tθ
. The second term cancels against
j uq
the first term of (B.6), by (B.2). The third and the fourth term cancel against
each other, by (B.3). The fifth term cancels against the second term of (B.6),
p
by (B.1). Finally, the sixth term is just ´ψpα,tη
. Hence, we find that
j uq
p
p
χp`1 dp ` dp´1 χp “ ψpβ,tθ
´ ψpα,tη
,
j uq
j uq
which gives the chain homotopy we were looking for.
Definition B.2.8. If V is a refinement of U , we will simply write ρp pV , U q for
the map ρppα,tηj uq defined above. By the lemma, it depends only on U , V , and
the fact that there exists a refining morphism from V to U . We will usually
drop the superscript where this does not lead to confusion.
It automatically follows that
ρpW , V qρpV , U q “ ρpW , U q
if V is a refinement of U and W of V .
Definition B.2.9. Let U and V be coverings of U P ob C . We write U ” V
if one is a refinement of the other and vice versa. This is clearly an equivalence
relation, since we can compose refinement morphisms.
Corollary B.2.10. If U ” V , then ρpV , U q is an isomorphism
„
Ȟ p pU , F q ÝÑ Ȟ p pV , F q
for any presheaf F on C .
Proof. If pα, tηj ujPJ q denotes a refining morphism from V to U and pβ, tθi uiPI q
one from U to V , then both the composite
pα ˝ β, tηβpiq ˝ θi uiPI q
and the identity are a refining morphism from U to itself. Hence, they induce
the same map on Čech cohomology, so
ρpU , V qρpV , U q “ 1.
Similarly for the other composition, hence ρpU , V q is the inverse of ρpV , U q.
Definition B.2.11. The set of open covers U “ tUi Ñ U uiPI of U up to the
equivalence relation above is denoted JU .
Remark B.2.12. The reader who is interested in such matters may convince
himself that in any of the cases we study (Zariski, étale, fppf), the collection
JU can indeed be taken to be a (small) set. This is however not the case for
any site; most notably the fpqc site (which we do not define) does not have this
property.
106
Corollary B.2.13. If V is a refinement of U , then the map ρpV , U q depends
only on the classes of U and V in JU .
Proof. This follows from the lemma and the previous corollary.
Remark B.2.14. Note that the ordering V ď U if V is a refinement of U
makes JU into a partially ordered set. Moreover, this set is actually directed,
as any two coverings tUi Ñ U uiPI , tVi Ñ U ujPJ have a common refinement
tUi ˆU Vj ÝÑ U upi,jqPIˆJ .
This inspires the following definition.
Definition B.2.15. Let C be a site, let U P ob C , and let F be a presheaf on
C . Then the (absolute) Čech cohomology groups of F are the groups
Ȟ p pU, F q “ colim
Ȟ p pU , F q,
ÝÑ
U PJU
with respect to the maps ρpV , U q of above. By the preceding remark, it is just
a direct limit in the classical sense.
Remark B.2.16. If f : V Ñ U is any morphism in C , then by the axioms of a
site, any covering U “ tUi Ñ U uiPI of U gives rise to a covering
U ˆU V “ tUi ˆU V Ñ V uiPI
of V . Let V “ tVi Ñ V uiPI denote this covering. Then
Vi0 ¨¨¨ip “ Vi0 ˆV . . . ˆV Vip
“ pUi0 ˆU V q ˆV . . . ˆV pUip ˆU V q
“ Ui0 ¨¨¨ip ˆU V,
and f induces morphisms
fi0 ,¨¨¨ ,ip : Vi0 ¨¨¨ip Ñ Ui0 ¨¨¨ip .
This determines a map
p
p
p
fU
,V : Č pU , F q Ñ Č pV , F q
psi qiPI p`1 ÞÑ presfi psi qqiPI p`1 .
p
p
p
It is clear that fU
,V commutes with all the dk , hence also with d , so it defines
a well-defined map
p
p
p
fU
,V : Ȟ pU , F q Ñ Ȟ pV , F q.
If U 1 is a refinement of U , then V 1 “ U 1 ˆU V is a refinement of V . It is
straightforward to check commutativity of the diagram
Ȟ p pU , F q
p
fU
,V
ρpU 1 , U q
Ȟ p pU 1 , F q
Ȟ p pV , F q
ρpV 1 , V q
p
fU
1 ,V 1
107
Ȟ p pV 1 , F q.
p
Hence the maps fU
,V give rise to a map
f p : Ȟ p pU, F q Ñ Ȟ p pV, F q,
making U ÞÑ Ȟ p pU, F q into a presheaf.
Definition B.2.17. The presheaf U ÞÑ Ȟ p pU, F q is denoted Hˇ p pF q.
Remark B.2.18. If U is the trivial covering tU Ñ U u, then each Č p pU , F q
řp`1
is just F pU q. The maps dp are just i“0 p´1qp ¨ 1F pU q , i.e. 0 if p is even and
the identity if p is odd. Hence,
´
¯
0
Ȟ 0 pU , F q “ ker F pU q ÝÑ F pU q “ F pU q,
which gives a map
F pU q “ Ȟ 0 pU , F q Ñ colim
Ȟ 0 pV , F q “ Ȟ 0 pU, F q.
ÝÑ
V PJU
For f : V Ñ U , the pullback V “ U ˆU V is just the trivial covering tV Ñ V u,
hence we have a commutative diagram
F pU q
resf
F pV q
Ȟ 0 pU , F q
Ȟ 0 pU, F q
0
fU
,V
f0
Ȟ 0 pV , F q
Ȟ 0 pV, F q
That is, we get a morphism of presheaves F Ñ Hˇ 0 pF q.
Definition B.2.19. Let F be a presheaf on a site C . Then F is separated if
the morphism of presheaves F Ñ Hˇ 0 pF q is injective.
Lemma B.2.20. Let F be a presheaf. Then F is separated if and only if for
each U P ob C and for each covering U “ tUi Ñ U uiPI of U , the natural map
ź
F pU q ÝÑ
F pUi q
iPI
is injective.
Proof. Clearly, F is separated if and only if for each U P ob C the map
F pU q Ñ Ȟ 0 pU, F q
is injective. Let U P ob C be given. Since Ȟ 0 pU, F q is the direct limit
colim
Ȟ 0 pU , F q,
ÝÑ
U PJU
an element x P F pU q maps to zero in Ȟ 0 pU, F q if and only if there exists some
U P JU such that x maps to zero in Ȟ 0 pU , F q.
Hence, the map F pU q Ñ Ȟ 0 pU, F q is injective if and only if each F pU q Ñ
Ȟ 0 pU , F q is injective.śThe result follows since Ȟ 0 pU , F q “ ker d0 is a subgroup of Č 0 pU , F q “ iPI F pUi q.
108
Remark B.2.21. The usual definition of a separated presheaf found in the
literature is the one of the lemma. The word ‘separated’ refers to the fact that
one can distinguish elements of F pU q (“global sections”) by their restrictions to
each F pUi q (“local sections”). Note however that, in contrast to the Zariski site,
elements of F pU q on an abstract site need not be functions of any sort.
Lemma B.2.22. Let F be a presheaf. Then F is a sheaf if and only if the
natural morphism of presheaves F Ñ Hˇ 0 pF q is an isomorphism.
Proof. Let U “ tUi Ñ U uiPI be a cover of some U P ob C . Note that Ȟ 0 pU , F q
is, by definition, the kernel of
ź
F pUi q
d00 ´ d01
ź
F pUi ˆU Uj q.
pi,jqPI 2
iPI
Note also that the kernel of d00 ´ d01 is the same thing as the equaliser of the pair
ź
F pUi q
iPI
d00
ź
d01
pi,jqPI 2
F pUi ˆU Uj q.
