The Stacks project

Lemma 61.12.21. Let $S$ be an affine scheme. Let $S_{app}$ denote the full subcategory of $S_{pro\text{-}\acute{e}tale}$ consisting of affine objects $U$ such that $\mathcal{O}(S) \to \mathcal{O}(U)$ is ind-étale. A covering of $S_{app}$ will be a standard pro-étale covering, see Definition 61.12.6. Then restriction

\[ \mathcal{F} \longmapsto \mathcal{F}|_{S_{app}} \]

defines an equivalence of topoi $\mathop{\mathit{Sh}}\nolimits (S_{pro\text{-}\acute{e}tale}) \cong \mathop{\mathit{Sh}}\nolimits (S_{app})$.

Proof. By Lemma 61.12.20 we may replace $S_{pro\text{-}\acute{e}tale}$ by $S_{affine, {pro\text{-}\acute{e}tale}}$. The lemma follows from Sites, Lemma 7.29.1 by checking that the inclusion functor $S_{app} \to S_{affine, {pro\text{-}\acute{e}tale}}$ is a special cocontinuous functor, see Sites, Definition 7.29.2. The conditions of Sites, Lemma 7.29.1 follow immediately from the definition and the facts (a) any object $U$ of $S_{affine, {pro\text{-}\acute{e}tale}}$ has a covering $\{ V \to U\} $ with $V$ ind-étale over $X$ (Proposition 61.9.1) and (b) the functor $u$ is fully faithful. $\square$


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