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Tag 098N

Chapter 55: Pro-étale Cohomology > Section 55.11: The pro-étale site

Lemma 55.11.15. Let $S$ be a scheme. Let $\textit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site containing $S$. The functor $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale} \to (\textit{Sch}/S)_{pro\text{-}\acute{e}tale}$ is a special cocontinuous functor. Hence it induces an equivalence of topoi from $\mathop{\textit{Sh}}\nolimits((\textit{Aff}/S)_{pro\text{-}\acute{e}tale})$ to $\mathop{\textit{Sh}}\nolimits((\textit{Sch}/S)_{pro\text{-}\acute{e}tale})$.

Proof. The notion of a special cocontinuous functor is introduced in Sites, Definition 7.28.2. Thus we have to verify assumptions (1) – (5) of Sites, Lemma 7.28.1. Denote the inclusion functor $u : (\textit{Aff}/S)_{pro\text{-}\acute{e}tale} \to (\textit{Sch}/S)_{pro\text{-}\acute{e}tale}$. Being cocontinuous just means that any pro-étale covering of $T/S$, $T$ affine, can be refined by a standard pro-étale covering of $T$. This is the content of Lemma 55.11.5. Hence (1) holds. We see $u$ is continuous simply because a standard pro-étale covering is a pro-étale covering. Hence (2) holds. Parts (3) and (4) follow immediately from the fact that $u$ is fully faithful. And finally condition (5) follows from the fact that every scheme has an affine open covering. $\square$

    The code snippet corresponding to this tag is a part of the file proetale.tex and is located in lines 2259–2267 (see updates for more information).

    \begin{lemma}
    \label{lemma-affine-big-site-proetale}
    Let $S$ be a scheme. Let $\Sch_\proetale$ be a big pro-\'etale
    site containing $S$.
    The functor $(\textit{Aff}/S)_\proetale \to (\Sch/S)_\proetale$
    is a special cocontinuous functor. Hence it induces an equivalence
    of topoi from $\Sh((\textit{Aff}/S)_\proetale)$ to
    $\Sh((\Sch/S)_\proetale)$.
    \end{lemma}
    
    \begin{proof}
    The notion of a special cocontinuous functor is introduced in
    Sites, Definition \ref{sites-definition-special-cocontinuous-functor}.
    Thus we have to verify assumptions (1) -- (5) of
    Sites, Lemma \ref{sites-lemma-equivalence}.
    Denote the inclusion functor
    $u : (\textit{Aff}/S)_\proetale \to (\Sch/S)_\proetale$.
    Being cocontinuous just means that any pro-\'etale covering of
    $T/S$, $T$ affine, can be refined by a standard pro-\'etale
    covering of $T$. This is the content of
    Lemma \ref{lemma-proetale-affine}.
    Hence (1) holds. We see $u$ is continuous simply because a standard
    pro-\'etale covering is a pro-\'etale covering. Hence (2) holds.
    Parts (3) and (4) follow immediately from the fact that $u$ is
    fully faithful. And finally condition (5) follows from the
    fact that every scheme has an affine open covering.
    \end{proof}

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