## Tag `0991`

## 55.12. Points of the pro-étale site

We first apply Deligne's criterion to show that there are enough points.

Lemma 55.12.1. 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 $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}$ 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. $\square$Let $S$ be a scheme. Let $\overline{s} : \mathop{\rm Spec}(k) \to S$ be a geometric point. We define a

pro-étale neighbourhoodof $\overline{s}$ to be a commutative diagram $$ \xymatrix{ \mathop{\rm Spec}(k) \ar[r]_-{\overline{u}} \ar[rd]_{\overline{s}} & U \ar[d] \\ & S } $$ with $U \to S$ weakly étale. In exactly the same manner as in Étale Cohomology, Lemma 53.29.4 one shows that the category of pro-étale neighbourhoods of $\overline{s}$ is cofiltered. Moreover, if $(U, \overline{u})$ is a pro-étale neighbourhood, and if $\{U_i \to U\}$ is a pro-étale covering, then there exists an $i$ and a lift of $\overline{u}$ to a geometric point $\overline{u}_i$ of $U_i$. For $\mathcal{F}$ in $\mathop{\textit{Sh}}\nolimits(S_{pro\text{-}\acute{e}tale})$ define thestalk of $\mathcal{F}$ at $\overline{s}$by the formula $$ \mathcal{F}_{\overline{s}} = \mathop{\rm colim}\nolimits_{(U, \overline{u})} \mathcal{F}(U) $$ where the colimit is over all pro-étale neighbourhoods $(U, \overline{u})$ of $\overline{s}$ with $U \in \mathop{\rm Ob}\nolimits(S_{pro\text{-}\acute{e}tale})$. A formal argument using the facts above shows the functor $\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}$ defines a point of the topos $\mathop{\textit{Sh}}\nolimits(S_{pro\text{-}\acute{e}tale})$: it is an exact functor which commutes with arbitrary colimits. In fact, this functor has another description.Lemma 55.12.2. In the situation above the scheme $\mathop{\rm Spec}(\mathcal{O}_{S, \overline{s}}^{sh})$ is an object of $X_{pro\text{-}\acute{e}tale}$ and there is a canonical isomorphism $$ \mathcal{F}(\mathop{\rm Spec}(\mathcal{O}_{S, \overline{s}}^{sh})) = \mathcal{F}_{\overline{s}} $$ functorial in $\mathcal{F}$.

Proof.The first statement is clear from the construction of the strict henselization as a filtered colimit of étale algebras over $S$, or by the characterization of weakly étale morphisms of More on Morphisms, Lemma 36.53.11. The second statement follows as by Olivier's theorem (More on Algebra, Theorem 15.87.25) the scheme $\mathop{\rm Spec}(\mathcal{O}_{S, \overline{s}}^{sh})$ is an initial object of the category of pro-étale neighbourhoods of $\overline{s}$. $\square$Contrary to the situation with the étale topos of $S$ it is not true that every point of $\mathop{\textit{Sh}}\nolimits(S_{pro\text{-}\acute{e}tale})$ is of this form, and it is not true that the collection of points associated to geometric points is conservative. Namely, suppose that $S = \mathop{\rm Spec}(k)$ where $k$ is an algebraically closed field. Let $A$ be an abelian group. Consider the sheaf $\mathcal{F}$ on $S_{pro\text{-}\acute{e}tale}$ defined by the rule $$ \mathcal{F}(U) = \frac{\{\text{functions }U \to A\}}{\{\text{locally constant functions}\}} $$ Then $\mathcal{F}(U) = 0$ if $U = S = \mathop{\rm Spec}(k)$ but in general $\mathcal{F}$ is not zero. Namely, $S_{pro\text{-}\acute{e}tale}$ contains quasi-compact objects with infinitely many points. For example, let $E = \mathop{\rm lim}\nolimits E_n$ be an inverse limit of finite sets with surjective transition maps, e.g., $E = \mathop{\rm lim}\nolimits \mathbf{Z}/n\mathbf{Z}$. The scheme $\mathop{\rm Spec}(\mathop{\rm colim}\nolimits \text{Map}(E_n, k))$ is an object of $S_{pro\text{-}\acute{e}tale}$ because $\mathop{\rm colim}\nolimits \text{Map}(E_n, k)$ is weakly étale (even ind-Zariski) over $k$. Thus $\mathcal{F}$ is a nonzero abelian sheaf whose stalk at the unique geometric point of $S$ is zero.

The solution is to use the existence of quasi-compact, weakly contractible objects. First, there are enough quasi-compact, weakly contractible objects by Lemma 55.11.27. Second, if $W \in \mathop{\rm Ob}\nolimits(S_{pro\text{-}\acute{e}tale})$ is quasi-compact, weakly contractible, then the functor $$ \mathop{\textit{Sh}}\nolimits(S_{pro\text{-}\acute{e}tale}) \longrightarrow \textit{Sets},\quad \mathcal{F} \longmapsto \mathcal{F}(W) $$ is an exact functor $\mathop{\textit{Sh}}\nolimits(S_{pro\text{-}\acute{e}tale}) \to \textit{Sets}$ which commutes with all limits. The functor $$ \textit{Ab}(S_{pro\text{-}\acute{e}tale}) \longrightarrow \textit{Ab},\quad \mathcal{F} \longmapsto \mathcal{F}(W) $$ is exact and commutes with direct sums (as $W$ is quasi-compact, see Sites, Lemma 7.17.5), hence commutes with all limits and colimits. Moreover, we can check exactness of a complex of abelian sheaves by evaluation at the quasi-compact, weakly contractible objects of $S_{pro\text{-}\acute{e}tale}$, see Cohomology on Sites, Proposition 21.41.2.

