The Stacks project

Lemma 59.33.5. Let $S$ be a scheme. The small étale site $S_{\acute{e}tale}$ endowed with its structure sheaf $\mathcal{O}_ S$ is a locally ringed site, see Modules on Sites, Definition 18.40.4.

Proof. This follows because the stalks $(\mathcal{O}_ S)_{\overline{s}} = \mathcal{O}^{sh}_{S, \overline{s}}$ are local, and because $S_{\acute{e}tale}$ has enough points, see Lemma 59.33.1, Theorem 59.29.10, and Remarks 59.29.11. See Modules on Sites, Lemmas 18.40.2 and 18.40.3 for the fact that this implies the small étale site is locally ringed. $\square$


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