Lemma 66.18.3 (03G1)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 66: Properties of Algebraic Spaces
Section 66.18: The étale site of an algebraic space
Lemma 66.18.3 (cite)

numberstags

$X_{\acute{e}tale}\longrightarrow X_{spaces, {\acute{e}tale}}, \quad U/X \longmapsto U/X$

is a special cocontinuous functor (Sites, Definition 7.29.2) and hence induces an equivalence of topoi $\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (X_{spaces, {\acute{e}tale}})$ .

Proof. We have to show that the functor satisfies the assumptions (1) – (5) of Sites, Lemma 7.29.1. It is clear that the functor is continuous and cocontinuous, which proves assumptions (1) and (2). Assumptions (3) and (4) hold simply because the functor is fully faithful. Assumption (5) holds, because an algebraic space by definition has a covering by a scheme. $\square$

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