## 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}
```

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