THE BRAUER-MANIN OBSTRUCTION
FRANK GOUNELAS
Contents
Introduction
1. Brauer Groups in Algebra
1.1. Central Simple Algebras and the Brauer Group as Group Cohomology
1.2. A Primer on Class Field Theory
1.3. Azumaya Algebras Over a Local Ring
2. The Grothendieck-Brauer group
2.1. Some Comments on Étale Morphisms
2.2. A Primer on Étale and Flat Cohomology
2.3. Grothendieck-Azumaya Algebras
2.4. The Cohomological Brauer Group
3. The Manin Obstruction
3.1. Adelic Points and the Manin Pairing
3.2. Rational Varieties and the Case of Surfaces
References
1
1
1
4
6
6
6
7
10
11
12
12
14
16
Introduction
The idea behind this note is to attempt to set up enough basic background theory and
references so as to state the Brauer-Manin obstruction and give a brief idea on how one
computes it. I unfortunately ended up cutting corners on the last section since things get
quite technical and are treated individually for particular classes of varieties in the literature
thus including them would deem this note as unsuitable for an introduction to the material.
Notation 1. Throughout, an algebraic variety (or just variety) will mean an integral
separated scheme of finite type over a field k which will be specified accordingly in each
section.
1. Brauer Groups in Algebra
1.1. Central Simple Algebras and the Brauer Group as Group Cohomology. The
bibliography for this section is ample. For example one can resort to [GS06], [Mil08b] or
[Ser79]. It is worth mentioning that much of the material here is also treated more classically
without much mention of group cohomology in [Pie82].
Date: June 11, 2009.
1
2
FRANK GOUNELAS
Let k be any field. For the rest of this section note that all k-algebras will be finite
dimensional left algebras over k. We say that a k-algebra is simple if it contains no nontrivial proper two-sided ideals. Let SA be the category of simple algebras over k. One can
classify objects in SA using the following
Theorem 1.1. (Wedderburn’s Theorem) For every simple k-algebra A, there exists an
integer n and a division k-algebra D which is unique up to isomorphism such that
A∼
= Mn (D).
Moreover, we have that for every k-algebra A, A ⊗k Mn (k) ∼
= Mn (A).
Proof. See [Mil08b] IV.1.15 and IV.2.1.
ὄ.ἕ.δ.
We say that a k-algebra is central if its centre is exactly k. Denote by CSA the subcategory
of SA of central simple k-algebras. Denote by Aopp the opposite of a k-algebra A, that is
the k-algebra which has the same underlying set and addition, but multiplication is defined
as a · b = ba for a, b ∈ Aopp . We have that for all A ∈ CSA, A ⊗k Aopp ∼
= Mn (k).
Theorem 1.2. Let k be a field and A a (finite dimensional) k-algebra. A is a central
simple algebra if and only if there exists an integer n > 0 and a finite Galois field extension
K/k so that A ⊗k K ∼
= Mn (K).
Proof. [GS06] 2.2.6.
ὄ.ἕ.δ.
We call the above field K the splitting field of A. Note that a corollary to the above is that
the dimension of a central simple k-algebra is a square, namely n2 using the notation of the
theorem. One calls n the degree of the algebra. Given the statement A ⊗k K ∼
= Mn (K) we
say that A is a twisted form of Mn (k) exactly because when we tensor A with a suitable
extension K of k we get Mn (k) ⊗k K ∼
= Mn (K).
Knowing that PGLn (k) is the automorphism group of Mn (k) and using Galois descent one
proves
Theorem 1.3. There is a base point preserving bijection between central simple algebras
of degree n which are split by K (i.e. elements of CSAK (n)) and elements of H 1 (Gal(K/k),
PGLn (K)).
Proof. [GS06] 2.4.3.
ὄ.ἕ.δ.
Remark 1.4. The above statement falls in a line of arguments which attempt to classify
twisted forms of various objects. It turns out that isomorphism classes of twisted forms of a
vector space with a tensor, scheme or sheaf of modules are in bijection with H 1 cohomology
classes under an action of some general linear or projective linear group or sheaf of groups.
More mention of these later in the case of Severi-Brauer varieties but also for a general sheaf
of modules in the setting of the étale topology.
If two central simple algebras are split by K then so is their tensor product, and the resulting
degree will be the product of the corresponding degrees.
THE BRAUER-MANIN OBSTRUCTION
3
The aim now is to first link all central simple k-algebras split by K (that is of all degrees
n) with a suitable cohomology group and subsequently all of them (that is of all splitting
fields). We start by defining a binary relation for objects A, B ∈ CSA. We say that A ∼ B
if A ⊗k Mn (k) ∼
= B ⊗k Mm (k) for some positive integers m, n, which can easily be proven
to be an equivalence relation.
Definition 1.5. Let Br(K/k) be the set of equivalence classes of central simple k-algebras
split by K modulo the equivalence relation ∼. One defines an abelian group law on Br(K/k)
as [A][B] = [A ⊗k B] and in which the inverse element of [A] is [Aopp ]. One defines the
Brauer group Br(k) of k as the union of the Br(K/k) over all K.
Remark 1.6. Serre in his book Local Fields [Ser79] describes the Brauer group as “one
of the principal invariants available for measuring the “degree of complexity” of a field k”.
Note that from Wedderburn’s theorem one could also have defined the Brauer group to be the
set of isomorphism classes of central division k-algebras. The group law for D1 , D2 ∈ Br(k)
would then be defined by D1 ⊗k D2 ∼
= Mn (D3 ) for some central division k-algebra D3 and
integer n. Next, note that we have injective maps PGLn (K) → PGLnm (K) for all m, n > 0
which in turn induce injections H 1 (G, PGLn (K)) → H 1 (G, PGLnm (K)) for G = Gal(K/k).
