brauer groups, tamagawa measures, and rational points

8 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
2.8. Remark. –––– Non-abelian group cohomology may easily be extended to
the case where G is a profinite group and A a discrete G-set (respectively
G-group) on which G operates continuously. Indeed, put
H i (G, A) := lim H i (G/G ′ , A G′ )
−→G
′
fori = 0and1. Here, thedirectlimitistakenover theinflationmapsandG ′ runs
through the open normal subgroups G ′ ⊆ G with finite quotient.
3. Galois descent
3.1. Definition. –––– Let L/K be a finite Galois extension and σ ∈ Gal(L/K ).
a) Then, a map i : V 1 → V 2 of L-vector spaces is said to be σ-linear if, for every
v, w ∈ V 1 and λ ∈ L, one has
i(v + w) = i(v) + i(w)
and
i(λv) = σ(λ)i(v) .
b) Let π 1 : X 1 → SpecL and π 2 : X 2 → SpecL be L-schemes. We say that
f : X 1 → X 2 is a morphism of L-schemes twisted by σ if the diagram
X 1
f
X 2
π 2
π 1
SpecL
S(σ)
SpecL
commutes. Here, S(σ): SpecL → SpecL denotes the morphism of affine
schemes induced by σ −1 : L → L.
3.2. Theorem. –––– Let L/K be a finite Galois extension and G := Gal(L/K ).
Then,
i) there are the following equivalences of categories,
⎧
⎫
⎨L-vector spaces
⎬
{K-vector spaces} −→ with a G-operation from the left
⎩
⎭ ,
such that every σ ∈ G operates σ-linearly
⎧
⎫
⎨L-algebras
⎬
{K-algebras} −→ with a G-operation from the left
⎩
⎭ ,
such that every σ ∈ G operates σ-linearly