Hence, F is a sheaf if and only if F pU q Ñ Ȟ 0 pU , F q is an isomorphism for any
U P ob C and any covering U of U . The result now follows since for any direct
system tAj ujPJ of abelian groups over a poset J containing an initial object j0 ,
the natural map
Aj0 ÝÑ colim
Aj
ÝÑ
jPJ
is an isomorphism if and only if each Aj0 Ñ Aj is an isomorphism.
Lemma B.2.23. Let F be a presheaf on C . Then Hˇ 0 pF q is separated.
Proof. Let U P ob C , and let U “ tUi Ñ U uiPI be a covering of U . We want
to show that
ź
φ : Ȟ 0 pU, F q Ñ
Ȟ 0 pUi , F q
iPI
is injective. Denote by φi the i-th component of this map, for all i P I.
Suppose x P ker φ, say x P Ȟ 0 pV , F q for some covering V “ tVj Ñ U ujPJ of
U . Put Vi “ V ˆU Ui for all i P I. Then φi pxq is the image under
Ȟ 0 pVi , F q Ñ Ȟ 0 pUi , F q
0
of the element fU
,Vi pxq (with the notation defined above). Since φi pxq “ 0, the
standard properties of direct limits give some covering Wi “ tWik Ñ Ui ukPKi of
Ui refining Vi such that
` 0
˘
ρpWi , Vi q fU
,Vi pxq “ 0,
as elements of Ȟ 0 pWi , F q.
109
Composing with the inclusion
Ȟ 0 pWi , F q Ď
ź
F pWik q,
iPKi
we find that each of the restrictions of x to the Wik must be 0.
But the Wik for fixed i cover Ui , hence by the axioms of a site, the set of
composites
tWik Ñ Ui Ñ U uiPI,kPKi
is a covering of U , which we will denote W . We have seen that x maps to 0
under the natural map
ź
ρpW , V q : Ȟ 0 pV , F q Ñ Ȟ 0 pW , F q Ď
F pWik q.
iPI
kPKi
This says exactly that x “ 0 in the direct limit
Ȟ 0 pU 1 , F q,
Ȟ 0 pU, F q “ colim
ÝÑ
U 1 PJU
i.e. that x “ 0. Hence, φ is injective.
We also have the following:
Proposition B.2.24. Let F be a separated presheaf. Then Hˇ 0 pF q is a sheaf.
Proof. Let U P ob C be given, and let U “ tUi Ñ U uiPI be a covering of U .
We already know that
ź
Ȟ 0 pUi , F q
Ȟ 0 pU, F q ÝÑ
iPI
is injective, by the lemma above. Also, we know that the image of this map is
actually inside
Ȟ 0 pU , Hˇ 0 pF qq “ ker d0 .
Hence, it suffices to show that it equals ker d0 . Let x P Ȟ 0 pU , Hˇ 0 pF qq be
given. Write xi for its component in Ȟ 0 pUi , F q. Since xi P Ȟ 0 pUi , F q, there is
a covering Vi “ tVik Ñ Ui ukPKi of Ui such that xi P Ȟ 0 pVi , F q.
Now for pi, jq P I 2 , let πij : Uij Ñ Ui be the first projection, where Uij denotes
Ui ˆU Uj . To ease notation, we shall identify Uij with Uji . We will also write
Vij “ tVijk Ñ Uij ukPKi for the covering Vi ˆU Uj “ tVik ˆU Uj Ñ Uij ukPKi of
Uij .
Now let pi, jq P I 2 . We know that the elements
pπij q0Vi ,Vij pxi q P Ȟ 0 pVij , F q
pπji q0Vj ,Vji pxj q P Ȟ 0 pVji , F q
become equal in Ȟ 0 pUij , F q, since x P ker d0 .
110
Since the morphism Viki ˆU Vjkj Ñ U factors through Uij , in particular the
images of xi and xj in Ȟ 0 pViki ˆU Vjkj , F q are the same.
Since each Vi covers Ui and the Ui cover U , the axioms of a site imply that
V “ tVik Ñ Ui Ñ U uiPI,kPKi
is a covering of U . Since any xi is an element of
ź
Ȟ 0 pVi , F q Ď Č 0 pVi , F q “
F pVik q,
kPKi
we can construct an element
ź
z P Č 0 pV , F q “
F pVik q
iPI
kPKi
by setting zik “ pxi qk . We have a commutative diagram
ź
ź
F pViki ˆU Vjkj q
F pVik q
pi,jqPI 2
ki PKi
kj PKj
iPI
kPKi
ź
ź
Ȟ 0 pVik , F q
Ȟ 0 pViki ˆU Vjkj , F q.
pi,jqPI 2
ki PKi
kj PKj
iPI
kPKi
induced by the morphism F Ñ Hˇ 0 pF q of presheaves. Moreover, both vertical
arrows are injective since F is separated. But z maps to 0 in the lower right
group, hence it is already 0 in the upper right group. Hence,
z P Ȟ 0 pV , F q.
Hence, z gives an element of Ȟ 0 pU, F q. It remains to show that the image of z
in Ȟ 0 pUi , F q equals xi for each i P I. By definition, this image is given by
´ ˇ
¯
ź
zjk ˇV ˆ Ui
P Ȟ 0 pV ˆU Ui , F q “
F pVjk ˆU Ui q.
jk
U
jPI,kPKj
jPI
kPKj
Since Hˇ 0 pF q is separated, the map
Ȟ 0 pUi , F q Ñ
ź
Ȟ 0 pVik , F q
kPKi
is injective. Now in each Ȟ 0 pVik , F q, the image of z equals the image of xi ,
since xi and xj become the same in Ȟ 0 pUij , F q for all j P I.
Corollary B.2.25. Let F be a presheaf. Then Hˇ 0 pHˇ 0 pF qq is a sheaf.
Proof. By Lemma B.2.23, Hˇ 0 pF q is separated, hence by Proposition B.2.24,
Hˇ 0 pHˇ 0 pF qq is a sheaf.
111
B.3
Sheafification
Definition B.3.1. Let F be a presheaf. Then we denote by F ` the sheaf
Hˇ 0 pHˇ 0 pF qq. It is called the sheaf associated to F , or the sheafification of F .
Remark B.3.2. There is a morphism of presheaves F Ñ F ` . It is easy to
check that this is natural in F , so sheafification becomes a functor.
Note also that if F is a sheaf, then F Ñ F ` is an isomorphism. Indeed, F Ñ
Hˇ 0 pF q is an isomorphism, hence Hˇ 0 pF q is a sheaf so also Hˇ 0 pF q Ñ F ` is
an isomorphism.
We will firstly prove a couple of easy but useful lemmata.
Lemma B.3.3. Let f : F Ñ G be an injective morphism of presheaves, and let
f 1 : Hˇ 0 pF q Ñ Hˇ 0 pG q be the induced morphism. Then f 1 is injective.
Proof. Let U P ob C , and let s P Ȟ 0 pU, F q such that f 1 psq “ 0. Then s is of
the form
ź
s “ psi qiPI P Ȟ 0 pU , F q Ď
F pUi q
iPI
for some covering U “ tUi Ñ U uiPI of U . Then f 1 psq is given by
ź
f 1 psq “ pf psi qqiPI P Ȟ 0 pU , G q Ď
G pUi q.
iPI
Hence, each f psi q is zero, so by injectivity of f , every si is zero, hence s “ 0.
Lemma B.3.4. Let F be a presheaf and G be a sheaf, and let f : F Ñ G be a
morphism of presheaves. Let ρ : F Ñ Hˇ 0 pF q be the natural morphism. Then
ker ρ Ď ker f.
In particular, if f is injective, then F is separated.
Proof. By naturality of F Ñ Hˇ 0 pF q, we have a commutative diagram
F
ρ
Hˇ 0 pF q
f
G
Hˇ 0 pG q.
The bottom arrow is an isomorphism since G is a sheaf, hence the result follows.
The last statement follows since ker ρ “ 0 in that case.
