CHAPTER 7 Lecture 7

1. Finite/Singular Structures

1.1. Local Structures. Let K be a local field with residue field k of characteristic p, and let M be a finite G_{K}-module of prime to p order. In any case we have exact sequences

0 → H^{1}(𝔤_{k}, M^{I_{K}}) → H^{1}(G_{K}, M) → H^{1}(I_{K}, M)^{𝔤_{k}} → 0

and

0 → H^{1}(𝔤_{k}, (M^{∗})^{I_{K}}) → H^{1}(G_{K}, M^{∗}) → H^{1}(I_{K}, M^{∗})^{𝔤_{k}} → 0.

If in addition M is only tamely ramified, then Theorem 2.2 of Lecture 6 applies. In this case we define the finite and singular parts of H^{1}(G_{K}, M) by

H_{f}^{1}(G_{K}, M) = H^{1}(𝔤_{k}, M^{I_{K}})

and

H_{s}^{1}(G_{K}, M) = H^{1}(I_{K}, M)^{𝔤_{k}}.

The definition also applies to M^{∗}. We will refer to these choices as the standard finite/singular structure on M. (We could of course make these definitions without any conditions on M, but they would not in general have the properties that we want.) Thus there is an exact commutative diagram

\xymatrix{0 \ar[r] & H_{s}^{1}(G_{K}, M)^{∨} \ar[r] & H^{1}(G_{K}, M)^{∨} \ar[r] & H_{f}^{1}(G_{K}, M)^{∨} \ar[r] & 0 \\ 0 \ar[r] & H_{f}^{1}(G_{K}, M^{∗}) \ar[r] \ar[u]^{≅} & H^{1}(G_{K}, M^{∗}) \ar[r] \ar[u]^{≅} & H_{s}^{1}(G_{K}, M^{∗}) \ar[r] \ar[u]^{≅} & 0}

with the vertical isomorphisms coming from Tate local duality. Put differently, Tate duality identifies H_{f}^{1}(G_{K}, M) and H_{s}^{1}(G_{K}, M^{∗}) as duals; it also identifies H_{s}^{1}(G_{K}, M) and H_{f}^{1}(G_{K}, M^{∗}) as duals.

Note that in the case that M is unramified as a G_{K}-module (meaning that I_{K} acts trivially on M), we have the slightly simpler definitions

H_{f}^{1}(G_{K}, M) = H^{1}(𝔤_{k}, M)

and

H_{s}^{1}(G_{K}, M) = Hom(I_{K}, M)^{𝔤_{k}}.

We will now briefly describe an interpretation of H_{f}^{1}(G_{K}, M) that helps to explain the terminology “finite”. We do this using the interpretation of H^{1}(G_{K}, M) in terms of torsors. (See Lecture 5, Section 2.) So let T be an M-torsor. In particular, T is a finite G_{K}-set. Now, there is an equivalence of categories between the category of finite G_{K}-sets and the category of finite, étale schemes over Spec K. (See

0Last modified March 7, 1998

Figure 1. Extending finite étale schemes over Spec K

[Mi-EC, Chapter 1, Section 5, Theorem 5.3].) So let T be the finite, étale scheme over Spec K associated with T.

Now, let O_{K} be the ring of integers of K. Spec O_{K} consists of the generic point Spec K and the closed point Spec k. A natural question is when the Spec K scheme T can be extended to a finite, étale scheme T′ over Spec O_{K}. It is not difficult to see that any finite, étale T over Spec K can be extended to a quasi-finite, étale scheme over Spec O_{K}; the difficult part is getting it to be finite. It turns out that, at least in the case where M is unramified, T′ can be taken to be finite if and only if the cohomology class associated to T lies in H_{f}^{1}(G_{K}, M). Thus the “finite” terminology is appropriate, at least in this case.

