Lemma 66.19.8 (04K1)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 66: Properties of Algebraic Spaces
Section 66.19: Points of the small étale site
Lemma 66.19.8 (cite)

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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 $X$.

The stalk functor $\textit{PAb}(X_{\acute{e}tale}) \to \textit{Ab}$, $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$ is exact.
We have $(\mathcal{F}^\# )_{\overline{x}} = \mathcal{F}_{\overline{x}}$ for any presheaf of sets $\mathcal{F}$ on $X_{\acute{e}tale}$.
The functor $\textit{Ab}(X_{\acute{e}tale}) \to \textit{Ab}$, $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$ is exact.
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. $\square$

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