Lemma B.3.5. Let F be a presheaf, and let ρ : F Ñ Hˇ 0 pF q be the natural
morphism. Let G be a sheaf, and g : Hˇ 0 pF q Ñ G a morphism such that gρ “ 0.
Then g “ 0.
112
Proof. Let U P ob C be given, and let s P Ȟ 0 pU, F q. Then s is of the form
ź
F pUi q
s “ psi qiPI P Ȟ 0 pU , F q Ď
iPI
for some covering U “ tUi Ñ U uiPI of U . Then gpρpsi qq “ 0 for all i P I. Now
s|Ui is given by
´ ˇ
¯
ˇ
P Ȟ 0 pU ˆU Ui , F q.
sˇU “ sj ˇU ˆ U
i
j
U
i
jPI
By definition of Ȟ 0 pU , F q, this is equal to
´ ˇ
¯
si ˇUj ˆ Ui
U
jPI
which is just the image of si P Ȟ pU0 , F q under the natural map Ȟ 0 pU0 , F q Ñ
Ȟ 0 pU ˆU Ui , F q, where U0 “ tUi Ñ Ui u is the trivial covering of Ui .
0
That is, ρpsi q “ s|Ui . Hence, gpsq|Ui “ gpρpsi qq “ 0. Since the Ui cover U and
since G is a sheaf, this forces gpsq “ 0.
We now come to one of the main theorems about sites.
Theorem B.3.6. Sheafification p´q` : PShpC q Ñ ShpC q is a left adjoint of the
inclusion ShpC q Ñ PShpC q.
Proof. We have to prove that there is a natural bijection
HomPSh pF , G q – HomSh pF ` , G q,
for any presheaf F and any sheaf G . Let f : F Ñ G be a morphism of
presheaves. By naturality of F Ñ Hˇ 0 pF q, we have a commutative diagram
F
Hˇ 0 pF q
F`
Hˇ 0 pG q
G `,
f
G
of which the arrows of the bottom row are isomorphisms since G is a sheaf.
Hence, every morphism F Ñ G factors through f ` : F ` Ñ G . By applying
the lemma above (twice), we find that this factorisation is unique.
Naturality in F and G is a formal consequence of naturality of F Ñ F ` .
Corollary B.3.7. The category ShpC q is complete, and limits are just pointwise
limits.
Proof. We note that the sheaf condition says that something is a kernel. Since
limits commute with limits, this shows that the presheaf limit of a diagram
D : J Ñ ShpC q is in fact a sheaf, so ShpC q is complete.
On the other hand, the inclusion functor ShpC q Ñ PShpC q is a right adjoint.
Hence, it preserves limits, so limits are just pointwise by Corollary B.1.14.
113
Corollary B.3.8. A morphism f : F Ñ G of sheaves is a monomorphism if
and only if f pU q : F pU q Ñ G pU q is injective for all U P ob C .
Proof. This follows from the description of a monomorphism as a limit, cf. Corollary B.1.15.
Remark B.3.9. Note that similar statements about colimits and epimorphisms
do not hold. In particular, for an epimorphism f : F Ñ G of sheaves, the maps
f pU q : F pU q Ñ G pU q are in general not surjective! Therefore, it of the greatest
importance to indicate in which category we work (PShpC q or ShpC q).
Corollary B.3.10. The category ShpC q is cocomplete. Moreover, for a diagram
D : J Ñ ShpC q, the colimit is the sheafification of the colimit in PShpC q.
Proof. This follows since left adjoints preserve colimits.
Lemma B.3.11. Let f : F Ñ G be a morphism of sheaves. Let H be the
presheaf cokernel of f , that is,
H pU q “ G pU q{f pF pU qq
for all U P ob C . Then f is an epimorphism in ShpC q if and only if H ` “ 0.
Proof. The sequence
F ÑG ÑH Ñ0
is exact in PShpC q. Let K be a sheaf, then left exactness of Homp´, K pU qq
for all U P ob C gives a short exact sequence of (not necessarily small) abelian
groups
f˚
0 Ñ HomPSh pH , K q Ñ HomPSh pG , K q Ñ HomPSh pF , K q.
By the sheafification adjunction, we can also describe this short exact sequence
as
f˚
0 Ñ HomSh pH ` , K q Ñ HomSh pG , K q Ñ HomSh pF , K q.
Now f is epic if and only if f ˚ is injective for any sheaf K . This is equivalent
to
HomSh pH ` , K q “ 0
for all sheaves K , which is in turn equivalent to H ` “ 0.
Remark B.3.12. One would be tempted to just use that f is an epimorphism
if and only if its sheaf cokernel is 0. This follows immediately once we know
that ShpC q is an abelian category. However, this is exactly what we are trying
to prove, which is why a different argument is needed.
Corollary B.3.13. Let f : F Ñ G be a morphism of sheaves. Then f is an
epimorphism in ShpC q if and only if for each U P ob C and for each s P G pU q,
there exists a covering U “ tUi Ñ U uiPI of U such that each s|Ui is in the
image of f pUi q.
114
Proof. Let H be the presheaf cokernel of f , as in the lemma. Since Hˇ 0 pH q
is separated, the morphism Hˇ 0 pH q Ñ H ` is injective. Hence, if H ` “ 0,
then Hˇ 0 pH q “ 0. Since H ` is the sheafification of Hˇ 0 pH q, the converse is
obvious.
Hence, H ` “ 0 if and only if Hˇ 0 pH q “ 0. But the latter is exactly equivalent
to the property stated.
Lemma B.3.14. Let f : F Ñ G be a monomorphism of sheaves. Then its
presheaf cokernel H is separated.
Proof. Let U P ob C be given, and let U “ tUi Ñ U uiPI be a covering of U .
Let s P H pU q be such that s|Ui “ 0 for all i P I. Let t P G pU q represent s.
Then each t|Ui is in the image of f , say t|Ui “ t1i . For i, j P I, it holds that
¯
´ ˇ
ˇ
ˇ
ˇ
´ t1j ˇ
“ ti ˇ
´ tj ˇ
“ 0,
f t1i ˇ
Ui ˆU Uj
Ui ˆU Uj
Ui ˆU Uj
Ui ˆU Uj
hence by injectivity of f , the t1i satisfy the glueing condition. Hence, there exists
t1 P F pU q with t1 |Ui “ t1i for all i P I. Then f pt1 q|Ui “ ti for all i P I, hence by
uniqueness of the glueing condition, f pt1 q “ t. Hence, s “ 0, so the map
ź
H pU q Ñ
H pUi q
iPI
is injective. By Lemma B.2.20, this is what we needed to prove.
Corollary B.3.15. The category ShpC q is balanced.
Proof. Let f : F Ñ G be both monic and epic. Since it is monic, we know that
each f pU q : F pU q Ñ G pU q is injective. On the other hand, we know that its
presheaf cokernel H is separated. Hence, it injects into the sheafification H ` ,
which is zero by Lemma B.3.11. Hence, H is already 0, so each f pU q is an
isomorphism, hence f is an isomorphism.
Theorem B.3.16. The category ShpC q is an abelian category.
Proof. We can clearly enrich ShpC q in abelian groups: if f, g : F Ñ G are two
morphisms, we define f ` g : F Ñ G by
ppf ` gqpU qq psq “ pf pU qq psq ` pgpU qq psq
for U P ob C , s P F pU q. One easily checks that composition becomes bilinear,
so indeed ShpC q is enriched in (not necessarily small) abelian groups.
Since Ab has a terminal object 0 and limits in ShpC q are pointwise, the constant
presheaf F defined by F pU q “ 0 is a sheaf, and it is the terminal object in
ShpC q. Since 0 is initial in Ab and the constant presheaf 0 is already a sheaf, it
is initial in ShpC q by Corollary B.3.10. Hence, ShpC q has a zero object 0.
Clearly ShpC q has binary products, so since it is enriched in abelian groups, it
has binary biproducts.