1.2. Global Structures. We now consider a global field F. For us we will simply take global field to mean number field; that is, a finite extension of ℚ. We will always consider F to come along with a choice of algebraic closure F̄, and we set G_{F} = Gal(F̄/F) as usual. For each place v of F (archimedean or non-archimedean) we let F_{v} be the completion of F at v; we fix an algebraic closure F̄_{v} of F_{v}. We also choose an embedding of F̄ into F̄_{v}; this is equivalent to choosing a place of F̄ above v, which in turn is equivalent to choosing a decomposition group of v in G_{F}. We will write G_{F_{v}} or just G_{v} for the Galois group Gal(F̄_{v}/F_{v}); this is (because of our choices) identified with the decomposition group of v in G_{F}. In the case where v is non-archimedean, we will write 𝔤_{v} for the absolute Galois group of the residue field of F_{v}.

[figure: F̄_{v}; G_{v}; F̄; G_{F}; F_{v}; F]

Now, let M be a finite G_{F} -module, with the G_{F} -action continuous for the discrete topology on M. Note that for every place v, M is also a G_{v}-module by restriction. Thus our choice of embedding

G_{v} ↪ G_{F}

induces maps on cohomology

H^{i}(G_{F} , M) ⟶^{res_{v}} H^{i}(G_{v}, M)

for all i ≥ 0. Here a remarkable thing happens: the maps res_{v} are independent of our choice of embedding G_{v} ↪ G_{F}. This follows from the fact that G_{v} is determined as a subgroup of G_{F} up to conjugation by elements of G_{F}; since conjugation induces the identity automorphism on cohomology ([Se-LF, Chapter 7, Section 5, Proposition 3]), res_{v} is thus independent of the choice of G_{v} in G_{F}.

We want to impose local structures on the first global cohomology group of M, H^{1}(G_{F} , M); we essentially want to use our finite/singular structures defined previously. However, we have only defined these local structures in the case that v is non-archimedean, M is at worst tamely ramified as a G_{v}-module, and the residue characteristic of F_{v} does not divide the order of M. (From now on we will say that the non-archimedean place v divides the order of M if its residue characteristic does.) We do not yet want to impose any particular structure at the other places of F, so we make the following definition.

Definition 3. A structured G_{F} -module is a finite, discrete G_{F} -module M together with a choice of finite/singular structure at each place v of F; by this we simply mean a choice of an exact sequence

0 → H_{f}^{1}(G_{v}, M) → H^{1}(G_{v}, M) → H_{s}^{1}(G_{v}, M) → 0

for each v. (That is, simply pick either a subgroup or a quotient group of H^{1}(G_{v}, M) and define the other group via the exact sequence.) We further require that there is a finite set S of places of F such that for all v ∉ S,

H_{f}^{1}(G_{v}, M) = H^{1}(𝔤_{v}, M^{I_{v}})

and

H_{s}^{1}(G_{v}, M) = H^{1}(I_{v}, M)^{𝔤_{v}}.

That is, we require that our choice of finite/singular structure agree with that defined previously, at least for v ∉ S.

We will always take S to contain all archimedean places, all places where M has wild ramification, and all places dividing the order of M, since we have not even defined finite/singular structures in these cases. It is easy to check that since M is finite and discrete there are only finitely many such places, so the definition makes sense. We can in fact improve this statement slightly; M is actually unramified as a G_{v}-module for all but finitely many v.

In order to exploit Tate local duality we want to be somewhat careful as to how the structure on M relates to that on its Cartier dual M^{∗}. In order to do this we must carefully consider the archimedean places for the first time. In fact, they behave much like the non-archimedean places. Since the case where v is complex is trivial (everything vanishes), the next proposition is all that we will need.

Proposition 1.1. Let M be a finite G = Gal(ℂ/ℝ)-module. Then the pairing

H^{1}(G, M) ⊗ H^{1}(G, M^{∗}) → H^{2}(G, M ⊗ M^{∗}) → H^{2}(G, μ(ℂ^{∗})) →^{∼} \frac{1}{2}ℤ/ℤ

given by cup product, Cartier duality and an easy computation, is perfect.

