The Stacks project

Lemma 61.18.4. In the situation above the scheme $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh})$ is an object of $X_{pro\text{-}\acute{e}tale}$ and there is a canonical isomorphism

\[ \mathcal{F}(\mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh})) = \mathcal{F}_{\overline{s}} \]

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 37.64.11. The second statement follows as by Olivier's theorem (More on Algebra, Theorem 15.104.24) the scheme $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh})$ is an initial object of the category of pro-étale neighbourhoods of $\overline{s}$. $\square$

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