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Tag 0994

52.13. Compact generation

Let $S$ be a scheme. The site $S_{pro\text{-}\acute{e}tale}$ has enough quasi-compact, weakly contractible objects $U$. For any sheaf of rings $\mathcal{A}$ on $S_{pro\text{-}\acute{e}tale}$ the corresponding objects $j_{U!}\mathcal{A}_U$ are compact objects of the derived category $D(\mathcal{A})$, see Cohomology on Sites, Lemma 21.42.5. Since every complex of $\mathcal{A}$-modules is quasi-isomorphic to a complex whose terms are direct sums of the modules $j_{U!}\mathcal{A}_U$ (details omitted). Thus we see that $D(\mathcal{A})$ is generated by its compact objects.

The same argument works for the big pro-étale site of $S$.

    The code snippet corresponding to this tag is a part of the file proetale.tex and is located in lines 2800–2823 (see updates for more information).

    \section{Compact generation}
    \label{section-compact-generation}
    
    \noindent
    Let $S$ be a scheme. The site $S_\proetale$ has enough quasi-compact,
    weakly contractible objects $U$. For any sheaf of rings $\mathcal{A}$
    on $S_\proetale$ the corresponding objects $j_{U!}\mathcal{A}_U$
    are compact objects of the derived category $D(\mathcal{A})$, see
    Cohomology on Sites, Lemma
    \ref{sites-cohomology-lemma-quasi-compact-weakly-contractible-compact}.
    Since every complex of $\mathcal{A}$-modules is quasi-isomorphic to
    a complex whose terms are direct sums of the modules
    $j_{U!}\mathcal{A}_U$ (details omitted). Thus we see that
    $D(\mathcal{A})$ is generated by its compact objects.
    
    \medskip\noindent
    The same argument works for the big pro-\'etale site of $S$.

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