# The Stacks Project

## Tag 098W

Lemma 55.11.23. 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 étale covering, see Definition 55.11.6. Then restriction $$\mathcal{F} \longmapsto \mathcal{F}|_{S_{affine, {\acute{e}tale}}}$$ defines an equivalence of topoi $\mathop{\textit{Sh}}\nolimits(S_{pro\text{-}\acute{e}tale}) \cong \mathop{\textit{Sh}}\nolimits(S_{affine, {pro\text{-}\acute{e}tale}})$.

Proof. This you can show directly from the definitions, and is a good exercise. But it also follows immediately from Sites, Lemma 7.28.1 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 7.28.2). $\square$

The code snippet corresponding to this tag is a part of the file proetale.tex and is located in lines 2534–2546 (see updates for more information).

\begin{lemma}
\label{lemma-alternative}
Let $S$ be a scheme. Let $S_{affine, \proetale}$ denote the full subcategory
of $S_\proetale$ consisting of affine objects. A covering of
$S_{affine, \proetale}$ will be a standard \'etale covering, see
Definition \ref{definition-standard-proetale}.
Then restriction
$$\mathcal{F} \longmapsto \mathcal{F}|_{S_{affine, \etale}}$$
defines an equivalence of topoi
$\Sh(S_\proetale) \cong \Sh(S_{affine, \proetale})$.
\end{lemma}

\begin{proof}
This you can show directly from the definitions, and is a good exercise.
But it also follows immediately from
Sites, Lemma \ref{sites-lemma-equivalence}
by checking that the inclusion functor
$S_{affine, \proetale} \to S_\proetale$
is a special cocontinuous functor (see
Sites, Definition \ref{sites-definition-special-cocontinuous-functor}).
\end{proof}

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