The Stacks project

Lemma 66.13.6. 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 √©tale 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 65.18.12 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 √Čtale Cohomology, Lemma 59.46.3. $\square$


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