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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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