brauer groups, tamagawa measures, and rational points

24 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
ii) Let L/K be a field extension. Then Azn
L/K will denote the set of all isomorphism
classes of central simple algebras A which are of dimension n 2 over K
and split over L. Obviously, Az K n := S Az L/K
L/K
n .
4.6. Theorem (cf. J.-P. Serre: Corps locaux [Se62, chap. X, §5]). —–
Let L/K be a finite Galois extension of fields, G := Gal(L/K ) its Galois group,
and n ∈Æ.
Then, there is a natural bijection of pointed sets
a = a L/K
n
: Az L/K
n
∼=
−→ H 1 (G, PGL n (L)) ,
A ↦→ a A .
Proof. Let A be a central simple algebra over K which splits over L,
The diagrams
A ⊗ K L ∼ =
−→ M n (L) .
f
A ⊗ K L f M n (L)
σ
σ
do not commute, in general.
A ⊗ K L f M n (L)
For each σ ∈ G, define a σ ∈ PGL n (L) by putting ( f ◦ σ) = a σ ◦ (σ ◦ f ).
It turns out that
f ◦ στ = ( f ◦ σ) ◦ τ
= a σ ◦ (σ ◦ f ) ◦ τ
= a σ ◦ σ ◦ ( f ◦ τ )
= a σ ◦ σ ◦ (a τ ◦ (τ ◦ f ))
= a σ ◦ σ a τ ◦ (στ ◦ f ) .
I.e., a στ = a σ · σa τ and (a σ ) σ∈G is a cocycle.
If one starts with another isomorphism f ′ : A⊗ K L −→ M n (L) then there exists
some b ∈ PGL n (L) such that f = b ◦ f ′ . The equality ( f ◦ σ) = a σ ◦ (σ ◦ f )
implies
f ′ ◦ σ = b −1 ◦ f ◦ σ = b −1 · a σ ◦ (σ ◦ (b ◦ f ′ )) = b −1 · a σ · σb ◦ (σ ◦ f ′ ) .
Thus, the isomorphism f ′ yields a cocycle cohomologous to (a σ ) σ∈G . The
mapping a is well-defined.