115
Since ShpC q is complete and cocomplete, every arrow has a kernel and a cokernel. So it remains to prove that every monomorphism is a kernel and every
epimorphism is a cokernel.
Firstly, let f : F Ñ G be a monomorphism of sheaves. Then for each U P ob C ,
the map f pU q : F pU q Ñ G pU q is injective, so we can define the presheaf quotient
H by
H pU q “ G pU q{F pU q.
It is the cokernel of f in PShpC q, since this holds pointwise. Then H ` is the
cokernel of f in ShpC q. Let g : G Ñ H be the quotient morphism of presheaves,
and h` : H Ñ H ` the sheafification morphism. Let g ` “ h` ˝g, then we want
to show that F “ ker g ` . Note that clearly F “ ker g Ď ker g ` .
Conversely, let U P ob C be given, and let s P ker g ` pU q. Then gpsq becomes
0 in H ` . By injectivity of Hˇ 0 pH q Ñ H ` , in fact gpsq has to become 0 in
Hˇ 0 pH q. Hence, there exists a covering U “ tUi Ñ U uiPI of U such that
gpsq|Ui “ 0 for all i P I. Hence, si :“ s|Ui P F pUi q for all i P I.
By the surjectivity criterion of Corollary B.3.13, this says exactly that F Ñ
ker g ` is an epimorphism. Hence, since ShpC q is balanced, F Ñ ker g ` is an
isomorphism, so F is the kernel of its cokernel.
Finally, let f : F Ñ G be an epimorphism of sheaves. Let e : E Ď F be the
kernel of f , and let h : F Ñ H be the presheaf cokernel of e, that is,
H pU q “ F pU q{E pU q
for all U P ob C . Since f e “ 0, we get a morphism of presheaves g : H Ñ G
such that gh “ f . Let g ` : H ` Ñ G be the associated morphism of sheaves,
and note that H ` is the sheaf cokernel of e. Write h` : F Ñ H ` for the
composition
h
F Ñ H Ñ H `.
Now g is injective since E pU q is the kernel of f pU q, by definition. Hence, by
applying Lemma B.3.3 twice, we see that g ` is injective. On the other hand,
we have f “ g ` h` , so g ` is epic since f is. Hence, as ShpC q is balanced, g ` is
an isomorphism, and f is the cokernel of its kernel.
Proposition B.3.17. The functor p´q` : PShpC q Ñ ShpC q is exact.
Proof. It is obviously additive. It is right exact since p´q` is the left adjoint
of the inclusion ShpC q Ñ PShpC q. On the other hand, it preserves monomorphisms by Lemma B.3.3 (applied twice). Hence, it is exact.
B.4
The étale site
In this section, we will give the main examples of sites we will study. Besides
the étale site, the two main examples are the Zariski site and the fppf site.
116
Definition B.4.1. Let X be a scheme. Then the category Ét{X is the full
subcategory of Sch{X of schemes f : U Ñ X over X for which the structure
morphism f is étale.
Lemma B.4.2. Every morphism in Ét{X is étale.
Proof. A morphism pU, f q Ñ pV, gq of schemes f : U Ñ X, g : V Ñ X over X
is just a morphism φ : U Ñ V such that gφ “ f . Since f is étale and g is
unramified, the result follows from Corollary 1.1.9.
We suggestively introduce the following.
Definition B.4.3. Let X be a scheme, and let U be a scheme over X. Then a
family tUi Ñ U uiPI of morphisms to U (in Sch{X) is called an étale covering of
U if all the maps Ui Ñ U are étale, and moreover the union of their images is
all of U (we will say that the morphisms Ui Ñ U are jointly surjective).
Remark B.4.4. We want to use this definition to define a Grothendieck pretopology on Ét{X. A priori, we need to restrict to all coverings tUi Ñ U uiPI
for which the structure morphisms Ui Ñ X are étale. However, they are given
by the compositions
Ui Ñ U Ñ X,
so they are automatically étale over X when U is.
Lemma B.4.5. Let tUi Ñ U uiPI be an étale covering of a scheme U . Let
V Ñ U be any morphism. Then
tUi ˆU V Ñ V uiPI
is an étale covering of V .
š
š
Proof. We write Vi for Ui ˆU V . Write U8 for i Ui , and V8 for i Vi . Then
V8 is the fibred product U8 ˆU V , by the construction of the fibred product.
Note that tUi Ñ U u is a covering if and only if U8 Ñ U is surjective.
By Lemma 1.1.7, each Vi Ñ V is étale. By Lemma 1.2.1, V8 Ñ V is surjective
since U8 Ñ U is. Hence, tVi Ñ V u is a covering.
Proposition B.4.6. Let X be a scheme. Then the collections tUi Ñ U uiPI that
are étale coverings of U define a Grothendieck pretopology on Ét{X.
Proof. If U0 Ñ U is an étale morphism and V Ñ U is any morphism of schemes
étale over X, then the fibred product U0 ˆU V exists in Sch{X. Moreover,
by Lemma 1.1.7, the morphism U0 ˆU V Ñ V is étale. Since the structure
morphism V Ñ X was étale by assumption, Lemma 1.1.6 asserts that the
composite morphism
U0 ˆU V Ñ V Ñ X
is étale as well.
117
But this is the structure morphism of U0 ˆU V , since the diagram
U0 ˆU V
V
U0
U
X
commutes. Hence, U0 ˆU V is an object of Ét{X. It is clearly the fibred product
of U0 and V along U in this category as well, since the inclusion Ét{X Ñ Sch{X
is full (and faithful).
Now let U “ tUi Ñ U uiPI be a covering of U P obpÉt{Xq, and let V Ñ U be
a morphism in Ét{X. Then tUi ˆU V Ñ V uiPI is an étale covering of V , by
Lemma B.4.5.
If tUi Ñ U uiPI is a covering of U , and for each i P I we have a covering
tUij Ñ Ui ujPJi , then clearly all the maps Uij Ñ U are étale. Since the Uij Ñ Ui
are jointly surjective and the Ui Ñ U are, so are the Uij Ñ U . Hence, the
family
tUij Ñ Ui Ñ U uiPI,jPJi
is a covering of U .
„
Finally, it is clear that the one object family tV Ñ U u is a covering of U .
Definition B.4.7. Let X be a scheme. Then the (small) étale site Xét is the
category Ét{X endowed with the Grothendieck pretopology described above.
Remark B.4.8. Despite the name, it is not a small category. The word small
is included to distinguish it from the big étale site, which is given by the same
Grothendieck topology, but with underlying category Sch{X. We will not study
this site in more detail. We do note that the proof that it is indeed a site is the
same as the proof above, except that existence of fibred products are automatic.
Definition B.4.9. Let X be a scheme, and let U be a scheme over X. Then a
family tUi Ñ U uiPI of morphisms to U (in Sch{X) is called an fppf covering of
U if all the maps Ui Ñ U are flat and locally of finite type, and moreover the
morphisms Ui Ñ U are jointly surjective.
Remark B.4.10. The term fppf is short for fidèlement plat de présentation
finie, which is French for ‘faithfully flat of finite presentation’. The faithful part
refers to the fact that the Ui Ñ U are jointly surjective. Since we assume all
schemes to be locally Noetherian, a morphism is locally of finite presentation if
and only if it is locally of finite type.
Proposition B.4.11. Let X be a scheme. Then the collections tUi Ñ U uiPI
that are fppf coverings of U P obpSch{Xq define a Grothendieck pretopology on
Sch{X.
Proof. Analogous to Proposition B.4.6.
118
Definition B.4.12. The category Sch{X together with the Grothendieck topology given by fppf coverings is called the big fppf site. It is denoted Xfppf .
Remark B.4.13. One does not usually define a small fppf site. One reason for
this is that there is no analogue of Corollary B.4.2, since a morphism U Ñ V
of schemes that are flat over X is not necessarily flat. For example, if X “ k is
(the spectrum of) a field, then both k and A1k are flat over k, but the morphism
k Ñ A1k mapping the single point to the origin in A1k is not flat.
