Lemma 66.22.3 (04KH)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 66: Properties of Algebraic Spaces
Section 66.22: Stalks of the structure sheaf
Lemma 66.22.3 (cite)

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Lemma 66.22.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The small étale 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 étale site is locally ringed. $\square$

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