The Stacks project

Lemma 61.12.20. Let $S$ be a scheme. Let $S_{affine, {pro\text{-}\acute{e}tale}}$ denote the full subcategory of $S_{pro\text{-}\acute{e}tale}$ consisting of affine objects. A covering of $S_{affine, {pro\text{-}\acute{e}tale}}$ will be a standard pro-étale covering, see Definition 61.12.6. Then restriction

\[ \mathcal{F} \longmapsto \mathcal{F}|_{S_{affine, {\acute{e}tale}}} \]

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

Proof. This you can show directly from the definitions, and is a good exercise. But it also follows immediately from Sites, Lemma 7.29.1 by checking that the inclusion functor $S_{affine, {pro\text{-}\acute{e}tale}} \to S_{pro\text{-}\acute{e}tale}$ is a special cocontinuous functor (see Sites, Definition 7.29.2). $\square$


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