Lemma 69.9.3 (0A4P)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 69: Cohomology of Algebraic Spaces
Section 69.9: Cohomology with support in a closed subspace
Lemma 69.9.3 (cite)

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Lemma 69.9.3. Let $S$ be a scheme. Let $f : X \to Y$ be an étale morphism of algebraic spaces over $S$ . Let $Z \subset Y$ be a closed subspace such that $f^{-1}(Z) \to Z$ is an isomorphism of algebraic spaces. Let $\mathcal{F}$ be an abelian sheaf on $X$ . Then

$\mathcal{H}^ q_ Z(\mathcal{F}) = \mathcal{H}^ q_{f^{-1}(Z)}(f^{-1}\mathcal{F})$

as abelian sheaves on $Z = f^{-1}(Z)$ and we have $H^ q_ Z(Y, \mathcal{F}) = H^ q_{f^{-1}(Z)}(X, f^{-1}\mathcal{F})$ .

Proof. Because $f$ is étale an injective resolution of $\mathcal{F}$ pulls back to an injective resolution of $f^{-1}\mathcal{F}$ . Hence it suffices to check the equality for $\mathcal{H}_ Z(-)$ which follows from the definitions. The proof for cohomology with supports is the same. Some details omitted. $\square$

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