# The Stacks Project

## Tag 0993

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 $\overline{s}$. $\square$

The code snippet corresponding to this tag is a part of the file proetale.tex and is located in lines 2722–2731 (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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