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Tag 0990

Chapter 52: Pro-étale Cohomology > Section 52.11: The pro-étale site

Lemma 52.11.27. 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 $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}$ and if $S$ is affine $S_{app}$ have enough quasi-compact, weakly contractible objects, see Sites, Definition 7.39.2.

Proof. Follows immediately from Lemma 52.11.10. $\square$

    The code snippet corresponding to this tag is a part of the file proetale.tex and is located in lines 2654–2661 (see updates for more information).

    \begin{lemma}
    \label{lemma-proetale-enough-w-contractible}
    Let $S$ be a scheme. The pro-\'etale sites
    $S_\proetale$, $(\Sch/S)_\proetale$, $S_{affine, \proetale}$, and
    $(\textit{Aff}/S)_\proetale$ and if $S$ is affine $S_{app}$
    have enough quasi-compact, weakly contractible
    objects, see Sites, Definition \ref{sites-definition-w-contractible}.
    \end{lemma}
    
    \begin{proof}
    Follows immediately from Lemma \ref{lemma-get-many-weakly-contractible}.
    \end{proof}

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