Lemma 69.9.2 (0A4N)—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 69.9.2 (cite)

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Lemma 69.9.2. Let $S$ be a scheme. Let $i : Z \to X$ be a closed immersion of algebraic spaces over $S$ . Let $\mathcal{G}$ be an injective abelian sheaf on $Z_{\acute{e}tale}$ . Then $\mathcal{H}^ p_ Z(i_*\mathcal{G}) = 0$ for $p > 0$ .

Proof. This is true because the functor $i_*$ is exact (Lemma 69.4.1) and transforms injective abelian sheaves into injective abelian sheaves (Cohomology on Sites, Lemma 21.14.2). $\square$

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