Taking a direct limit over all n one defines H 1 (G, PGL∞ ) (cohomology commutes with direct
limits). Note also that for L/K/k Galois extensions of fields, we have an induced injection
H 1 (Gal(K/k), PGL n (K)) → H 1 (Gal(L/k), PGLn (K)) and so corresponding injections over
PGL∞ . For ksep a separable closure of k we form the direct limit over all Galois extensions
K/k and call this H 1 (k, PGL∞ ) = H 1 (Gal(ksep /k), PGL∞ ).
Lemma 1.7. We have bijections
Br(K/k) ∼
= H 1 (Gal(K/k), PGL∞ ) and Br(k) ∼
= H 1 (k, PGL∞ )
Proof. [GS06] 2.4.10.
ὄ.ἕ.δ.
This cohomological form of the Brauer group is not easy to work with. Thankfully another
interpretation exists. By considering the short exact sequence
(1.1)
1 → (ksep )× → GLn (ksep ) → PGLn (ksep ) → 1
and noting that a generalisation of Hilbert’s theorem 90 gives H 1 (k, GLn (ksep )) = 0 we have
an induced injective map in cohomology groups
H 1 (k, PGLn (ksep )) → H 2 (k, (ksep )× )
which in turn gives rise to the following
Theorem 1.8. For every finite Galois extension K/k and ksep a separable closure of k
there are isomorphisms of abelian groups
∼
∼
Br(K/k) −
→ H 1 (Gal(K/k), PGL∞ ) −
→ H 2 (Gal(K/k), K × )
∼
∼
Br(k) −
→ H 1 (k, PGL∞ ) −
→ H 2 (k, (ksep )× ).
4
FRANK GOUNELAS
Proof. [GS06] 4.4.7.
ὄ.ἕ.δ.
The group Br(k) expressed as a Galois cohomology group (cohomology of a profinite group)
has various properties, most importantly it is torsion. One can also easily deduce that Br
is functorial.
There is much material on the Brauer group of various fields, elaborated in [Ser02]. If k
is a finite field or a field of transcendence degree 1 over an algebraically closed field, then
Br(k) = 0. Both finite fields and these function fields are C1 fields and all of these have
trivial Brauer group. The former statement is essentially Wedderburn’s theorem on finite
division algebras, whereas the second is known as Tsen’s theorem. For the case of local
fields, which is also of interest to us, one can fully classify the Brauer group of all local
fields using the invariant map invK/k : Br(K/k) → Q/Z of class field theory, some details
of which will be given in a later section. Here is a short list of some known cases.
Br(R)
Br(C)
Br(Fq )
Br(K)
=
=
=
=
Z/2Z
0
0
Q/Z
for any non-archimedean local field K. For further examples see [Poo08] page 18.
1.2. A Primer on Class Field Theory. In this section I will certainly not attempt to
embarrass anyone by making an inroad into the technical inner workings of class field theory,
I will more try to build towards stating the fundamental short exact sequence of global class
field theory. The invariant map will be covered along the way. References are [Mil08b] and
[Ser79].
Recall that the Hilbert symbol is a non-degenerate bimultiplicative symmetric pairing
(·, ·)p : Q× /Q×2 × Q× /Q×2 → {±1}
such that (a, b)p = 1 if V : z 2 = ax2 + by 2 has non-trivial solutions in Qp and −1 otherwise.
Theorem 1.9. (Hilbert product formula) For all a, b ∈ Q and p an integer prime
Y
(a, b)p = 1.
p
In the above, note that p runs through all primes of Q, including the archimedean ones.
The conic V defined above will have points locally (over the various Qp ) for almost all p.
More so, conics are among the class of varieties that satisfy the Hasse principle which
states the following for an algebraic variety X over a field k
(1.2)
If X has points over every completion of k then it does over k
Remark 1.10. I introduce here the ring of adeles of the number field k. First, let Ẑ =
lim Z/nZ be the inverse limit of the Z/nZ with the obvious morphisms. For a number field
←−
Q
Q
k define Ak = Ẑ ⊗ k. One can prove that this is isomorphic to ′v kv where ′v denotes the
restricted product (all but finitely many are in fact elements of Okv the ring of integers of
THE BRAUER-MANIN OBSTRUCTION
5
kv ) and v runs through all the valuations of k. If our variety X is proper (for example if it
is projective), then the valuative
criterion of properness ([Har77] II.4, see (3.3) for schemes)
Q
gives us that X(Ak ) = v X(kv ). The Hasse principle is said to hold for X if
X(Ak ) 6= ∅ ⇒ X(k) 6= ∅
For the remainder of this section let L/k be a Galois field extension with group G =
Gal(L/k) and let H 2 (L/k) denote the group cohomology H 2 (Gal(L/k), L× ). Taking cohomology of the short exact sequence
0 → Z → Q → Q/Z → 0
one derives
· · · → H 1 (G, Q) → H 1 (G, Q/Z) → H 2 (G, Z) → H 2 (G, Q) → · · ·
However Q is uniquely divisible and so H r (G, Q) = 0 for r > 0 (see III.1 in [Mil08b]). One
deduces that H 1 (G, Q/Z) ∼
= H 2 (G, Z). We thus have the following homomorphisms
∼
∼
∼
H 2 (L/k) −
→ H 2 (G, Z) −
→ H 1 (G, Q/Z) −
→ Homcts (G, Q/Z) → Q/Z.
The composition of the above maps gives us the invariant map invL/k : H 2 (L/k) → Q/Z.
If k is a local field, then this map is an isomorphism. We extend this map to the Brauer
group Br(k) of a general field k.