Some authors write pSch{Xqfppf for the big fppf site, to indicate that its definition is not analogous to the small étale site, but rather to the big étale site,
which is usually denoted pSch{Xqét . Since we will only use the small étale and
big fppf site, we will not make this distinction.
Remark B.4.14. Recall that the (small) Zariski site, from Definition B.1.10,
is defined as the small category
ToppXq
of open sets on X, together with the Grothendieck topology given by coverings
in the classical, topological sense: a covering of U Ď X is a family tUi Ď U uiPI
such that the union of all the Ui is U .
We will denote this site by XZar . As opposed to the two other sites we are
considering, its underlying category is actually a small category, being a full
subcategory of the power set of X, viewed as poset.
Remark B.4.15. Note that the underlying categories of XZar , Xét and Xfppf
have fibred products. For XZar , they are given by the intersection (which is
also the scheme theoretic fibred product). For Xét , this is proven in Proposition
B.4.6, bearing in mind that any morphism in Ét{X is étale (in order to assert
that the fibred product is again étale over X). For Xfppf , it is just the fibred
product in Sch{X, which coincides with the fibred product in Sch.
Hence, not only do fibred products exist in the categories XZar , Xét and Xfppf ,
but they are also preserved and reflected by the inclusion functor to Sch{X. In
particular, when writing a fibred product in any of the above categories, it will
be understood as the fibred product in the category of schemes.
B.5
Change of site
Definition B.5.1. Let u : C Ñ D be a functor. Then the functor
PShpDq Ñ PShpC q
F ÞÑ F ˝ uop
is denoted up . Here, uop : C op Ñ D op denotes the opposite functor of u.
Lemma B.5.2. Let u : C Ñ D be a functor. Then up preserves all limits and
colimits.
119
Proof. We will prove the statement about limits; the one about colimits follows
similarly.
Let F be a limit of a diagram D : J Ñ PShpDq. Since limits in presheaf
categories are pointwise, this implies that F pU 1 q is the limit of evU 1 D for all
U 1 P ob D. Hence, up pF qpU q “ F pupU qq is the limit of evupU q D for all U P
ob C , so up pF q is the limit of up D.
Corollary B.5.3. The functor up is exact.
Proof. It preserves finite limits and colimits.
In what follows, we will construct a left adjoint up for up , under certain conditions on u.
Definition B.5.4. Let u : C Ñ D be a functor, and let A P ob D. Then we
write IA for the comma category pA Ó uq. That is, an object of IA is an
object U P ob C together with a morphism f : A Ñ upU q, and a morphism
φ : pU, f q Ñ pV, gq is a morphism φ : U Ñ V making commutative the diagram
A
f
g
upU q
upφq
upV q.
Definition B.5.5. Let F be a presheaf on C . For A P ob D and pU, f q P ob IA ,
define
DA,F pU, f q “ F pU q.
If φ : pU, f q Ñ pV, gq is a morphism in IA , then define
DA,F φ : DpV, gq Ñ DpU, f q
as the restriction F pV q Ñ F pU q defined by φ. This clearly defines a functor
op
DA,F : IA
Ñ Ab.
The colimit of this diagram is denoted up pF qpAq. We will drop the F from the
subscript when it causes no confusion.
Remark B.5.6. The careful reader should convince himself that the category
IA is equivalent to a small category for each of the sites we study. Hence, the
limit can be seen as a small limit, and is thus well-defined.
Definition B.5.7. If a : A Ñ B is a morphism in D, then there is a functor
IB Ñ IA
defined on objects pU, f q P ob IB by pU, f aq, and on morphisms φ : pU, f q Ñ
pV, gq by φ, viewed as morphism pU, f aq Ñ pV, gaq.
120
In particular, for each pU, f q P ob IB , we get a morphism
DB pU, f q ÝÑ up pF qpAq “ colimop DA pV, gq
pV,gqPIA
by viewing DB pU, f q as DA pU, f aq.
Lemma B.5.8. The maps DB pU, f q Ñ up pF qpAq make up pF qpAq into a cocone under DB .
Proof. Given a morphism φ : pU, f q Ñ pV, gq in IB , we also have a morphism
φ : pU, f aq Ñ pV, gaq in IA . Hence, the diagram
DA pU, f aq
resφ
DA pV, gaq
up pF qpAq
commutes.
Corollary B.5.9. There is a unique homomorphism
up pF qpBq Ñ up pF qpAq
defined on DB pU, f q by the map DA pU, f aq Ñ up pF qpAq. Moreover, these maps
make up pF q a presheaf on D.
Proof. Only the last statement is new. But functoriality in A follows from the
uniqueness statement.
One easily sees that this construction is functorial in F . Hence, we obtain a
functor
up : PShpC q Ñ PShpDq.
Theorem B.5.10. The functor up is a left adjoint for up .
Proof. Let F be a presheaf on C . Let U P ob C . Then pU, 1q is an object of
IupU q , hence it gives rise to a morphism
F pU q Ñ up pF qpupU qq “ pup up F qpU q.
A simple inspection shows that these maps are compatible for different U P ob C ,
hence we get a morphism of presheaves
ηF : F Ñ up up F .
Conversely, let G be a presheaf on D. Let A P ob D, and let pU, f q P ob IA .
Then we get a morphism
resf : G pupU qq Ñ G pAq.
121
By the definition of the category IA , these maps are compatible for different
pU, f q P ob IA . Hence, they make G pAq into a cocone under DA,up G . Hence,
there is a unique morphism
pup up G qpAq Ñ G pAq
given by the resf . One checks that this is natural in A, giving a morphism of
presheaves
εG : up up G Ñ G .
The constructions above are clearly natural in F and G , so η and ε are natural
transformations. In order to check that they are the unit and counit of an
adjunction, one needs to check commutativity of the following two diagrams:
up F
u p ηF
up up up F
up G
ηup G
εup F
1
1
up F
up up up G
up εG
up G .
We omit the verification.
Corollary B.5.11. The functor up is right exact.
Proof. Any left adjoint is right exact.
We want to know in which cases up is also left exact. Since it is defined by a
colimit, it is natural to ask whether that colimit is a direct limit.
Lemma B.5.12. Suppose C has and u preserves finite limits. Let A P ob D.
Then IA is cofiltered.
Proof. Since C has finite limits, in particular it has a terminal object T . Moreover, upT q is terminal in D. Hence, there exists a unique f : A Ñ upT q. Hence,
pT, f q is an object of IA , so IA is nonempty.
Now let pU, f q, pV, gq be objects of IA . We can form the product U ˆ V in C ,
and we know that upU ˆ V q “ upU q ˆ upV q. In particular, we get a morphism
f ˆ g : A ÝÑ upU ˆ V q,
with projections π1 : upU ˆ V q Ñ upU q, π2 : upU ˆ V q Ñ upV q such that π1 ˝
pf ˆ gq “ f and π2 ˝ pf ˆ gq “ g.
Hence, we have an object pU ˆ V, f ˆ gq in IA , together with morphisms
π1 : pU ˆ V, f ˆ gq ÝÑ pU, f q
π2 : pU ˆ V, f ˆ gq ÝÑ pV, gq.
Finally, let a, b : pU, f q Ñ pV, gq be a pair of parallel morphisms in IA . Then let
w : W Ñ U be the equaliser of a and b in C . Then upW q is the equaliser of upaq
and upbq.
122
Since af “ g “ bf , there exists a unique morphism
h : A Ñ upW q
such that upwq ˝ h “ f . In particular, we get a morphism w : pW, hq Ñ pU, f q
such that aw “ bw.
Hence, IA is cofiltered.
Corollary B.5.13. Suppose C has and u preserves fibred products and a terminal object. Then up is exact.
Proof. It is a standard result from category theory that all finite limits can be
built from fibred products and a terminal object. Hence, C has and u preserves
finite limits.