Proof. Write G = {1, c}, where c is complex conjugation. c acts on μ(ℂ^{∗}) by inversion, so we can choose an identification of μ(ℂ^{∗}) with ℚ/ℤ on which c acts by negation. Cartier duality between M and M^{∗} now becomes a sort of Pontrjagin duality

M ⊗ M^{∗} → ℚ/ℤ,

where φ ∈ M^{∗} is considered as a map φ : M → ℚ/ℤ such that

(cφ)(m) = cφ(c^{−1}m)

for all m ∈ M. Given our action of c on ℚ/ℤ and the fact that c = c^{−1}, this becomes

(cφ)(m) = −φ(cm).

Recall that we have an identification

H^{1}(G, M) = CrossHom(G, M)/ PrincCrossHom(G, M).

A crossed homomorphism h must send 1 ∈ G to 0 ∈ M, so it is determined completely by where it sends c. The cocycle condition is just

c(h(c)) + h(c) = 0,

so we can identify CrossHom(G, M) with the subgroup of M on which c acts by negation; we write this as M^{−}. Under this identification the principal crossed homomorphisms are just (1 − c)M, so we have an identification

H^{1}(G, M) = M^{−}/(1 − c)M.

We have a similar expression for M^{∗}.

Now, we return to our “Pontrjagin” duality

M ⊗ M^{∗} → ℚ/ℤ.

The annihilator of M^{∗−} in M is easily seen to be (1 − c)M, since φ ∈ M^{∗−} if and only if

φ(cm) = φ(m)

for all m ∈ M. Similarly, the annihilator of M^{−} is just (1 − c)M^{∗}. Thus the above pairing restricts to a perfect pairing

M^{−}/(1 − c)M ⊗ M^{∗−}/(1 − c)M^{∗} → ℚ/ℤ.

Now our above observations and an easy computation using the definition of cup products (see [AW, Section 7]) identify this pairing with the cup product followed by Cartier duality, so the proof is complete.

We will refer to the above pairing simply as a local pairing.

Definition 4. Let M be a structured G_{F} -module. We define the Cartier dual of M as a structured G_{F} -module to be the G_{F} -module

M^{∗} = Hom_{ℤ}(M, μ(F̄))

together with choices of local finite/singular structures

0 → H_{f}^{1}(G_{v}, M^{∗}) → H^{1}(G_{v}, M^{∗}) → H_{s}^{1}(G_{v}, M^{∗}) → 0

such that H_{f}^{1}(G_{v}, M^{∗}) is the exact dual of H_{s}^{1}(G_{v}, M) under the local pairing for all v. Since the local pairings are perfect, it follows that H_{s}^{1}(G_{v}, M^{∗}) is the exact dual of H_{f}^{1}(G_{v}, M).

Note that Theorem 2.2 of Lecture 6 insures that the above definition really does give a finite/singular structure on M^{∗}.

2. Generalized Selmer Groups

2.1. Definitions. We continue to let F be a number field and we let M be a finite, structured G_{F} -module. We now define finite and singular parts of H^{1}(G_{F} , M) using local conditions. First note that under the maps

H^{1}(G_{F} , M) ⟶^{res_{v}} H^{1}(G_{v}, M),

each element h ∈ H^{1}(G_{F} , M) maps to H_{f}^{1}(G_{v}, M) for all but finitely many v. To see this, recall that H^{1}(G_{F} , M) is defined as a direct limit, so we can consider h as an element of H^{1}(Gal(L/F), M^{G_{L}}) for some finite, Galois extension L of F. Let v be a place of F and let w be the place of L above v determined by our choice G_{v} ↪ G_{F}. Further suppose that v is non-archimedean and that L/F is unramified at v; this is true for almost all v. But in this situation, letting ℓ_{w} and f_{v} be the residue fields of L_{w} and F_{v} respectively, the inflation-restriction sequence gives an isomorphism

H^{1}(Gal(ℓ_{w}/f_{v}), M^{G_{L}}) ⟶^{≅} H^{1}(Gal(L_{w}/F_{v}), M^{G_{L}}),

since the inertia group is trivial. The commutative diagram

\xymatrix{H^{1}(Gal(L/F), M^{G_{L}}) \ar[r]^{^{res^{v}}} \ar[dd]_{inf} & H^{1}(Gal(ℓ_{w}/f_{v}), M^{G_{L}}) \ar[d]^{inf} \\  & H_{f}^{1}(G_{v}, M) \ar@{^{(}->}[d]^{inf} \\ H^{1}(G_{F} , M) \ar[r]^{_{res_{v}}} & H^{1}(G_{v}, M)}

now completes the proof.