Let k be a number field. Global class field theory proves (see [Mil08b] VIII.4.2) the exactness
of the sequence
P
M
invv
(1.3)
Br(kv ) −−−−→ Q/Z → 0.
0 → Br k →
v
It is worth mentioning here the connection between Severi-Brauer varieties, the Hilbert
symbol and central simple algebras. For a, b ∈ kv for some field k and place v, one can form
a quaternion kv -algebra
ha, biv = kv [x, y]/(x2 − a, y 2 − b, xy + yx)
or simply ha, bi for the same algebra defined using k. It can be proven, [Mil08b] exercise
IV.5.1, that each such algebra is central simple and self-opposite, and thus represents a class
of order 2 in Br(kv ). By choosing a, b ∈ k, we have ha, bi ⊗k kv = ha, biv . For k = Q, we
can in fact prove that for the Hilbert symbol (·, ·)v where v is any valuation (archimedean
or not)
ha, biv ∼
= M2 (Qv ) if and only if (a, b)v = 1.
This allows us to pinpoint which elements of Br(Qv ) come from algebras of the form ha, biv .
One can go on to prove the Hilbert product formula using the fundamental sequence of
global class field theory (1.3).
The above construction of a quaternion algebra corresponding to a conic generalises to
elements of Br(L/k) of higher order than two. The equivalent of a conic in higher dimensions
is defined as follows ([GS06])
6
FRANK GOUNELAS
Definition 1.11. An n-dimensional projective variety X over a field k is called Severin−1
.
Brauer if for some finite field extension L/k we have that X ×k L is isomorphic to PL
1.3. Azumaya Algebras Over a Local Ring. The first generalisation of the above ideas
was provided by Azumaya, who defined the notion of a Brauer group for local rings. Auslander and Goldman generalised this to arbitrary rings whereas Grothendieck defined the
notion for schemes as we will discuss later on. I will briefly discuss the case of commutative local rings, since this is the case of interest if one considers generalisations to schemes,
following [Mil80].
Definition 1.12. Let R be a local ring and let A be a not necessarily commutative Ralgebra. We say that A is Azumaya if it is free of finite rank as an R-module and there is
an isomorphism A ⊗R Aopp → EndR−mod (A) sending a ⊗ a′ 7→ (x 7→ axa′ ).
One can prove that if R is a field then A is an Azumaya algebra if and only if it is a central
simple R-algebra. More so, we have similar statements to the case of fields about the tensor
product of two Azumaya algebras being Azumaya and also that the matrix ring Mn (R) is
Azymaya. One sets up an equivalence relation on these algebras as A ∼ A′ if and only if
A ⊗R Mn (R) ∼
= A′ ⊗R Mm (R) just as in the case of fields. We call Br(R) the resulting
quotient of all Azumaya algebras over R modulo this equivalence relation and it is again
functorial. An R-algebra S is said to split an Azumaya R-algebra A if A ⊗R S ∼
= Mn (S) for
some n.
2. The Grothendieck-Brauer group
My main sources in this section are [Mil80], [Poo08], Grothendieck’s original papers [Gro68a],
[Gro68b], [Gro68c] and Vistoli’s notes [Vis08].
2.1. Some Comments on Étale Morphisms. Let X, Y be two schemes. A morphism
f : X → Y is flat if we have that for all x ∈ X, tensoring with OX,x as an OY,f (x) -module
preserves exactness of sequences. What flatness aims to make precise is a sense of continuity
in the fibers Xy = f −1 (y) of the morphism f . Some applications of flatness are more explicit
in the case where X and Y are locally Noetherian schemes. Here we have that if f is flat
then
dim OXy ,x = dim OX,x − dim OY,y
More so, if X and Y were varieties (we have assumed these to be integral and thus have
a unique generic point) and f was surjective (see [Liu02] 4.3.1), then the above would say
that the dimension of all the fibers of f are exactly dim X − dim Y .
A morphism locally of finite type f : X → Y is unramified if for every point x ∈ X,
OX,x /my OX,x is a finite separable field extension of k(y) = OY,y /my and my = f (mx ),
where y = f (x) and my is the unique maximal ideal in OY,y .
Example 2.1. If L/K is a finite field extension then Spec L → Spec K is unramified if
and only if the extension L/K is separable. Pre-empting the definition of étale morphisms
one should note that an above morphism is always flat and so the same statement holds if
we replace unramified with étale.
THE BRAUER-MANIN OBSTRUCTION
7
If X and Y are locally Noetherian and f is of finite type, then f is unramified if and only if for
all y ∈ Y , Xy is finite and k(x)/k(y) is a separable field extension ([Liu02] 4.3.2). Assuming
f is locally of finite type, f being unramified is equivalent to the sheaf of relative differentials
Ω1X/Y being zero which is equivalent to the diagonal morphism ∆X/Y : X → X ×Y X being
an open immersion ([Mil80] I.3.5).
Definition 2.2. A morphism of schemes is étale if it is flat and unramified.
An étale morphism is the algebraic geometry equivalent of “local isomorphism” one encounters for manifolds. I will make this precise
Theorem 2.3. Let φ : W → V be a regular morphism of varieties over an algebraically
closed field. Then φ is étale if for every point w ∈ W the induced map Cv (V ) → Cw (W ), on
the tangent cones, is an isomorphism. This is equivalent to saying that the map ÔV,φ(w) →
ÔW,w , on the ring-theoretic completions of the stalks, is an isomorphism.
Proof. [Mil08a] Chapter 1.
ὄ.ἕ.δ.
Remark 2.4. In the above, if W, V were also smooth varieties then note that the tangent
cone is simply the tangent space at the point w.