Hence, by the lemma above, for every A P ob D, the category IA is cofiltered.
This makes up pF qpAq a filtered colimit, and we know that filtered colimits in
Ab are exact. The result follows since limits (and hence exactness) in presheaf
categories are pointwise.
Definition B.5.14. Let C , D be sites. Then a functor u : C Ñ D is continuous
if it preserves fibred products that exist in C , and for every covering tUi Ñ U uiPI
of some U P ob C , the image tupUi q Ñ upU quiPI is a covering of upU q.
Example B.5.15. If X 1 Ñ X is a morphism of schemes and U Ñ X is étale,
then by Lemma 1.1.7, the base change U 1 “ U ˆX X 1 Ñ X 1 is étale as well.
Hence, we get a functor
1
u : Xét Ñ Xét
U ÞÑ U ˆX X 1 .
Moreover, by Lemma B.4.5, we see that, for any covering tUi Ñ U u of U P
obpÉt{Xq, the associated family tUi ˆU U 1 Ñ U 1 u is also a covering of U 1 .
Finally, if U1 , U2 P obpÉt{Xq are two schemes over a third scheme U P obpÉt{Xq,
we have isomorphisms
pU1 ˆU U2 q ˆX X 1 – pU1 ˆU U2 q ˆU U 1
– pU1 ˆU U 1 q ˆU 1 pU2 ˆU U 1 q
– pU1 ˆX X 1 q ˆU 1 pU2 ˆX X 1 q.
Hence, u preserves fibred products, so u is continuous.
Example B.5.16. Similarly, if X 1 Ñ X is a morphism of schemes, then it
defines continuous functors
1
XZar Ñ XZar
1
Xfppf Ñ Xfppf
on the Zariski and fppf sites.
123
Example B.5.17. If X is a scheme, we get inclusion functors XZar Ñ Xét Ñ
Xfppf . It is clear that they preserve coverings, and they preserve fibred products
by Remark B.4.15. Hence, they are continuous.
In particular, if X 1 Ñ X is a morphism of schemes, we also get continuous
functors
1
XZar Ñ Xét
1
XZar Ñ Xfppf
1
Zét Ñ Xfppf
,
obtained by the composition Xτ Ñ Xτ1 Ñ Xτ1 1 , for τ, τ 1 P tZar, ét, fppfu. One
easily sees that it is also given by the composition Xτ Ñ Xτ 1 Ñ Xτ1 1 .
Remark B.5.18. Note that in each of the given examples, the site C has and
the functor u : C Ñ D preserves fibred products and terminal objects. Hence,
Corollary B.5.13 applies, and there is an adjunction
PShpC q ÝÑ
ÐÝ PShpDq
of exact functors.
Lemma B.5.19. Let u : C Ñ D be a continuous functor of sites. Let F be a
sheaf on D. Then up F is a sheaf on C .
Proof. Since up is the right adjoint of up , it preserves all limits. In particular, it
preserves products and equalisers. Moreover, since u is continuous, it preserves
fibred products that exist in C and it preserves coverings. Hence, the sheaf
condition of up F on the covering tUi Ñ U u of U P ob C is just the sheaf
condition of F on the covering tupUi q Ñ upU qu of upU q.
Definition B.5.20. The restriction of up to ShpDq Ñ ShpC q is denoted us .
Definition B.5.21. The composite functor
up
p´q`
ShpC q Ñ PShpC q ÝÑ PShpDq ÝÑ ShpDq
is denoted us .
Theorem B.5.22. The functor us is a left adjoint of us .
Proof. This follows from the chain of adjunctions
PShpC q ÝÑ
ÐÝ PShpDq ÝÑ
ÐÝ ShpDq,
noting that the composition from right to left lands inside ShpC q by Lemma
B.5.19.
Proposition B.5.23. Suppose C has and u preserves fibred products and a
terminal object. Then us is exact.
124
Proof. It is clearly right exact, being a left adjoint. Moreover, the functors
ShpC q Ñ PShpC q Ñ PShpDq Ñ ShpDq preserve finite limits by Theorem B.3.6,
Corollary B.5.13 and Proposition B.3.17, respectively.
Definition B.5.24. A morphism of sites f : D Ñ C is a continuous functor
u : C Ñ D such that us is exact.
Remark B.5.25. Note that f and u go in opposite directions. This is to
emphasise the geometrical nature, as illustrated by the following examples.
Example B.5.26. Let f : X Ñ Y be a morphism of schemes. Then f defines
a continuous functor
u : Yét Ñ Xét
as above. This gives a morphism of sites Xét Ñ Yét , which we will denote by
fét . We will drop the subscript and confusingly write f if the site is understood.
For all the sites we are interested in, we indeed get a morphism of sites, according
to Proposition B.5.23 and Remark B.5.18.
Definition B.5.27. Let f : D Ñ C be a morphism of sites. Then we denote by
f˚ : ShpDq Ñ ShpC q the functor us . It is called the direct image functor.
Definition B.5.28. Let f : D Ñ C be a morphism of sites. Then we denote by
f ´1 : ShpC q Ñ ShpDq the functor us . It is called the inverse image functor.
Theorem B.5.29. Let f : D Ñ C be a morphism of sites. There is a natural
isomorphism
HomShpC q pG , f˚ F q – HomShpDq pf ´1 G , F q.
Moreover, f ´1 is exact.
Proof. This is a reformulation of the above.
Remark B.5.30. For the Zariski site, this is just the well-known adjunction
from basic sheaf theory (cf. Hartshorne [10], Exercise II.1.18).
B.6
Cohomology
Definition B.6.1. Let A be an abelian category. Then an object I P A is
injective if the functor A p´, Iq is exact.
Definition B.6.2. Let A be an abelian category. Then A has enough injectives
if for each object A P ob A there exists an injective object I P ob A together
with a monomorphism A Ñ I.
Definition B.6.3. Let A be an abelian category. Then an injective resolution
of an object A P ob A is an exact sequence
0 Ñ A Ñ I0 Ñ I1 Ñ . . . ,
where each I i is injective.
125
Remark B.6.4. Suppose A has enough injectives, and let A P ob A . Write
A0 “ A, and let I 0 “ I. Then inductively also Ai “ I i´1 {Ai´1 injects into some
injective object I i , and we get an injective resolution
0 Ñ A Ñ I0 Ñ I1 Ñ . . . .
Hence, A has enough injectives if and only if every object has an injective
resolution.
We recall the following procedure:
Definition B.6.5. Let A and B be abelian categories, and let F : A Ñ B be
a left exact functor. Assume A has enough injectives. Then the right derived
functors Ri F : A Ñ B of F are defined as follows:
Let A P ob A . Choose an injective resolution
0 Ñ A Ñ I0 Ñ I1 Ñ I2 Ñ . . .
of A. Then we get a truncated chain complex
0 Ñ I0 Ñ I1 Ñ I2 Ñ . . . .
Applying F to the complex, we obtain a chain complex
0 Ñ F I0 Ñ F I1 Ñ F I2 Ñ . . .
in B. Then we denote by pRi F qA the i-th cohomology of this chain complex.
Remark B.6.6. Since F is left exact, the sequence
0 Ñ F A Ñ F I0 Ñ F I1
is exact. Hence, F A is the kernel of F I 0 Ñ F I 1 , which is the same thing as
pR0 F qA.
The standard results then show that this definition depends only on the chosen
injective resolution of A up to isomorphism.
Definition B.6.7. Let A be an abelian category. Then A satisfies:
• (AB3) if A is cocomplete;
• (AB4) if A is cocomplete and direct sums are exact;
• (AB5) if A is cocomplete and direct limits are exact.
Dually, A satisfies (AB3*), (AB4*) or (AB5*) if the abelian category A op
satisfies (AB3), (AB4) or (AB5) respectively.
Remark B.6.8. Note that A has (AB3) if and only if it has coproducts, i.e.
direct sums. This is since any abelian category has coequalisers, and arbitrary
colimits can be constructed from colimits and coequalisers.
