The Stacks project

Lemma 81.4.1. Let $S$ be a scheme. Let $\{ f_ i : X_ i \to X\} $ be an étale covering of algebraic spaces. The functor

\[ \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \longrightarrow \text{descent data for étale sheaves wrt }\{ f_ i : X_ i \to X\} \]

is an equivalence of categories.

Proof. In Properties of Spaces, Section 66.18 we have defined a site $X_{spaces, {\acute{e}tale}}$ whose objects are algebraic spaces étale over $X$ with étale coverings. Moreover, we have a identifications $\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) = \mathop{\mathit{Sh}}\nolimits (X_{spaces, {\acute{e}tale}})$ compatible with morphisms of algebraic spaces, i.e., compatible with pushforward and pullback. Hence the statement of the lemma follows from the much more general discussion in Sites, Section 7.26. $\square$

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