Remark 2.5. Étale morphisms satisfy all the conditions one would hope for such as closed
under composition, closed under base change and so on. Another noteworthy consequence
is that if f : X → Y is an étale morphism of locally Noetherian schemes, then we have
dim OX,x = dim OY,y for y = f (x). For example, if X and Y are integral this implies
dim X = dim Y .
2.2. A Primer on Étale and Flat Cohomology. The idea behind a Grothendieck topology is to loosen the notions of “open subset” and “open covering” which are abundantly used
in Grothendieck’s Éléments de Géométrie Algébriques. One does this by formally considering open subsets U of a scheme X as inclusion morphisms U → X. It now does not seem
far-fetched to consider morphisms other than inclusions and see what changes occur for the
notions of covering, sheaf, cohomology and so on.
A site, is a category C with a Grothendieck topology on it. This includes the notion of
a covering. A Grothendieck topology is a system of coverings with the following data, for
each object U in C a distinguished set of families {Ui → U }i∈I called a covering of U which
satisfy the following three axioms
i. If U ′ → U is an isomorphism then it is a covering of U ;
ii. If {Ui → U }i∈I is a covering of U , and for each i, {Uij → Ui }j∈J is a covering of Ui ,
then {Uij → U }i,j is a covering of U ;
iii. For any covering {Ui → U }i∈I and morphism V → U , then the fibre products Ui ×U V
exist and {Ui ×U V → V }i∈I is a covering of V .
As far as algebraic geometry is concerned we want to vary two things in the above scenario.
First of all, what the morphisms in our coverings will be and secondly what the category C
is. Varying the former leads to various famous Grothendieck topologies such as the étale or
8
FRANK GOUNELAS
flat whereas within these topologies, varying the latter leads to distinctions between a big
or small site.
The basic example is the Zariski site Xzar for a scheme X. Let X be considered as a
category whose objects are the open subsets U ⊂ X and morphisms are only the inclusions.
A covering in the Grothendieck topology sense is simply an open covering in the usual
topological sense. Note that since the only maps are the inclusions, we have that for U, U ′
open subsets of V in X, U ×V U ′ = U ∩ U ′ .
The small étale site Xét of a scheme X is the category Ét/X whose objects are schemes
U with étale morphisms U → X and whose arrows are X-morphisms from the category of
X-schemes. The coverings
S in Xét are surjective families of étale morphisms {φi : Ui → U }.
Surjective means that i φi (Ui ) = U . The big étale site XÉt is similar, only we consider the
underlying category Sch/X of all schemes over X as opposed to just schemes with an étale
morphism to X.
Similarly, one defines the big flat site XFl for a scheme X by considering the category Sch/X
of all X-schemes and then as coverings for the Grothendieck topology all surjective families
of X-morphisms (Ui → U )i which are flat and of finite type.
A continuous map of sites f : C1 → C2 is a functor between the underlying category of C2 to
C1 (note how they are reversed) which takes a covering to a covering. For a scheme X, the
sites we discussed above can be put in the following diagram of continuous maps of sites
XFl → XÉt → Xét → Xzar
This is more clear if one thinks that every Zariski open inclusion is certainly étale, every
étale covering on the small site is certainly one on the big site and finally that every étale
morphism is certainly flat.
Remark 2.6. So far we have taken the notion of an open inclusions and replaced it by étale
or flat morphisms, we have also substituted an equivalent notion to open covering which we
call a surjective family. The idea is to now develop a theory of sheaves and cohomology of
sheaves which is very similar to the classical settings, but encompassing a slightly higher
level of abstraction which replaces subsets with morphisms, intersections as fibre products
and so on. The actual statements of the results turn out to be quite similar.
I will now stick to the small étale site, as it provides just the right amount of fineness (that is
finer than the Zariski topology, but not so fine as to be losing cohomological information).
Without getting into too many details, one defines the category of presheaves of abelian
groups on Xét as the category whose objects are contravariant functors from Xét to the
category of abelian groups. A presheaf F in this category is called a sheaf if for all coverings
{Ui → U } in Xét , we have that
h // Q
f Q
//
// i,j F(Ui ×X Uj )
F(U )
i F(Ui )
g
is exact. That is to say that f is injective and f (F(U )) agrees with the equaliser of g and
h. Grothendieck proved that the category of sheaves of abelian groups on Xét is an abelian
THE BRAUER-MANIN OBSTRUCTION
9
category with enough injectives (see [Mil08a] section 8). The notions of stalks, morphisms of
sheaves (including the functors f∗ , f ∗ , f! , f −1 ), sheafification and so on, generalise straightforwardly to our new settings. Some examples of sheaves for the étale topology are the
structure sheaf OXét whose sections for étale U → X are defined as Γ(U, OXét ) = Γ(U, OU )
where OU is the structure sheaf in the standard Zariski topology. Another sheaf which we
will be using a lot of is the following.
Definition 2.7. Let Gm be the sheaf of multiplicative abelian groups for a scheme Xét ,
such that for étale U → X, we have Gm (U ) = OU (U )× .
Remark 2.8. To generalise the above, one considers for U → X étale, the functor
GLn (U ) = GLn (Γ(U, OU )). We thus recover GL1 = Gm . The above sheaves are representable functors, we have that Gm is represented by the multiplicative group scheme
Spec Z[t, t−1 ] and GLn is similarly given in terms of the following representable functor
Z[T11 , . . . , Tnn , T ]
∼
.
GLn = Hom −, Spec
(T det(Tij ) − 1)
As with sheaf cohomology on a scheme ([Har77] III), we have a left exact functor Γ(X, −)
of global sections from the category of sheaves for Xét to the category of abelian groups.