Definition B.6.9. Let A be a category. Then an object U P ob A is a generator
if for every monomorphism A Ñ B in A , there exists a morphism U Ñ B that
does not factor through A.
126
Theorem B.6.10. Let A be an abelian category. Suppose A satisfies (AB5)
and (AB3*), and that there exists a generator U P ob A . Then A has enough
injectives.
Proof. See Grothendieck’s Tōhoku paper [9], Théorème 1.10.1. The same proof
is also included in the Stacks Project [11], Tag 079H.
Theorem B.6.11. Let C be a site, and suppose C is equivalent to a small
category. Then the category ShpC q has enough injectives.
Proof. We know that ShpC q is complete and cocomplete, by Corollary B.3.7 and
B.3.10. Hence, it satisfies (AB3) and (AB3*). Moreover, (AB5) follows since
colimits in ShpC q are the sheafification of the corresponding colimit in PShpC q,
following Corollary B.3.10, and since Ab satisfies (AB5).
It remains to exhibit a generator for ShpC q. This is done in [15], after Lemma
III.1.3.
Definition B.6.12. Let C be a site, such that C is equivalent to a small category. Let U P ob C , and let F be a sheaf on C . Then the cohomology of F on
U is
H i pU, F q “ Ri ΓpU, F q.
The cohomology presheaf H i pF q is defined as the right derived functor of the
inclusion ShpC q Ñ PShpC q. Note that
ΓpU, H i pF qq “ H i pU, F q,
since the functor evU : PShpC q Ñ Ab is exact and evU H 0 pF q “ H 0 pU, F q.
B.7
Examples of sheaves
So far, we haven’t seen a single sheaf for the étale or fppf topologies. There is
an easy way to check whether a presheaf is a sheaf:
Lemma B.7.1. Let F be a presheaf for the étale (of fppf ) topology. Then F
is a sheaf if and only if F |UZar is a sheaf on UZar for every U P obpÉt{Xq
(resp. obpSch{Xq), and for any covering tV Ñ U u with U and V both affine,
the sequence
d0
0 Ñ F pU q Ñ F pV q ÝÑ F pV ˆU V q
is exact.
Proof. It is clear that the two properties hold when F is a sheaf. Conversely,
let F satisfy the two properties above.
š
If tVi Ñ U u is a covering, and V “
Vi is the disjoint union, then the first
condition asserts that
ź
F pVi q.
F pV q “
127
š
Moreover, V ˆU V is the disjoint union Vi ˆU Vj . Hence, in the commutative
diagram
ś
ś
0
F pU q
F pVi q
F pVi ˆU Vj q
0
F pU q
F pV q
F pV ˆU V q,
all vertical arrows are isomorphisms.
Hence, if the index set I is finite, and all the Vi as well as U are affine, our
second assumption on F implies that the top row of this diagram is exact, since
the bottom row is.
Now let tVi Ñ U u be an arbitrary covering. By the above, to check the sheaf
f
condition onštVi Ñ U u, it suffices to check the sheaf condition on tV Ñ U u,
where V “ Vi is the disjoint union.
Ť
Now write U “ Ui as the union (not necessarily disjoint) of affine schemes Ui ,
and cover the inverse image f ´1 pUi q with affines Vik . Since f is flat, it is open,
so the image of Vik is open. Since Ui is affine, it is compact, hence there are
finitely many Vik such that their images cover Ui , so we can assume that there
are finitely many Vik for any i.
We have a commutative diagram
0
0
0
F pU q
ś
0
F pUi q
i
0
F pV q
F pV ˆU V q
śś
F pUi ˆU Uj q
i,j
śś
i
ś
F pVik q
śś
F pVik ˆU Vjl q
i,j k,l
k
F pVik ˆU Vil q.
i k,l
The top two rows are exact since F is a sheaf on the respective Zariski sites,
and the middle column is exact since tVik Ñ Ui u is a finite covering of the affine
Ui by the affines Vik . Then F pU q Ñ F pV q is injective, so F is separated.
Since F is separated, the right column is exact. If x P F pV q maps to zero in
F pV ˆU V q, then a simple diagram chase shows that x must come from some
element in F pU q. Hence, the left column is exact, so F is a sheaf.
Lemma B.7.2. Let f : A Ñ B be a faithfully flat ring homomorphism, and let
M be an A-module. Then the chain complex
1bd0
1bd1
0 ÝÑ M ÝÑ M bA B ÝÑ M bA B b2 ÝÑ . . .
řn
is exact, where the maps dn “ i“0 p´1qi dni are given by
dni : B bn ÝÑ B bn`1
b1 b . . . bn ÞÝÑ b0 b . . . b bi´1 b 1 b bi`1 b . . . b bn .
128
Proof. The standard argument shows that it is a chain complex (compare the
Čech complex of Definition B.2.2).
Assume firstly that f has a retraction g : B Ñ A (that is, gf “ 1A ).
Then we define
hn : B bn ÝÑ B bn´1
b1 b . . . b bn ÞÝÑ gpb1 qb2 b b3 b . . . b bn .
Then one easily sees that
hn`1 dni`1 ` dn´1
hn “ 0
i
for all i P t0, . . . , n ´ 1u. Hence, only the term hn`1 dn0 remains, so
hn`1 dn ` dn´1 hn “ hn`1 dn0 “ 1.
Hence, h is a contraction for pB bn q, so the same goes for 1 b h on pM bA B bn q.
Now in the general case, we tensor everything over A with B. Since B is
faithfully flat over A, the sequence of M is exact if and only if the same holds
for the sequence of M bA B over B (with respect to the B-algebra B bA B). But
the ring homomorphism B Ñ B bA B has a section, given by b1 bb2 ÞÑ b1 b2 .
Proposition B.7.3. Let f : Spec B Ñ Spec A be a faithfully flat morphism of
finite type of affine schemes, and let Z be any scheme. Then the diagram
HompSpec A, Zq Ñ HompSpec B, Zq ÝÑ
ÝÑ HompSpec B bA B, Zq
is an equaliser diagram (in Set).
Proof. The lemma asserts that the diagram
AÑB
Ñ
Ñ
B bA B
is an equaliser in ModA , hence also in Set (since ModA Ñ Set preserves limits).
Then it is an equaliser in Ring as well: if C Ñ B is a ring homomorphism such
that the compositions C Ñ B Ñ
Ñ B bA B agree, then it factors set-theoretically
through A. Since A is a subring of B, the obtained map C Ñ A has to be a
ring homomorphism, since C Ñ B is.
Hence, if Z “ Spec C is affine, the result is true. Now for general Z, we will
firstly prove that the map
HompSpec A, Zq Ñ HompSpec B, Zq
is injective, i.e. that Spec B Ñ Spec A is an epimorphism. Let g1 , g2 : Spec A Ñ
Z be such that g1 f “ g2 f . Since f is surjective, the topological maps g1 and g2
have to coincide.
If x P Spec A is a point, and z “ g1 pxq “ g2 pxq, let U be an affine open
neighbourhood of z. Let V Ď g1´1 pU q “ g2´1 pU q be an affine open containing x;
without loss of generality of the form V “ Spec Aa for some a P A.
129
Note that Spec Ba is faithfully flat over Spec Aa . Since
ˇ
ˇ
pg1 f qˇSpec B “ pg2 f qˇSpec B ,
a
a
the above shows that g1 |Spec Aa “ g2 |Spec Aa , since U is affine. Since x was
arbitrary, this shows g1 “ g2 , so the map
HompSpec A, Zq Ñ HompSpec B, Zq
is injective.
Now let h : Spec B Ñ Z be such that hπ1 “ hπ2 , where πi : Spec B bA B Ñ
Spec B is the i-th projection. Let x P Spec A be given, and let y P Spec B be in
its fibre. Let z “ hpyq, and let U be an affine open neighbourhood of z. Since
f is flat, it is open, so f ph´1 pU qq is an open neighbourhood of x.
