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Tag 0992

Chapter 52: Pro-étale Cohomology > Section 52.12: Points of the pro-étale site

Lemma 52.12.1. Let $S$ be a scheme. The pro-étale sites $S_{pro\text{-}\acute{e}tale}$, $(\textit{Sch}/S)_{pro\text{-}\acute{e}tale}$, $S_{affine, {pro\text{-}\acute{e}tale}}$, and $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}$ have enough points.

Proof. The big topos is equivalent to the topos defined by $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}$, see Lemma 52.11.15. The topos of sheaves on $S_{pro\text{-}\acute{e}tale}$ is equivalent to the topos associated to $S_{affine, {pro\text{-}\acute{e}tale}}$, see Lemma 52.11.23. The result for the sites $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}$ and $S_{affine, {pro\text{-}\acute{e}tale}}$ follows immediately from Deligne's result Sites, Proposition 7.38.3. $\square$

    The code snippet corresponding to this tag is a part of the file proetale.tex and is located in lines 2673–2678 (see updates for more information).

    \begin{lemma}
    \label{lemma-points-proetale}
    Let $S$ be a scheme. The pro-\'etale sites
    $S_\proetale$, $(\Sch/S)_\proetale$, $S_{affine, \proetale}$, and
    $(\textit{Aff}/S)_\proetale$ have enough points.
    \end{lemma}
    
    \begin{proof}
    The big topos is equivalent to the topos defined by
    $(\textit{Aff}/S)_\proetale$, see
    Lemma \ref{lemma-affine-big-site-proetale}.
    The topos of sheaves on $S_\proetale$ is equivalent to the topos
    associated to $S_{affine, \proetale}$, see
    Lemma \ref{lemma-alternative}.
    The result for the sites $(\textit{Aff}/S)_\proetale$ and
    $S_{affine, \proetale}$ follows immediately from Deligne's result
    Sites, Proposition \ref{sites-proposition-criterion-points}.
    \end{proof}

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