By choosing an injective resolution F → I • of a sheaf F on Xét and applying the functor
Γ(X, −) to this complex, we derive a new complex
Γ(X, I 0 ) → Γ(X, I 1 ) → Γ(X, I 2 ) → · · ·
whose cohomology groups are defined as H r (Xét , F). This is expressed as a right derived
functor H r (Xét , −) from the category of sheaves of Xét to the category of abelian groups.
r (X, F) and H r (X , F) to mean the same thing.
Authors interchangeably use Hét
ét
There also exists a notion of Čech cohomology which under certain conditions (for example
if our scheme is a quasi-projective variety) agrees with normal étale sheaf cohomology (see
[Mil80] III.2).
Proposition 2.9. For a scheme X, we have
∼
∼
∼
∼
H 0 (Xét , Gm ) −
→ H 0 (XFl , Gm ) −
→ OX (X)×
H 1 (Xét , Gm ) −
→ H 1 (XFl , Gm ) −
→ Pic X.
Proof. [Mil80] III.4.9 or [Poo08] 6.6.1.
ὄ.ἕ.δ.
Remark 2.10. As discussed in [Mil08a] 11.4, the above proposition generalises to the
following
Ln (Xzar ) ↔ Ȟ 1 (Xzar , GLn ) ↔ Ȟ 1 (Xét , GLn ) ↔ Ȟ 1 (XFl , GLn )
where Ln (Xzar ) denotes isomorphism classes of rank n vector bundles on X (see [Har77]
exercise II.5.18), or equivalently, isomorphism classes of locally free sheaves of rank n on X.
10
FRANK GOUNELAS
I will digress slightly to discuss cohomological vanishing comparisons between the various
sites. Recall that Grothendieck’s vanishing theorem states that for any Noetherian topological space X and F a sheaf of abelian groups on X, H r (Xzar , F) = 0 for r > dim X.
If one were to expect to have some kind of Brauer group as H 2 of a dimension 1 scheme
(for example a curve), then something more refined than the Zariksi topology is required.
All is not lost when one extends to the étale or flat topologies though. One can establish
comparison isomorphisms between the Zariski, étale and flat cohomologies as is done in
[Mil80] III.3. It turns out that for quasi-coherent sheaves of OX -modules F on Xzar , the
topologies agree (for example Ga ). Note however that Gm is not an OX -module so as we
will see below, it is natural to expect Brauer groups of curves to be non-trivial.
A large class of sheaves is covered in the following. Recall the notion of cohomological
dimension: we say that cd(Xét ) ≤ n if H r (Xét , F) = 0 for r > n and F a torsion sheaf (i.e.
F(U ) is torsion for all U → X étale).
Theorem 2.11. Let X be a variety over an algebraically closed field k. Then we have
cd(Xét ) ≤ 2 dim X and if X is affine cd(Xét ) ≤ dim X, with equality if X is also complete.
Proof. See [Mil08a] 15.1.
ὄ.ἕ.δ.
2.3. Grothendieck-Azumaya Algebras. For a scheme X and an OX -algebra A, we recall
that Aopp is the sheaf of OX -algebras which sends an open subset U ⊆ X to the opposite
algebra (A(U ))opp , as defined in the previous section. The following definition summarises
various equivalent definitions of an Azumaya algebra on a scheme X.
Definition 2.12. Let A be a coherent OX -algebra over X, then A is Azumaya if any of
the following equivalent conditions are true
i. A is locally free and of finite rank as an OX -module and for any point x ∈ X, Ax is
an Azumaya algebra over OX,x .
ii. A is locally free as a OX -module and A(x) = Ax ⊗ k(x) is a central simple algebra
over the residue field k(x) for all x ∈ X.
iii. A is locally free as a OX -module and there is a canonical isomorphism of sheaves
∼
A ⊗OX Aopp −
→ HomOX (A, A) = EndOX (A)
iv. There is a covering (Ui → X) in the étale topology on X such that for each i there
exists an ri such that A ⊗OX OUi ∼
= Mri (OUi ).
Proof. [Mil80] IV.2.
ὄ.ἕ.δ.
Remark 2.13. The last statement in the above proposition resembles (1.2) in the sense
that it tells us that Grothendieck’s Azumaya algebras are twisted forms of Mn (OXét ). For
the equivalent to (1.3) see (2.14) below.
One can generalise the group law on central simple k-algebras to Azumaya algebras, by
defining an equivalence relation between two Azumaya algebras A1 , A2 for a scheme X as
THE BRAUER-MANIN OBSTRUCTION
11
A1 ∼ A2 if there exist locally free coherent OX -modules M1 , M2 and an isomorphism of
sheaves
∼
A1 ⊗OX EndOX (M1 ) −
→ A2 ⊗OX EndOX (M2 ).
Next, for the set of equivalence classes of Azumaya algebras of a scheme X, one trivially
showes that the tensor product over OX gives closure as a group law and that Aopp is the
inverse of A. This is called the Brauer group Br(X) of X and it is not to be confused
with the cohomological Brauer group which will be defined later. We can define a functor
Br : Schopp → Ab generalising the functorial property we saw for fields and local rings.
Proposition 2.14. The set of isomorphism classes of Azumaya algebras of rank n2 over
X is equal to Ȟ 1 (Xét , PGLn ).
Proof. [Mil80] IV.2.5.
ὄ.ἕ.δ.
2.4. The Cohomological Brauer Group. Arguably, this section is harder than the rest
of the theory so far. One can simply skip this section, retaining however definition (2.15)
and keeping in mind the statements of theorems (2.18) and (2.19).
Definition 2.15. For a scheme X, the cohomological Brauer group of X is defined as
2
Br′ (X) := Hét
(X, Gm ).