We want to define H_{f}^{1}(G_{F} , M) to be the set of global cohomology classes which are always locally finite. Precisely we define

H_{f}^{1}(G_{F} , M) = ker(H^{1}(G_{F} , M) → ⨁_{v} H_{s}^{1}(G_{v}, M)),

where the direct sum is over all places v of F. (Our argument above shows that this map really does land in the direct sum.) That is,

H_{f}^{1}(G_{F} , M) = {h ∈ H^{1}(G_{F} , M) | res_{v}(h) ∈ H_{f}^{1}(G_{v}, M) for all v}.

We will call H_{f}^{1}(G_{F} , M) the Selmer group of the structured G_{F} -module M.

We define

H_{s}^{1}(G_{F} , M) = H^{1}(G_{F} , M)/H_{f}^{1}(G_{F} , M),

so that there is an exact sequence

0 → H_{f}^{1}(G_{F} , M) → H^{1}(G_{F} , M) → H_{s}^{1}(G_{F} , M) → 0.

We will call H_{s}^{1}(G_{F} , M) the Kolyvagin group of the structured G_{F} -module M.

We should note that our choice of notation is not intended to imply that there is any sort of duality between the Selmer groups and Kolyvagin groups of M and M^{∗}. The statement of Poitou-Tate global duality is significantly more complicated than that of Tate local duality.

2.2. A First Example. In order to illuminate the meaning of a Selmer group slightly, we now consider a simple example. Let M be a finite abelian group considered as a G_{F} -module with trivial G_{F} -action. We define a structure on M as follows: Let S be a set of places of F containing all of the bad places. That is, S contains the archimedean places and the set of places dividing the order of M. For v ∈ S, we simply take

H_{f}^{1}(G_{v}, M) = H^{1}(G_{v}, M)

so that

H_{s}^{1}(G_{v}, M) = 0.

For v ∉ S, we take the standard finite/singular structure. Since M is unramified everywhere, this takes the simple form

0 → H^{1}(𝔤_{v}, M) → H^{1}(G_{v}, M) → Hom(I_{v}, M)^{𝔤_{v}} → 0.

Now, consider H^{1}(G_{F} , M). Since G_{F} acts trivially on M, we have

H^{1}(G_{F} , M) = Hom(G_{F} , M) = Hom(G_{F}^{ab}, M).

(The homomorphisms are continuous, of course.) A cocycle h ∈ H^{1}(G_{F} , M) lies in the Selmer group H_{f}^{1}(G_{F} , M) if it maps to 0 in H_{s}^{1}(G_{v}, M) for all places v. If v ∈ S, this is automatic. If v ∉ S, this simply asks that h vanishes on the inertia group I_{v}. (We get these simple interpretations since h is just a homomorphism.) Thus H_{f}^{1}(G_{F} , M) is simply the subgroup of Hom(G_{F}^{ab}, M) of homomorphisms vanishing on all inertia groups of v ∈ S. In other words, letting F^{ab,S} be the maximal abelian extension of F unramified outside of S, we have

H_{f}^{1}(G_{F} , M) = Hom(Gal(F^{ab,S}/F), M).

(Hom(Gal(F^{ab,S}/F), M) is considered as a subgroup of Hom(G_{F}^{ab}, M) via the natural surjection G_{F}^{ab} ↠ Gal(F^{ab,S}/F).) Thus knowledge of H_{f}^{1}(G_{F} , M) should yield information about F^{ab,S}.

If instead we had imposed the finite/singular structure with

H_{f}^{1}(G_{v}, M) = 0

and

H_{s}^{1}(G_{v}, M) = H^{1}(G_{v}, M)

for v ∈ S, one can easily check in the same way as above that we would have obtained

H_{f}^{1}(G_{F} , M) = Hom(Gal(F′/F), M),

where F′ is the maximal abelian extension of F which is unramified everywhere and in which all places in S split completely. (The key fact here is that all of the local Galois groups for a place v vanish if and only if v splits completely.)