The code snippet corresponding to this tag is a part of the file `proetale.tex` and is located in lines 2667–2794 (see updates for more information).

```
\section{Points of the pro-\'etale site}
\label{section-points}
\noindent
We first apply Deligne's criterion to show that there are enough points.
\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}
\noindent
Let $S$ be a scheme. Let $\overline{s} : \Spec(k) \to S$ be a geometric
point. We define a {\it pro-\'etale neighbourhood} of $\overline{s}$
to be a commutative diagram
$$
\xymatrix{
\Spec(k) \ar[r]_-{\overline{u}} \ar[rd]_{\overline{s}} & U \ar[d] \\
& S
}
$$
with $U \to S$ weakly \'etale. In exactly the same manner as in
\'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-cofinal-etale}
one shows that the category of pro-\'etale
neighbourhoods of $\overline{s}$ is cofiltered. Moreover, if
$(U, \overline{u})$ is a pro-\'etale neighbourhood, and if $\{U_i \to U\}$
is a pro-\'etale covering, then there exists an $i$ and a lift of
$\overline{u}$ to a geometric point $\overline{u}_i$ of $U_i$.
For $\mathcal{F}$ in $\Sh(S_\proetale)$ define the {\it stalk
of $\mathcal{F}$ at $\overline{s}$} by the formula
$$
\mathcal{F}_{\overline{s}} = \colim_{(U, \overline{u})} \mathcal{F}(U)
$$
where the colimit is over all pro-\'etale neighbourhoods $(U, \overline{u})$
of $\overline{s}$ with $U \in \Ob(S_\proetale)$.
A formal argument using the facts above shows the functor
$\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}$
defines a point of the topos $\Sh(S_\proetale)$:
it is an exact functor which commutes with arbitrary colimits.
In fact, this functor has another description.
\begin{lemma}
\label{lemma-classical-point}
In the situation above the scheme $\Spec(\mathcal{O}_{S, \overline{s}}^{sh})$
is an object of $X_\proetale$ and there is a canonical isomorphism
$$
\mathcal{F}(\Spec(\mathcal{O}_{S, \overline{s}}^{sh})) =
\mathcal{F}_{\overline{s}}
$$
functorial in $\mathcal{F}$.
\end{lemma}
\begin{proof}
The first statement is clear from the construction of the strict henselization
as a filtered colimit of \'etale algebras over $S$, or by the characterization
of weakly \'etale morphisms of
More on Morphisms, Lemma
\ref{more-morphisms-lemma-weakly-etale-strictly-henselian-local-rings}.
The second statement follows as by Olivier's theorem
(More on Algebra, Theorem \ref{more-algebra-theorem-olivier})
the scheme $\Spec(\mathcal{O}_{S, \overline{s}}^{sh})$
is an initial object of the category of pro-\'etale neighbourhoods
of $\overline{s}$.
\end{proof}
\noindent
Contrary to the situation with the \'etale topos of $S$ it is not true
that every point of $\Sh(S_\proetale)$ is of this form, and it is not
true that the collection of points associated to geometric points is
conservative. Namely, suppose that $S = \Spec(k)$ where $k$ is an
algebraically closed field. Let $A$ be an abelian group.
Consider the sheaf $\mathcal{F}$ on $S_\proetale$ defined by the rule
$$
\mathcal{F}(U) = \frac{\{\text{functions }U \to A\}}{\{\text{locally constant functions}\}}
$$
Then $\mathcal{F}(U) = 0$ if $U = S = \Spec(k)$ but in general $\mathcal{F}$
is not zero. Namely, $S_\proetale$ contains quasi-compact objects
with infinitely many
points. For example, let $E = \lim E_n$ be an inverse limit of
finite sets with surjective transition maps,
e.g., $E = \lim \mathbf{Z}/n\mathbf{Z}$.
The scheme $\Spec(\colim \text{Map}(E_n, k))$
is an object of $S_\proetale$ because $\colim \text{Map}(E_n, k)$
is weakly \'etale (even ind-Zariski) over $k$.
Thus $\mathcal{F}$ is a nonzero abelian sheaf whose stalk at the
unique geometric point of $S$ is zero.
\medskip\noindent
The solution is to use the existence of quasi-compact, weakly contractible
objects. First, there are enough quasi-compact, weakly contractible objects by
Lemma \ref{lemma-proetale-enough-w-contractible}.
Second, if $W \in \Ob(S_\proetale)$ is quasi-compact, weakly contractible,
then the functor
$$
\Sh(S_\proetale) \longrightarrow \textit{Sets},\quad
\mathcal{F} \longmapsto \mathcal{F}(W)
$$
is an exact functor $\Sh(S_\proetale) \to \textit{Sets}$ which commutes
with all limits. The functor
$$
\textit{Ab}(S_\proetale) \longrightarrow \textit{Ab},\quad
\mathcal{F} \longmapsto \mathcal{F}(W)
$$
is exact and commutes with direct sums (as $W$ is quasi-compact, see
Sites, Lemma \ref{sites-lemma-directed-colimits-sections}), hence
commutes with all limits and colimits. Moreover, we can check exactness of
a complex of abelian sheaves by evaluation at the
quasi-compact, weakly contractible objects of $S_\proetale$, see
Cohomology on Sites, Proposition
\ref{sites-cohomology-proposition-enough-weakly-contractibles}.
```

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