brauer groups, tamagawa measures, and rational points

10 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
denotes the morphism induced by S(σ): SpecL → SpecL.
Proof. In each case we have to prove that the functor given is fully faithful and
essentially surjective. Full faithfulness is proven in Propositions 3.7, 3.8, and
3.9, respectively. Propositions 3.3, 3.5, and 3.6 show essential surjectivity. □
3.3. Proposition (Galois descent-algebraic version). —–
Let L/K be a finite Galois extension of fields and G := Gal(L/K ).
Let W be a vector space (an algebra, a central simple algebra, a commutative
algebra, a commutative algebra with unit, ... ) over L together with an operation
T : G × W → W of G from the left. Assume that T respects all extra structures
and that for each σ ∈ G the action of σ is a σ-linear map T σ : W → W .
Then, there is a vector space V (an algebra, a central simple algebra, a commutative
algebra, a commutative algebra with unit, ... ) over K such that there is an
isomorphism
V ⊗ K L −→ b
∼=
W .
Here, V ⊗ K L is equipped with the G-operation induced by the canonical operation
on L. b respects all extra structures and the operation of G.
Proof. Define V := W G . This is clearly a K-vector space (a K-algebra, a
commutative K-algebra, a commutative K-algebra with unit). If W is a central
simple algebra over L then V is a central simple algebra over K. The latter can
not be seen directly but follows immediately from the formula W G ⊗ K L = W
which we prove below. For that, let {l 1 , . . . , l n } be a K-basis of L.
The assertion follows from the claim below.
□
3.4. Claim. –––– There exist an index set J and a subset {x j | j ∈ J } ⊂ W G such
that {l i x j | i ∈ {1, . . . , n}, j ∈ J } is a K-basis of W .
Proof. By Zorn’s Lemma, there exists a maximal subset {x j | j ∈ J } ⊂ W G
such that {l i x j | i ∈ {1, . . . , n}, j ∈ J } ⊂ W is a system of K-linearly
independent vectors. Assume, that system is not a basis of W . Then,
〈l i x j | i ∈ {1, . . . , n}, j ∈ J 〉 K
is a proper G-invariant L-sub-vector space of W and one can choose an element
x ∈ W \ 〈l i x j | i ∈ {1, . . . , n}, j ∈ J 〉 K . For every l ∈ L, the sum
∑ T σ (lx) = ∑ σ(l) · T σ (x)
σ∈G σ∈G