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 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 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 Étale Cohomology, Lemma 59.46.3.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)