The short exact sequence of (1.1) generalises to sheaves for the étale or flat topologies as
follows
1 → Gm → GLn → PGLn → 1
Theorem 2.16. For any scheme X, we have an injection
Br X ֒→ Br′ X.
Proof. If X satisfies the conditions necessary for Čech étale cohomology to agree with
normal étale cohomology (i.e. X is quasi-compact and every finite subset of X is contained
in an open affine) then the proof follows without too much effort from (2.14), remark (2.10)
and the fact that we can use explicit cocycle calculations in Čech cohomology which will
then agree with the normal cohomology.
The general case is more difficult as one would have to develop the theory of non-abelian
cohomology. One associates to every Azumaya algebra a gerb which in turn gives us a
non-abelian cohomology 2-cocycle and finally one is left with checking that the non-abelian
étale dimension 2 cohomology of Gm (which is abelian!) is isomorphic with the standard
étale dimension 2 cohomology. Milne sketches the proof on page 144, 145 of [Mil80] but all
the particulars are to be found in Giraud’s book [Gir71].
ὄ.ἕ.δ.
2 (Spec k, G ) ∼ H 2 (k, G ) = Br(k).
Remark 2.17. Note, that for a field k, we have Hét
m =
m
12
FRANK GOUNELAS
Corollary 2.18. Let X be a regular Noetherian integral scheme and let k(X) be its
function field. We then have that the map Br(X) → Br(k(X)) is injective and so that
Br(X) is a torsion abelian group.
Proof. [Mil80] IV.2.6 or originally from [Gro68b] 1.10. For the first claim one uses the
Leray spectral sequence induced from the map g : Spec k(X) → X
H p (X, Rq g∗ Gm,k(X) ) ⇒ H p+q (Spec k(X), Gm )
and the later follows from H p (Spec k(X), Gm ) = Br(k(X)) being a Galois cohomology group
and thus torsion (See for example Serre’s book [Ser02]).
ὄ.ἕ.δ.
The search for cases when the morphism Br(X) → Br′ (X) = H 2 (Xét , Gm ) is also surjective
was initiated by Grothendieck but it was many years later (1980s?) when Ofer Gabber in
an unpublished theorem, which was reproved by de Jong in [dJ04], stated the following
Theorem 2.19. If X has an ample invertible sheaf (e.g. X a quasi-projective scheme
over Spec A for some Noetherian ring A) then we have an isomorphism
Br(X) → H 2 (Xét , Gm )
To summarise what we have so far, in the suitable setting of quasi-projective varieties over a
field k note that we have two descriptions of the Brauer group, namely via Azumaya algebras
but also as a cohomology group, each giving us information about our variety in its own
unique way. Both these interpretations will be put to use when discussing the Brauer-Manin
obstruction in the next section.
3. The Manin Obstruction
3.1. Adelic Points and the Manin Pairing. The material in this section was originally
established by Manin in [Man86] and various papers before that, but developed and applied
to many cases in a myriad of papers in the 1980’s and 1990’s by Colliot-Thélène, Sansuc,
Swinnerton-Dyer, Kavensky... I will however be following from [Poo08], [Sko01] and a nice
modern account is also given in [KT08].
For this section assume that X is a projective variety over the number field k and let A
be an Azumaya algebra representing a class in Br(X). For valuations v of k, the injection
k ֒→ kv lifts to an injection of X(k) ֒→ X(kv ). For xv ∈ X(kv ) we can apply the functor Br
defined in previous sections to the map Spec kv → X defined by xv and retrieve a map on
the corresponding Brauer groups Br X → Br kv . Define A(xv ) toQbe the image of A under
this map. From [Poo08] 8.3.1, we have that if {xv } ∈ X(Ak ) = v X(kv ), then A(xv ) = 0
for all but finitely many v. Considering the definitions in the previous section we get the
adelic Brauer-Manin pairing defined as in [Sko01] 5.2.
(·, ·) : Br X × X(Ak ) → Q/Z
X
(A, {xv }) 7→
invv A(xv ).
v
Note that the pairing, when restricted to elements of X(k) is 0. That is for x ∈ X(k)
we have (A, {xv }) = 0 for all A ∈ Br(X). This is not hard to see if one considers the
THE BRAUER-MANIN OBSTRUCTION
13
commutativity of the following diagram (see [Poo08] 8.3.2)
X(k)
// Br k
0
// X(Ak )
//
L
v Br(kv )
// Q/Z
// 0
We next define X(Ak )Br X as the right side kernel of the adelic Brauer-Manin pairing. That
is to say
\
X(Ak )Br X =
X(Ak )A
A∈Br X
=
\
{{xv } ∈ X(Ak ) : (A, {xv }) = 0}
A∈Br X
=
(
{xv } ∈ X(Ak ) :
X
)
invv A(xv ) = 0 for all A ∈ Br(X) .
v
Since (A, {xv }) = 0 for all x ∈ X(k), we have that X(k) ⊆ X(Ak )Br X . The above yield
∅ ⊆ X(k) ⊆ X(Ak )Br X ⊆ X(Ak )A ⊆ X(Ak )
and so one is lead to the following definition.
Definition 3.1. A variety X over a number field k is said to have a Brauer-Manin
obstruction if X(Ak )Br X = ∅ and X(Ak ) 6= ∅. One says that the Brauer-Manin obstruction
is the only one to the Hasse principle if X(Ak )Br X 6= ∅ implies that X(k) 6= ∅. Finally,
one says that the Brauer-Manin obstruction is empty (or that there is no Brauer-Manin
obstruction) if X(Al )Br X 6= ∅.
