The Stacks project

Lemma 66.19.8. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\overline{x}$ be a geometric point of $X$.

  1. The stalk functor $\textit{PAb}(X_{\acute{e}tale}) \to \textit{Ab}$, $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$ is exact.

  2. We have $(\mathcal{F}^\# )_{\overline{x}} = \mathcal{F}_{\overline{x}}$ for any presheaf of sets $\mathcal{F}$ on $X_{\acute{e}tale}$.

  3. The functor $\textit{Ab}(X_{\acute{e}tale}) \to \textit{Ab}$, $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$ is exact.

  4. Similarly the functors $\textit{PSh}(X_{\acute{e}tale}) \to \textit{Sets}$ and $\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \to \textit{Sets}$ given by the stalk functor $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$ are exact (see Categories, Definition 4.23.1) and commute with arbitrary colimits.

Proof. This result follows from the general material in Modules on Sites, Section 18.36. This is true because $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$ comes from a point of the small étale site of $X$, see Lemma 66.19.7. See the proof of Étale Cohomology, Lemma 59.29.9 for a direct proof of some of these statements in the setting of the small étale site of a scheme. $\square$

Comments (0)

There are also:

  • 4 comment(s) on Section 66.19: Points of the small étale site

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Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 04K1. Beware of the difference between the letter 'O' and the digit '0'.