Lecture 13: Brauer Groups

William Stein

Feb 10, 2010

This lecture is about Brauer groups. Reference: Chapter X of Serre’s Local Fields.

1 The Definition

Let k be a field, and fix a separable closure k^{sep} of k.

Definition 1.1. The Brauer group of k is

Br_{k} = H^{2}(k, (k^{sep})^{∗}).

The Brauer group of a field is a measure of the complexity of the field. It also plays a central role in duality theorems, and in class field theory.

1.1 Some Motivating Examples

1. Let E be an elliptic curve over k and n a positive integer coprime to char(k).

Consider the Weil pairing

E[n] ⊗ E[n] → μ_{n}.

Cup product defines a map

H^{1}(k, E[n]) ⊗ H^{1}(k, E[n]) → H^{2}(k, μ_{n}).

The inclusion μ_{n} ↪ (k^{sep})^{∗} defines a homomorphism

H^{2}(k, μ_{n}) → H^{2}(k, (k^{sep})^{∗}) = Br_{k} .

We thus have a pairing on H^{1}(k, E[n]) with values Br_{k}. It would thus be very handy to understand Brauer groups better.

2. If A is a simple abelian variety over k, then R = End(A) ⊗ k is a division algebra over k. Its center is an extension F of k, and R is a central simple F -algebra. As we will see later, the isomorphism classes of central simple F -algebras are in natural bijection with the elements of Br_{F}. It would thus be very handy, indeed, to understand Brauer groups better.

2 Examples

Recall that if G is a finite cyclic group and A is a G-module, then Ĥ^{2q}(G, A) ≈ Ĥ^{0}(G, A) and Ĥ^{2q+1}(G, A) ≈ Ĥ^{1}(G, A), a fact we proved by explicitly writing down the following very simple complete resolution of G:

⋯ → ℤ[G] ⟶^{s−1} ℤ[G] ⟶^{N} ℤ[G] ⟶^{s−1} ℤ[G] ⟶^{N} ℤ[G] → ⋯,

where N = ∑ s^{i} is the norm.

Proposition 2.1. The Brauer group of the field ℝ of real numbers has order 2.

Proof. We have ℂ = ℝ^{sep}, and G = Gal(ℂ/ℝ) is cyclic of order 2. Thus

Br_{ℝ} = H^{2}(G, ℂ^{∗}) ≅ Ĥ^{0}(G, ℂ^{∗}) ≈ (ℂ^{∗})^{G}/Nℂ^{∗} ≅ ℝ^{∗}/ℝ_{+}^{∗} ≅ {±1}.

Lemma 2.2. Suppose G is a finite cyclic group and A is a finite G-module. Then

#Ĥ^{q}(G, A) = #Ĥ^{0}(G, A)

for all q ∈ ℤ, i.e., #Ĥ^{q}(G, A) is independent of q.

Proof. Since, as was mentioned above, Ĥ^{2q}(G, A) ≈ Ĥ^{0}(G, A) and Ĥ^{2q+1}(G, A) ≈ Ĥ^{1}(G, A), it suffices to show that #Ĥ^{−1}(G, A) = #Ĥ^{0}(G, A). Let s be a generator of G. We have an exact sequence

0 → A^{G} → A ⟶^{s−1} A → A/(s − 1)A → 0.

Since every term in the sequence is finite,

#A^{G} = #(A/(s − 1)A).

Letting N_{G} = ∑ s^{i} be the norm, we have by definition an exact sequence

0 → Ĥ^{−1}(G, A) → A/(s − 1)A ⟶^{N_{G}} A^{G} → Ĥ^{0}(G, A) → 0.

The middle two terms in the above sequence have the same cardinality, so the outer two terms do as well, which proves the lemma.

Proposition 2.3. If k is a finite field, then Br_{k} = 0.

Proof. By definition,

Br_{k} = H^{2}(k, \overline{k}^{∗}) = \underrightarrow{lim}_{F} H^{2}(F/k, F^{∗}),

where F runs over finite extensions of k. Because G = Gal(F/k) is a finite cyclic group, Lemma 2.2 and triviality of the first cohomology of the multiplicative group of a field together imply that

# H^{2}(F/k, F^{∗}) = #Ĥ^{1}(F/k, F^{∗}) = 1.

Example 2.4. The following field all have Br_{k} = 0.

