## Tag `0992`

Chapter 55: Pro-étale Cohomology > Section 55.12: Points of the pro-étale site

Lemma 55.12.1. Let $S$ be a scheme. The pro-étale sites $S_{pro\text{-}\acute{e}tale}$, $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$, $S_{affine, {pro\text{-}\acute{e}tale}}$, 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 55.11.15. 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 55.11.23. 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, Lemma 7.38.4. $\square$

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, Lemma \ref{sites-lemma-criterion-points}.
\end{proof}
```

## Comments (0)

## Add a comment on tag `0992`

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).

All contributions are licensed under the GNU Free Documentation License.

There are no comments yet for this tag.

There are also 7 comments on Section 55.12: Pro-étale Cohomology.