A Short Course on Galois Cohomology - William Stein - University of ...

19.2 Some Motivating Examples
(i) Let E be an elliptic curve over k and n a positive integer coprime to
char(k). Consider the Weil pairing
Cup product defines a map
E[n] ⊗ E[n] → µn.
H 1 (k, E[n]) ⊗ H 1 (k, E[n]) → H 2 (k, µn).
The inclusion µn ↩→ (k sep ) ∗ defines a homomorphism
H 2 (k, µn) → H 2 (k, (k sep ) ∗ ) = Brk .
We thus have a pairing on H 1 (k, E[n]) with values Brk. It would thus
be very handy to understand Brauer groups better.
(ii) 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 BrF . It would thus be very handy, indeed, to understand Brauer
groups better.
19.3 Examples
Recall that if G is a finite cyclic group and A is a G-module, then ˆ H 2q (G, A) ≈
ˆH 0 (G, A) and ˆ H 2q+1 (G, A) ≈ ˆ H 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 19.2. 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
BrR = H 2 (G, C ∗ ) ∼ = ˆ H 0 (G, C ∗ ) ≈ (C ∗ ) G /NC ∗ ∼ = R ∗ /R ∗ + ∼ = {±1}.
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