2.3. Interpretation in terms of Étale Cohomology. We now give an interpretation of (some) Selmer groups in terms of étale cohomology. We let M be any finite G_{F} -module, and we again take S to be a set of places containing all of the bad places. (This time it also contains those places where M is wildly ramified.) We take the usual finite/singular structure at v ∉ S, and we set

H_{f}^{1}(G_{v}, M) = H^{1}(G_{v}, M)

for v ∈ S. We let S′ be the subset of S of non-archimedean places.

Recall (see Lecture 1, Section 1) that to M we can associate an étale sheaf M over Spec F. Let O_{F} be the ring of integers of F, and consider the natural open immersions

Spec F ↪^{j} Spec O_{F} − S′ ↪^{i} Spec O_{F} .

Here Spec F maps to the generic point. Note that Spec O_{F} − S′ can be realized as Spec O_{F} [1/N] where N is any element of O_{F} divisible by every place of S′ and no other places. In this situation it can be shown that

H_{f}^{1}(G_{F} , M) = H_{et}^{1}(Spec O_{F} − S′, j_{∗}M).

We sketch a proof. Set X = Spec O_{F} − S′. The Leray spectral sequence ([Mi-EC, Chapter 3, Theorem 1.18]) for j : Spec F → X takes the form

H_{et}^{p}(X, R^{q}j_{∗}M) ⇒ H_{et}^{p+q}(Spec F, M).

The exact sequence of low degree terms begins

0 → H_{et}^{1}(X, j_{∗}M) → H_{et}^{1}(Spec F, M) → R^{1}j_{∗}M(X).

But H_{et}^{1}(Spec F, M) is just H^{1}(G_{F} , M) (see [Mi-EC, Chapter 3, Example 1.7]), so we have identified H_{et}^{1}(X, j_{∗}M) with the subgroup of elements of H^{1}(G_{F} , M) which vanish in R^{1}j_{∗}M(X).

To check that an element of this last group vanishes it is enough to check on the stalks. So let \overline{v} ↪ X be a geometric point sitting over some prime v of F, v ∉ S′. The stalk of O_{X} at \overline{v} is just the strict Henselization of the local ring of O_{X} at v (recall that the strict Henselization of a local ring A is the smallest local ring containing A which has algebraically closed residue field and satisfies Hensel’s lemma); in this case, this is just the ring of integers of the maximal unramified extension of F_{v}. The base change of Spec O_{X,\overline{x}} up to Spec F is then just Spec F_{v}^{ur}, so by [Mi-EC, Chapter 3, Theorem 1.15] we have

R^{1}j_{∗}(M)\overline{v} = H^{1}(Spec F_{v}^{ur}, M) = H^{1}(I_{v}, M),

where I_{v} is the inertia group at v. A similar calculation shows that the stalk at the generic point is 0.

Combining this with our above exact sequence, we see that an element h ∈ H^{1}(F, M) lies in H_{et}^{1}(X, j_{∗}M) if and only if h maps to 0 in each H^{1}(I_{v}, M) for v ∉ S. But this is exactly the definition of H_{f}^{1}(G_{F} , M), so

H_{f}^{1}(G_{F} , M) = H_{et}^{1}(Spec O_{F} − S′, j_{∗}M),

as claimed.

2.4. Other Notation. Our notation differs somewhat from that in [Ru-ES]. The translation is easy; for convenience, and so the reader can go through [Ru-ES, Chapter 1, Section 6], we record it here.

First, Rubin uses the standard notations

H^{i}(K, M) = H^{i}(Gal(K̄/K), M)

for any field K, and

H^{i}(L/K, M) = H^{i}(Gal(L/K), M)

for any Galois extension L/K. He defines finite/singular structures for archimedean v and for all v not dividing the order of M; they agree with ours when both are defined. He writes H_{f}^{1}(F_{v}, M) for the finite part and H_{/f}^{1}(F_{v}, M) for the singular part (H_{s}^{1}(G_{v}, M) in our notation).

