The Stacks project

Lemma 68.16.3. In Situation 68.16.1.

  1. Given an affine morphism $X' \to X$ of algebraic spaces, we have $H^1(X', \mathcal{F}') = 0$ for every quasi-coherent $\mathcal{O}_{X'}$-module $\mathcal{F}'$.

  2. Given an $A$-algebra $A'$ setting $X' = X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A')$ the morphism $X' \to X$ is affine and $\Gamma (X', \mathcal{O}_{X'}) = A'$.

Proof. Part (1) follows from Lemma 68.8.2 and the Leray spectral sequence (Cohomology on Sites, Lemma 21.14.5). Let $A \to A'$ be as in (2). Then $X' \to X$ is affine because affine morphisms are preserved under base change (Morphisms of Spaces, Lemma 66.20.5) and the fact that a morphism of affine schemes is affine. The equality $\Gamma (X', \mathcal{O}_{X'}) = A'$ follows as $(X' \to X)_*\mathcal{O}_{X'} = A' \otimes _ A \mathcal{O}_ X$ by Lemma 68.11.1 and thus

\[ \Gamma (X', \mathcal{O}_{X'}) = \Gamma (X, (X' \to X)_*\mathcal{O}_{X'}) = \Gamma (X, A' \otimes _ A \mathcal{O}_ X) = A' \]

by Lemma 68.16.2. $\square$


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