brauer groups, tamagawa measures, and rational points

12 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
and affine K-schemes X ij , 1 ≤ i < j ≤ n, such that there are isomorphisms
X ij × SpecK SpecL ∼ =
−→ Y i ∩ Y j .
It is to be shown that every X ij admits canonical, open embeddings into X i
and X j .
In each case, on the level of rings we have a homomorphism A⊗ K L −→ B ⊗ K L
and an isomorphism B ⊗ K L ∼ =
−→ (A ⊗ K L) f such that their composition is the
localization map. Clearly, f may be assumed to be G-invariant. I.e., we may
suppose f ∈ A. Consequently, B ⊗ K L ∼ = A f ⊗ K L and, by consideration of
the G-invariants on both sides, B ∼ = A f .
The cocycle relations are obvious. Hence, we can glue the affine schemes
X 1 , . . . , X n along the affine schemes X ij for 1 ≤ i < j ≤ n to obtain the
scheme X desired. Lemma 3.12.ix) below completes the proof. □
3.6. Proposition (Galois descent for quasi-coherent sheaves). —–
Let L/K be a finite Galois extension of fields and G := Gal(L/K ).
Further, let X be a K-scheme, π : X × SpecK SpecL → X the canonical morphism
and x σ : X × SpecK SpecL → X × SpecK SpecL be the morphism induced by
S(σ): SpecL → SpecL.
Let M bea quasi-coherentsheaf over X × SpecK SpecL together with a system (ι σ ) σ∈G
of isomorphisms ι σ : x ∗ σM → M which are compatible in the sense that for each
σ, τ ∈ G there is the relation ι τ ◦ x ∗ τ(ι σ ) = ι στ .
Then, there exists a quasi-coherent sheaf F over X such that there is an isomorphism
under which the canonical isomorphism
π ∗ F ∼ =
−→ M
i σ : x ∗ σπ ∗ F = (πx σ ) ∗ F = π ∗ F id
−→ π ∗ F
is identified with ι σ for each σ. I.e., the diagrams
b
x ∗ σπ ∗ F
x∗ σ(b)
x ∗ σM
are commutative.
i σ
π ∗ F
b
M
Proof. First step. Assume X ∼ = SpecA to be an affine scheme.
Then, M = ˜M for some (A ⊗ K L)-module M. We have
x ∗ σ M =
˜
M ⊗ (A⊗K L) (A ⊗ K L σ−1 ) = M ˜ ⊗ L L σ−1 .
ι σ