Remark 3.2. We have thus defined an obstruction to the Hasse principle to the existence
of rational points, given that we have points everywhere locally (1.2).
A secondary question in the same line of thought asks whether X(k) is dense as a subset of
X(Ak ). If so we say that X satisfies weak approximation. This gives rise to Brauer-Manin
obstructions to weak approximations which require that X(k) 6= 0 and X(Ak )Br(X) 6=
X(Ak ).
Remark 3.3. Everything in this section can readily be defined in the more general
setting
Q
of a Noetherian proper scheme. It is noteworthy that the fact that X(Ak ) = v X(kv ) in
the general case follows easily from the valuative criterion of properness found in [Har77]
II.4.7.
It is unfortunately the case that the Brauer-Manin obstruction is not the only one for all
smooth projective varieties. For example there is an example of a surface in [Sko01] chapter
8 which violates the Hasse principle but the violation is not explained by the Brauer-Manin
obstruction.
For smooth proper curves the situation is slightly better. Namely, there is a result of Manin
14
FRANK GOUNELAS
that states that for an elliptic curve, under the assumption that X is finite, the BrauerManin obstruction is the only one. For a general smooth proper curve the obstruction is
the only one if the curve has no rational divisor classes of degree 1 or if it has such a divisor
class but the Jacobian over the field of definition of the curve is finite, assuming again in
both cases that X of the Jacobian is finite (see [Sko01] pages 127 and 114 respectively).
Conjecture 1. (Colliot-Thélène [CT03]) For a rationally connected smooth projective
geometrically integral variety over a number field k, the Brauer-Manin obstruction is the
only one.
Remark 3.4. The rationally connected surfaces are precisely the rational surfaces ([Har01]).
Remark 3.5. Other obstructions do exist, but would form the content of a different set
of notes. For example in [Poo08] and [Sko01] one finds reference to the descent obstruction,
the elementary obstruction and various others.
One might also wonder what a good substitute to the notion of a rational point might be
for the Brauer-Manin obstruction to be the only one to. I mention in passing that it is
conjectured that this is the case for 0-cycles of degree 1 on a smooth projective variety over
a number field, extending the previous construction by linearity. See [CT99] for a discussion
and [ES08] for a proof in the case of curves assuming the finiteness of the Tate-Shafarevich
group of the Jacobian.
3.2. Rational Varieties and the Case of Surfaces.
Lemma 3.6. (Hochschild-Serre Spectral Sequence) Let X be a smooth connected projective
variety over a field k of characteristic 0 and let K/k be a Galois extension where G =
Gal(K/k) and write XK = X ×k K. For F an étale sheaf on X we have
q
p+q
E2pq = H p (G, Hét
(XK , F)) ⇒ Hét
(X, F)
Proof. This is a special case of the Hochschild-Serre spectral sequence for Galois covers
X ′ → X proven in [Mil80] III.2.20.
ὄ.ἕ.δ.
The above sequence gives rise to a long exact sequence in low degree terms (see [Wei94]
5.8.3) for G = Gal(K/k)
0 → Pic(X) → H 0 (G, Pic(XK )) → H 2 (G, K ∗ ) → ker(Br(X) → Br(XK ))
→ H 1 (G, Pic(XK )) → H 3 (G, K ∗ )
We denote Br0 (X) = im(Br(k) → Br(X)) and Br1 (X) = ker(Br(X) → Br(X).
Proposition 3.7. Let X be a smooth projective variety over a number field k, then
Q
i. If v X(kv ) 6= 0 the map Br(k) → Br(X) is injective and for any Galois K/k
(Pic(XK ))Gal(K/k) = Pic(X)
ii. there is an isomorphism
Br1 (X)/ Br0 (X) → H 1 (k, Pic(X))
THE BRAUER-MANIN OBSTRUCTION
15
Proof. Can be found in [Sko01]. The first part is mostly trivial and follows from the long
exact sequence of low degree terms of the Hochschild-Serre spectral sequence. The second
follows again from the same spectral sequence argument in combination with a theorem of
∗
ὄ.ἕ.δ.
Tate that states that for a number field k, H 3 (k, k ) is trivial.
I will now restrict to the case of rational varieties since this is where one expects the BrauerManin obstruction to be the only one.
Given a rational variety X over a number field k, one first checks whether X(Ak ) is empty or
not. This is a finite computation and can be performed using a computer with software such
as MAGMA. If it is empty, then our search for rational points x ∈ X(k) is finished, since
we have the inclusion X(k) ֒→ X(Ak ). If there are points everywhere locally, then from the
above proposition, the map Br(k) → Br(X) is an inclusion and so Br0 (X) = im(Br(k) →
Br(X)) ∼
= Br(k). Since X is a rational variety X is birational to projective space and from
theorem 48.2 of [Man86] we have that Br(X) is trivial and so Br1 (X) = Br(X). From the
above proposition we thus derive the following corollary
Corollary 3.8. Let X be a rational variety over a number field k which has points
everywhere locally, then
∼
Br(X)/ Br(k) −
→ H 1 (Gal(k/k), Pic(X))
This isomorphism is key in the computation of the Brauer-Manin obstruction. It is the case
that the group on the right is often computable and small in order, however the proof of
this is no trivial task and for many cases of rational varieties constitutes the material of
several research articles. What is however very difficult is finding suitable representatives
of Br(X) under this isomorphism. It is often the case that H 1 (k, Pic(X)) is a cyclic group
of small order and so one needs to find one single representative of Br(X) to compute the
obstruction. Computing the local invariants invv is also a laborious task however a range
of tricks are in place to make these computations easier. For example there are various
well known types of Azumaya algebras such as cyclic and quaternion algebras (see [GS06])
which one works with. One usually expresses these as elements of Br(k(X)) in terms of some
rational function f in the function field of X since in this case, computing local invariants
is a simpler task.