1. Let k be any algebraically or separably closed field. Then Br_{k} = 0, obviously, since k^{sep} = k.

2. Let k be any extension of transcendance degree 1 of an algebraically closed field. Then Br_{k} = 0. (See §X.7 of Serre’s Local Fields for references.)

3. Let k be the maximal unramified extension K^{ur} of a local field K with perfect residue field (e.g., the maximal unramified extension of a finite extension of ℚ_{p}). Then Br_{k} = 0. (See §X.7 of Serre’s Local Fields for references.)

4. Let k be any algebraic extension k of ℚ that contains all roots of unity (thus k is necessarily an infinite degree extension of ℚ). Then Br_{k} = 0.

The following theorem is one of the main results of local class field theory.

Theorem 2.5. Let k be a local field with perfect residue field (e.g., a finite extension of ℚ_{p}). Then Br_{k} ≅ ℚ/ℤ.

The following theorem is one of the main results of global class field theory.

Theorem 2.6. Let k be a number field, and for any place v of k, let k_{v} be the completion of k at v, so k_{v} is a p-adic local field, ℝ, or ℂ. We have a natural exact sequence

0 → Br_{k} → ⨁_{v} Br_{k_{v}} \underrightarrow{(x_{v})↦∑ x_{v}} ℚ/ℤ → 0,

We obtain the map to ℚ/ℤ by using Theorem 2.5 to view each Br_{k_{v}} as ℚ/ℤ, and we view Br_{ℝ} = \frac{1}{2}ℤ/ℤ.

3 Brauer Groups and Central Simple Algebras

Definition 3.1. Let k be a field. Then a central simple k-algebra is a finite dimensional k-algebra A that satisfies any one of the following equivalent conditions:

1. A has no nontrivial two-sided ideals, and A has center k.

2. The algebra A\overline{k} = A ⊗_{k} \overline{k} is isomorphic to a matrix algebra over \overline{k}.

3. There is a finite extension F/k such that A_{F} is isomorphic to a matrix algebra over F.

4. A is isomorphic to a matrix algebra over a division algebra D with center k.

We say that two central simple k-algebras are equivalent if the corresponding division algebras D in 4 above are k-isomorphic. Tensor product endows the set of equivalence classes of central simple k-algebras with the structure of abelian group.

Theorem 3.2. The group B_{k} of equivalence classes of central simple k-algebras is isomorphic to the Brauer group Br_{k}.

The proof of Theorem 3.2 is somewhat involved. We will content ourselves with sketching some of the main ideas; in particular, we will explicitly construct the homomorphism B_{k} → Br_{k}, but will not prove that it is an isomorphism (the argument, which uses descent, is given in Serre’s Local Fields).

Fix a finite Galois extension F of k and let B(n, F/k) be the set of equivalence classes of central simple k-algebras A such that A_{F} ≈ M_{n}(F), where M_{n}(F) is the algebra of n × n matrices over F. Then B is the union of all B(n, F/k) over all n and F.

Given A ∈ B(n, F/k), let φ : A_{F} → M_{n}(F) be a fixed choice of isomorphism. Define a set-theoretic map

f : Gal(F/k) → Aut_{F} (M_{n}(F)) ≈ PGL_{n}(F)

by

f(s) = φ^{−1} ∘ s(φ) = φ^{−1} ∘ s ∘ φ ∘ s^{−1}

Then

[f] ∈ H^{1}(F/k, PGL_{n}(F)),

where this H^{1} is a cohomology set (!).

Proposition 3.3. The above construction A ↦ [f] defines a bijection between B(n, F/K) and H^{1}(F/k, PGL_{n}(F)).

(The above proposition is proved in Serre’s Local Fields.)

Consider the exact sequence

1 → F^{∗} → GL_{n}(F) → PGL_{n}(F) → 1.

There is a well-defined connecting homorphism

H^{1}(F/k, PGL_{n}(F)) → H^{2}(F/k, F^{∗}).

Since H^{2}(F/k, F^{∗}) ⟶^{inf} Br_{k}, we thus obtain a natural map

B(n, F/K) → Br_{k} .

This induces the claimed isomorphism B → Br_{k}.

Next time: Galois cohomology of abelian varieties. Principal homogenous spaces. Hopefully, a complete proof that for any abelian variety over a finite field k, we have H^{q}(k, A) = 0 for all q ≥ 1.