## Tag `0993`

Chapter 52: Pro-étale Cohomology > Section 52.12: Points of the pro-étale site

Lemma 52.12.2. 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 15.85.25) the scheme $\mathop{\rm Spec}(\mathcal{O}_{S, \overline{s}}^{sh})$ is an initial object of the category of pro-étale neighbourhoods of $\overline{s}$. $\square$

The code snippet corresponding to this tag is a part of the file `proetale.tex` and is located in lines 2728–2737 (see updates for more information).

```
\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}
```

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