Our Selmer group H_{f}^{1}(G_{F} , M), assuming our finite/singular structures agree with his for v not dividing the order of M and given our choices for those v which do divide the order of M, would be written as S(F, M). Rubin also defines “partial” Selmer groups where not all local conditions are enforced, and “strict” Selmer groups where some H_{f}^{1}(G_{F} , M) are set to be 0 (in our terminology). For details, see [Ru-ES, Chapter 1, Section 5].

In [Ru-ES, Chapter 1, Section 6] he gives several interesting interpretations of Selmer groups, relating them to ideal class groups, units and abelian varieties. The reader is strongly encouraged to look at that section.

We should also note that Rubin’s Galois modules are rarely finite; they tend to be p-adic. We will deal with this later in our course; for the time being this distinction can probably be safely ignored.

3. Brauer Groups

3.1. Definitions. We begin by recalling the various interpretations of the Brauer group. For proofs of the claims in this section, see [FD, Chapter 4]. For any field L, we take as our definition

Br_{L} = H^{2}(G_{L}, L̄^{∗}).

As a first alternate definition, Br_{L} can be considered as the set of finite dimensional central simple algebras over L modulo the equivalence relation where two central simple algebras are considered to be equivalent if they become isomorphic as L-algebras after tensoring each with (possibly different) total matrix algebras M_{n}(L). Br_{L} is made into a group by tensor product. It can be shown that every central simple algebra over L is a matrix algebra over a central divison algebra over L, and that central simple algebras are equivalent in our sense if and only if they come from the same division algebra. Thus Br_{L} also classifies central division algebras over L. Lastly, it is also true that an L-algebra is central simple over L if and only if it becomes isomorphic to M_{n}(L̄), for some n, after tensoring with L̄. In particular, the dimension of any central simple algebra over L is a perfect square.

To recall a few examples, if k is a finite field we have Wedderburn’s theorem (see [FD, Chapter 3, Theorem 3.18])

Br_{k} = 0,

and for the real numbers we have

Br_{ℝ} = ℤ/2ℤ

with the non-trivial division algebra being the quaternions ℍ. (See [FD, Chapter 3, Theorem 3.20]) We also note that

BrL̄ = 0

for any algebraically closed field L̄; the proof of this is easy.

3.2. Brauer Group of a Local Field. Recall that if K is a local field, then local class field theory gives an isomorphism

Br_{K} = H^{2}(G_{K}, K̄^{∗}) ⟶^{inv_{K}} ℚ/ℤ.

(See [Se-LF, Chapter 13, Section 3].) For completeness, we include here the proof that

H^{2}(G_{K}, μ(K̄^{∗})) ≅ H^{2}(G_{K}, K̄^{∗}) ≅ ℚ/ℤ.

We use the Kummer exact sequence

1 → μ_{N}(K̄^{∗}) → K̄^{∗} ⟶^{N} K̄^{∗} → 1

for every N. The long exact sequence in cohomology yields

H^{1}(G_{K}, K̄^{∗}) → H^{2}(G_{K}, μ_{N}(K̄^{∗})) → H^{2}(G_{K}, K̄^{∗})[N] → 0,

where by H^{2}(G_{K}, K̄^{∗})[N] we mean the N-torsion. But by Hilbert’s Theorem 90 the first term is zero, and by the invariant map the last term is \frac{1}{N} ℤ/ℤ. Taking the direct limit over N now completes the proof.

We recall the explicit description of division algebras over K. Let m/n ∈ ℚ/ℤ be in lowest terms, with 0 < m < n. (0 ∈ ℚ/ℤ corresponds to K, of course.) We will construct the central division algebra A mapping to m/n under the invariant map. Let L be the unique unramified extension of K of degree n. A has dimension n^{2} over K, and contains L as a maximal commutative subalgebra. In fact, it requires only one more generator: A = L[γ], where γ normalizes L (that is, γLγ^{−1} = L), and for any x ∈ L, γxγ^{−1} = Fr^{m} x, where Fr is the Frobenius automorphsim for K. See [Pi, Chapter 17, Section 10] for details.

