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Tag 06VX

Chapter 53: Étale Cohomology > Section 53.30: Points in other topologies

Lemma 53.30.1. Let $S$ be a scheme. All of the following sites have enough points $S_{Zar}$, $S_{\acute{e}tale}$, $(\textit{Sch}/S)_{Zar}$, $(\textit{Aff}/S)_{Zar}$, $(\textit{Sch}/S)_{\acute{e}tale}$, $(\textit{Aff}/S)_{\acute{e}tale}$, $(\textit{Sch}/S)_{smooth}$, $(\textit{Aff}/S)_{smooth}$, $(\textit{Sch}/S)_{syntomic}$, $(\textit{Aff}/S)_{syntomic}$, $(\textit{Sch}/S)_{fppf}$, and $(\textit{Aff}/S)_{fppf}$.

Proof. For each of the big sites the associated topos is equivalent to the topos defined by the site $(\textit{Aff}/S)_\tau$, see Topologies, Lemmas 33.3.10, 33.4.11, 33.5.9, 33.6.9, and 33.7.11. The result for the sites $(\textit{Aff}/S)_\tau$ follows immediately from Deligne's result Sites, Lemma 7.38.4.

The result for $S_{Zar}$ is clear. The result for $S_{\acute{e}tale}$ either follows from (the proof of) Theorem 53.29.10 or from Lemma 53.21.2 and Deligne's result applied to $S_{affine, {\acute{e}tale}}$. $\square$

    The code snippet corresponding to this tag is a part of the file etale-cohomology.tex and is located in lines 3775–3784 (see updates for more information).

    \begin{lemma}
    \label{lemma-points-fppf}
    Let $S$ be a scheme. All of the following sites have enough points
    $S_{Zar}$, $S_\etale$,
    $(\Sch/S)_{Zar}$, $(\textit{Aff}/S)_{Zar}$,
    $(\Sch/S)_\etale$, $(\textit{Aff}/S)_\etale$,
    $(\Sch/S)_{smooth}$, $(\textit{Aff}/S)_{smooth}$,
    $(\Sch/S)_{syntomic}$, $(\textit{Aff}/S)_{syntomic}$,
    $(\Sch/S)_{fppf}$, and $(\textit{Aff}/S)_{fppf}$.
    \end{lemma}
    
    \begin{proof}
    For each of the big sites the associated topos is equivalent to the
    topos defined by the site $(\textit{Aff}/S)_\tau$, see
    Topologies, Lemmas \ref{topologies-lemma-affine-big-site-Zariski},
    \ref{topologies-lemma-affine-big-site-etale},
    \ref{topologies-lemma-affine-big-site-smooth},
    \ref{topologies-lemma-affine-big-site-syntomic}, and
    \ref{topologies-lemma-affine-big-site-fppf}.
    The result for the sites $(\textit{Aff}/S)_\tau$ follows immediately
    from Deligne's result
    Sites, Lemma \ref{sites-lemma-criterion-points}.
    
    \medskip\noindent
    The result for $S_{Zar}$ is clear. The result for $S_\etale$
    either follows from (the proof of)
    Theorem \ref{theorem-exactness-stalks}
    or from
    Lemma \ref{lemma-alternative}
    and Deligne's result applied to $S_{affine, \etale}$.
    \end{proof}

    Comments (3)

    Comment #2585 by Ingo Blechschmidt on May 31, 2017 a 12:40 am UTC

    Deligne's referenced result requires that the site contains finite limits. But does $\mathrm{Aff}/S$ contain a terminal object? I'm under the impression that $\mathrm{Aff}/S$ is the category of $S$-schemes which are affine as schemes over $\operatorname{Spec} \mathbb{Z}$, not the category of $S$-schemes whose structural morphism to $S$ is affine.

    Comment #2586 by Johan (site) on May 31, 2017 a 2:01 pm UTC

    Yes, that is a mistake. Thanks very much. The point is that it locally has the right structure. I've fixed this here.

    Comment #2587 by Ingo Blechschmidt on June 1, 2017 a 3:41 pm UTC

    Perfect, that covers it. Thank you!

    There are also 2 comments on Section 53.30: Étale Cohomology.

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