Lemma 59.79.5 (0A46)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 59: Étale Cohomology
Section 59.79: Cohomology with support in a closed subscheme
Lemma 59.79.5 (cite)

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Lemma 59.79.5. Let $X$ be a scheme. Let $Z \subset X$ be a closed subscheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$ -module and denote $\mathcal{F}^ a$ the associated quasi-coherent sheaf on the small étale site of $X$ (Proposition 59.17.1). Then

$H^ q_ Z(X, \mathcal{F})$ agrees with $H^ q_ Z(X_{\acute{e}tale}, \mathcal{F}^ a)$ ,
if the complement of $Z$ is retrocompact in $X$ , then $i_*\mathcal{H}^ q_ Z(\mathcal{F}^ a)$ is a quasi-coherent sheaf of $\mathcal{O}_ X$ -modules equal to $(i_*\mathcal{H}^ q_ Z(\mathcal{F}))^ a$ .

Proof. Let $j : U \to X$ be the inclusion of the complement of $Z$ . The statement (1) on cohomology groups follows from the long exact sequences for cohomology with supports and the agreements $H^ q(X_{\acute{e}tale}, \mathcal{F}^ a) = H^ q(X, \mathcal{F})$ and $H^ q(U_{\acute{e}tale}, \mathcal{F}^ a) = H^ q(U, \mathcal{F})$ , see Theorem 59.22.4. If $j : U \to X$ is a quasi-compact morphism, i.e., if $U \subset X$ is retrocompact, then $R^ qj_*$ transforms quasi-coherent sheaves into quasi-coherent sheaves (Cohomology of Schemes, Lemma 30.4.5) and commutes with taking associated sheaf on étale sites (Descent, Lemma 35.9.5). We conclude by applying Lemma 59.79.3. $\square$

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