3.3. Global Brauer Groups. We now consider the Brauer group of a number field F. Recall that we have canonical maps

H^{2}(G_{F} , F̄^{∗}) ⟶^{res_{v}} H^{2}(G_{v}, F̄_{v}^{∗})

for every place v of F. We claim that for any h ∈ H^{2}(G_{F} , F̄^{∗}), the image of h in H^{2}(G_{v}, F̄_{v}^{∗}) is zero for almost all v.

Proposition 3.1. For any h ∈ H^{2}(G_{F} , F̄^{∗}), res_{v}(h) = 0 for all but finitely many places v of F.

Proof. Note that as usual, since H^{2}(G_{F} , F̄^{∗}) is a direct limit, we can assume that h lies in H^{2}(Gal(L/F), L^{∗}) for some finite Galois extension L/F. Now, let c be a cocycle representing h. Since Gal(L/F) is finite, c takes on only finitely many values; in particular, its image is divisible by only finitely many primes. So let T be the set of all such places, together with the archimedean places and the places where L/K is ramified; T is finite.

Now consider res_{v}(h) ∈ H^{2}(Gal(L_{w}/F_{v}), L_{w}^{∗}) for any v ∉ T, where w is the place of L above v induced by our choice of embedding F̄ ↪ F̄_{v}. By the definition of the restriction map, res_{v}(h) is represented by the restriction of c. In particular, c takes values in O_{w}^{∗}, the units of the ring of integers of L_{w}. This means that res_{v}(h) lies in the image of the natural map

H^{2}(Gal(L_{w}/F_{v}), O_{w}^{∗}) → H^{2}(Gal(L_{w}/F_{v}), L_{w}^{∗}).

Consider the exact sequence

0 → U_{1} → O_{w}^{∗} → ℓ^{∗} → 0,

where U_{1} are the units congruent to 1 modulo 𝔪_{w}, and ℓ^{∗} is the residue field of L_{w}. This is an exact sequence of Gal(L_{w}/F_{v})-modules, and using the fact (see [Se-LF, Chapter 12, Section 3, Lemma 2]) that H^{i}(Gal(L_{w}/F_{v}), U_{1}) = 0 for all i ≥ 1, we get an isomorphism

H^{2}(Gal(L_{w}/F_{v}), O_{w}^{∗}) ≅ H^{2}(Gal(L_{w}/F_{v}), ℓ^{∗}).

Finally, since L_{w}/F_{v} is unramified, the cohomology of a finite cyclic group ([Se-LF, Chapter 8, Section 4]) and the surjectivity of the norm on a finite field show that

H^{2}(Gal(L_{w}/F_{v}), ℓ^{∗}) = 0,

which implies that the image of H^{2}(Gal(L_{w}/F_{v}), O_{w}^{∗}) in H^{2}(Gal(L_{w}/F_{v}), L_{w}^{∗}) is zero, and thus that res_{v}(h) = 0 for all places not in T. This proves the claim.

For a proof of this fact using the cohomology of the ideles, see [Ta, Section 7, Proposition 7.3 and Section 9.6]. (Note that he doesn’t actually prove that the maps he is considering are the restriction map, but it is not too difficult to see this.) For a proof in terms of divison algebras, see [Pi, Chapter 18, Section 5].

It follows that the restriction maps combine to give a map

H^{2}(G_{F} , F̄^{∗}) ⟶^{res} ⨁_{v} H^{2}(G_{v}, F̄_{v}^{∗}),

the direct sum being over all places v of F. One of the fundamental theorems of global class field theory is the determination of the kernel and cokernel of this map. Before we state it, we consider

⨁_{v} H^{2}(G_{v}, F̄_{v}^{∗}).

This can be evaluated as

⨁_{v non-arch} ℚ/ℤ ⊕ ⨁_{v real} \frac{1}{2}ℤ/ℤ,

given the results of the previous two sections.

3. BRAUER GROUPS 59

Theorem 3.2. There is an exact sequence

0 → Br_{F} ⟶^{res} ⨁_{v} Br_{F_{v}} → ℚ/ℤ → 0,

where the last map is summation of elements of ℚ/ℤ in the identification above.

Proof. This is proved along with the fundamental results of global class field theory; see [Ta, Section 9.6] for the statement.