Let a P A such that Spec Aa Ď Spec A contains x and is contained in f ph´1 pU qq.
Now if y1 , y2 P Spec B are two points with f py1 q “ f py2 q, then the fibred product
ty1 u ˆSpec A ty2 u is nonempty, so there exists a point y P Spec B bA B with
πi pyq “ yi for i P t1, 2u. Hence,
hpy1 q “ hπ1 pyq “ hπ2 pyq “ hpy2 q,
so y1 P h´1 pU q if and only if y2 P h´1 pU q. In particular,
f ´1 pSpec Aa q Ď f ´1 pf ph´1 pU qqq “ h´1 pU q.
But f ´1 pSpec Aa q is Spec Ba . Then by the affine case treated above, there exists
ga : Spec Aa Ñ U such that
ˇ
ˇ
ga ˝ f ˇSpec Ba “ hˇSpec Ba .
By the uniqueness statement above, the restrictions of ga and ga1 have to coincide on Spec Aaa1 “ Spec Aa XSpec Aa1 , so they glue to a morphism g : Spec A Ñ
Z satisfying gf “ h.
Theorem B.7.4. Let S be a scheme. Let X be an S-scheme, and let G be
a commutative group scheme over S. Then the presheaf F of abelian groups
defined by U ÞÑ HomS pU, Gq is a sheaf for the étale and fppf topologies on X.
Proof. For every U P obpÉt{Xq (resp. obpSch{Xq), the restriction of F to UZar
is a sheaf, by ‘glueing morphisms’. Moreover, if tV Ñ U u is a one-object
covering with both U “ Spec A and V “ Spec B affine, then the proposition
above shows that the diagram
F pU q Ñ F pV q ÝÑ
ÝÑ F pV ˆU V q
is an equaliser in Set. That is, the sequence
d0
0 Ñ F pU q Ñ F pV q ÝÑ F pV ˆU V q
is exact, so the result follows from Lemma B.7.1.
130
Definition B.7.5. Let X be a scheme. Let F be a sheaf of OX -modules on
XZar . Then define the presheaf W pF q on Xét (or Xfppf ) by
W pF qpU q “ ΓpU, f ˚ F q,
for any f : U Ñ X étale (or any morphism f : U Ñ X, respectively).
Theorem B.7.6. Let F be a quasi-coherent sheaf of OX -modules on XZar .
Then W pF q is a sheaf on Xét (or Xfppf ).
Proof. Clearly its restriction to UZar is a sheaf for every U P obpÉt{Xq. If
tf : V Ñ U u is a one-object covering with U “ Spec A and V “ Spec B both
affine, then W pF q|UZar corresponds to an A-module M . If g : U Ñ X denotes
the structure map, then
´
¯
ˇ
ˇ
W pF qˇ
“ pgf q˚ F “ f ˚ W pF qˇ
.
VZar
UZar
By Hartshorne [10], Proposition II.5.2(e), the latter is just pM bA Bq˜. The
sequence
0 Ñ M Ñ M bA B Ñ M bA B bA B
is exact by Lemma B.7.2, hence W pF q is a sheaf by Lemma B.7.1.
B.8
The étale site of a field
In this section, we will have X “ Spec K, where K is a field. If F is a sheaf
on Xét and L{K is a finite separable extension, then we will simply write F pLq
for F pSpec Lq.
We will fix a separable closure K̄ of K, with absolute Galois group ΓK . We
denote by x the unique point in X, and we will write x̄ for Spec K̄.
Definition B.8.1. Let F be a sheaf on Xét . Then the restriction of the functor
F : pÉt{Xqop Ñ Ab
to the subcategory consisting of Spec L for L Ď K̄ finite over K gives a functor
F : tL Ď K̄ | L{K finiteu Ñ Ab.
We denote its colimit by AF .
Remark B.8.2. As the category over which the colimit is taken is a directed
set, the above is just a direct limit. In particular, AF is the union of the images
of F pLq in it. Since F is a sheaf, the maps F pLq Ñ F pM q associated to an
extension L Ď M are injective, hence the direct limit is a union
ď
AF “
F pLq.
(B.7)
LĎK̄
rL:Ksă8
131
Now each F pLq comes with a ΓK -action, and the actions are compatible as L
varies. This defines a ΓK -module structure on AF . It is a discrete ΓK -module
H
by (B.7), as AH
(where H Ď ΓK denotes an open
F “ F pLq whenever L “ K̄
subgroup). Finally, note that the association
F ÞÝÑ AF
is functorial in F , since F pLq Ñ G pLq commutes with the ΓK -actions for every
finite L{K contained in K̄.
Definition B.8.3. Let A be a discrete ΓK -module. Then define the presheaf
FA : Ét{X Ñ Ab
by setting FA pSpec Lq “ AH when L “ K̄ H for H Ď ΓK open, and
˜
¸
ź
ž
FA pSpec Li q.
FA
Spec Li “
iPI
iPI
Lemma B.8.4. Let A be a discrete ΓK -module. Then FA is a sheaf.
š
Proof. Whenever U Ñ Spec K is étale, we have U “ Spec Li for certain Li {K
finite and contained in K̄. Hence, by the second part of the definition, FA |UZar
is a sheaf.
Now let K Ď L Ď M be a tower of finite extensions contained in K̄. Let M 1 be
a finite extension of M such that M 1 {L is Galois with group G “ tσ1 , . . . , σn u.
We have a commutative diagram
0
FA pLq
FA pM q
FA pM bL M q
0
FA pLq
FA pM q
FA pM bL M q.
1
1
(B.8)
1
Since M 1 {L is Galois, there exists α P M 1 such that σ1 pαq, . . . , σn pαq form an
L-basis for M 1 . This induces an isomorphism of L-algebras
„
M 1 bL M 1 ÝÑ M 1n ,
`
˘n
m1 b m2 ÞÝÑ m1 σi pm2 q i“1 .
„
1
1
1n
In particular, the compositions M 1 ÝÑ
are given by m ÞÑ
ÝÑ M bL M ÝÑ M
pm, . . . , mq and m ÞÑ pσ1 pmq, . . . , σn pmqq. The bottom row of diagram (B.8) is
given by
0 Ñ BG Ñ B Ñ Bn,
where B “ AΓM 1 . The last map is given by
b ÞÑ pb ´ σ1 pbq, . . . , b ´ σn pbqq,
hence its kernel is exactly B G , so the bottom row of (B.8) is exact. Since all
maps in the left hand square are injective, one easily sees that this forces the top
row to be exact as well. Hence, the sheaf condition is satisfied for the covering
tSpec M Ñ Spec Lu.
132
To proceed to the general affine case, observe that if U is étale over K and
affine, then (by compactness of affine schemes) U is a finite union of Spec Li . If
tV Ñ U u is a one-element covering with U and V affine, the first argument of
the proof of Lemma B.7.1 deduces the sheaf condition of tV Ñ U u from that of
Spec M Ñ Spec L for various L, M . Hence, Lemma B.7.1 gives the result.
Remark B.8.5. Just like the construction F ÞÑ AF , also the construction
A ÞÑ FA is functorial.
Theorem B.8.6. The functors ShpXét q Ø ΓK ´ Mod given by F ÞÑ AF and
A ÞÑ FA give an equivalence of the two categories.
Proof. We already remarked that both are indeed functors. It is clear from the
definition that AFA – A for any discrete ΓK -module A, and conversely that
FAF – F for any sheaf F on Xét .
Corollary B.8.7. For any sheaf F on Xét and any i P Zě0 , there is an isomorphism
H i pXét , F q – H i pK, AF q,
where the right hand side is the Galois cohomology of K.
Proof. Under the correspondence F ÞÑ AF , taking global sections corresponds
to taking ΓK -invariants. The result follows since Galois cohomology is defined
as the right derived functors of
A ÞÑ AΓK .
133
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135