Lemma 67.13.6 (0DK1)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 67: Morphisms of Algebraic Spaces
Section 67.13: Closed immersions
Lemma 67.13.6 (cite)

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Lemma 67.13.6. Let $S$ be a scheme. Let $i : Z \to X$ be a closed immersion of algebraic spaces over $S$ . Let $\overline{z}$ be a geometric point of $Z$ with image $\overline{x}$ in $X$ . Then $(i_{small, *}\mathcal{F})_{\overline{z}} = \mathcal{F}_{\overline{x}}$ for any sheaf $\mathcal{F}$ on $Z_{\acute{e}tale}$ .

Proof. Choose an étale neighbourhood $(U, \overline{u})$ of $\overline{x}$ . Then the stalk $(i_{small, *}\mathcal{F})_{\overline{z}}$ is the stalk of $i_{small, *}\mathcal{F}|_ U$ at $\overline{u}$ . By Properties of Spaces, Lemma 66.18.12 we may replace $X$ by $U$ and $Z$ by $Z \times _ X U$ . Then $Z \to X$ is a closed immersion of schemes and the result is Étale Cohomology, Lemma 59.46.3. $\square$

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