The Stacks project

Lemma 66.22.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The small ├ętale site $X_{\acute{e}tale}$ endowed with its structure sheaf $\mathcal{O}_ X$ is a locally ringed site, see Modules on Sites, Definition 18.40.4.

Proof. This follows because the stalks $\mathcal{O}_{X, \overline{x}}$ are local, and because $S_{\acute{e}tale}$ has enough points, see Lemmas 66.22.1 and Theorem 66.19.12. See Modules on Sites, Lemma 18.40.2 and 18.40.3 for the fact that this implies the small ├ętale site is locally ringed. $\square$


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