brauer groups, tamagawa measures, and rational points

14 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
i) to give a homomorphism r : V → V ′ of K-vector spaces (of algebras over K,
of central simple algebras over K, of commutative K-algebras, of commutative
K-algebras with unit, ... ),
ii) to give a homomorphism r L : V ⊗ K L → V ′ ⊗ K L of L-vector spaces (of algebras
over L, ofcentralsimplealgebrasover L, ofcommutative L-algebras, ofcommutative
L-algebras with unit, ... ) which is compatible with the G-operations. I.e., such that
for each σ ∈ G the diagram
V ⊗ K L
r L
V ′ ⊗ K L
commutes.
σ
V ⊗ K L
r L
σ
V ′ ⊗ K L
Proof. If r is given then one defines r L := r ⊗ K L. Clearly, if r is a ring
homomorphism then r L is, too.
Conversely, in order to construct r from r L , note that the commutativity of
the diagrams implies that r L is compatible with G-invariants. Further, we
know (V ⊗ K L) G = V and (V ′ ⊗ K L) G = V ′ . Thus, we obtain a K-linear
map r : V −→ V ′ . If r L is a ring homomorphism then its restriction r is, too.
It is clear that the two procedures described are inverse to each other.
□
3.8. Proposition (Galois descent for morphisms of schemes). —–
Let L/K be a finite Galois extension of fields and G := Gal(L/K ).
Then, it is equivalent
i) to give a morphism of K-schemes f : X → X ′ ,
ii) to give a morphism of L-schemes f L : X × SpecK SpecL → X ′ × SpecK SpecL
which is compatible with the G-operations. I.e., such that for each σ ∈ G the
diagram
X × SpecK SpecL
f L
X ′ × SpecK SpecL
commutes.
σ
X × SpecK SpecL
f L
σ
X ′ × SpecK SpecL
Proof. Let f be given. Then, one defines f L := f × SpecK SpecL.
Conversely, in order to construct f from f L , observe that the question is
local in X ′ and X. Thus, we may assume we are given a homomorphism