The case of surfaces has been covered extensively but all cases are not complete yet. Fano
varieties (with ample anti-canonical divisor) of dimension 2 are called del Pezzo surfaces
and these have been studied extensively in relation to the Brauer-Manin obstruction since
the mid 1980’s. One can look at the recent thesis of Patrick Corn [Cor05] which summarises
what is known so far for del Pezzo surfaces and examines each case extensively.
The conjectures in place should not make one think that one is restricted to rational varieties
only. Many explicit cases of varieties are still accounted for by the Brauer-Manin obstruction
but it is simply not the case that all of them can be handled in this manner. For example
diagonal quartic surfaces (a special type of a K3 surface) are discussed in Martin Bright’s
thesis [Bri02].
16
FRANK GOUNELAS
References
[Bri02]
[Cor05]
[CT99]
[CT03]
[dJ04]
[ES08]
[Gir71]
[Gro68a]
[Gro68b]
[Gro68c]
[GS06]
[Har77]
[Har01]
[KT08]
[Liu02]
[Man86]
[Mil80]
[Mil08a]
[Mil08b]
[Pie82]
[Poo08]
[Ser79]
[Ser02]
[Sko01]
[Vis08]
Martin Bright. Computations on diagonal quartic surfaces. PhD Dissertation at the University of
Cambridge, 2002.
Patrick Kenneth Corn. Del pezzo surfaces and the brauer-manin obstruction. PhD Dissertation at
the University of Berkeley, 2005.
Jean-Louis Colliot-Thélène. Conjectures de type local-global sur l’image des groupes de Chow dans
la cohomologie étale. In Algebraic K-theory (Seattle, WA, 1997), volume 67 of Proc. Sympos. Pure
Math., pages 1–12. Amer. Math. Soc., Providence, RI, 1999.
Jean-Louis Colliot-Thélène. Points rationnels sur les fibrations. In Higher dimensional varieties and
rational points (Budapest, 2001), volume 12 of Bolyai Soc. Math. Stud., pages 171–221. Springer,
Berlin, 2003.
A.J. de Jong. A result of gabber, preprint, http://www.math.columbia.edu/ dejong/papers/2gabber.pdf. 2004.
Dennis Eriksson and Victor Scharaschkin. On the Brauer-Manin obstruction for zero-cycles on
curves. Acta Arith., 135(2):99–110, 2008.
Jean Giraud. Cohomologie non abélienne. Springer-Verlag, Berlin, 1971. Die Grundlehren der
mathematischen Wissenschaften, Band 179.
Alexander Grothendieck. Le groupe de Brauer. I. Algèbres d’Azumaya et interprétations diverses.
In Dix Exposés sur la Cohomologie des Schémas, pages 46–66. North-Holland, Amsterdam, 1968.
Alexander Grothendieck. Le groupe de Brauer. II. Théorie cohomologique. In Dix Exposés sur la
Cohomologie des Schémas, pages 67–87. North-Holland, Amsterdam, 1968.
Alexander Grothendieck. Le groupe de Brauer. III. Exemples et compléments. In Dix Exposés sur
la Cohomologie des Schémas, pages 88–188. North-Holland, Amsterdam, 1968.
Philippe Gille and Tamás Szamuely. Central simple algebras and Galois cohomology, volume 101
of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2006.
Robin Hartshorne. Algebraic geometry. Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, No. 52.
Joe Harris. Lectures on rationally connected varieties, notes for a school course given in trento,
2001. Available at http://www.mat.uab.es/ kock/RLN/rcv.pdf.
Andrew Kresch and Yuri Tschinkel. Effectivity of Brauer-Manin obstructions. Adv. Math.,
218(1):1–27, 2008.
Qing Liu. Algebraic geometry and arithmetic curves, volume 6 of Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford, 2002. Translated from the French by Reinie Erné, Oxford
Science Publications.
Yu. I. Manin. Cubic forms, volume 4 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam, second edition, 1986. Algebra, geometry, arithmetic, Translated from the
Russian by M. Hazewinkel.
James S. Milne. Étale cohomology, volume 33 of Princeton Mathematical Series. Princeton University Press, Princeton, N.J., 1980.
James S. Milne. Lectures on etale cohomology (v2.10), 2008. Available at www.jmilne.org/math/.
J.S. Milne. Class field theory (v4.00), 2008. Available at www.jmilne.org/math/.
Richard S. Pierce. Associative algebras, volume 88 of Graduate Texts in Mathematics. SpringerVerlag, New York, 1982. Studies in the History of Modern Science, 9.
Bjorn Poonen. Rational points on varieties, notes from 18.787 massachusetts institute of technology, 2008. Available at http://math.mit.edu/ poonen/.
Jean-Pierre Serre. Local fields, volume 67 of Graduate Texts in Mathematics. Springer-Verlag, New
York, 1979. Translated from the French by Marvin Jay Greenberg.
Jean-Pierre Serre. Galois cohomology. Springer Monographs in Mathematics. Springer-Verlag,
Berlin, english edition, 2002. Translated from the French by Patrick Ion and revised by the author.
Alexei Skorobogatov. Torsors and rational points, volume 144 of Cambridge Tracts in Mathematics.
Cambridge University Press, Cambridge, 2001.
Angelo Vistoli. Grothendieck topologies, fibered categories and descent theory. 2008.
THE BRAUER-MANIN OBSTRUCTION
[Wei94]
17
Charles A. Weibel. An introduction to homological algebra, volume 38 of Cambridge Studies in
Advanced Mathematics. Cambridge University Press, Cambridge, 1994.