Lemma 61.18.4 (0993)—The Stacks project

Loading web-font TeX/Main/Regular

Table of contents
Part 3: Topics in Scheme Theory
Chapter 61: Pro-étale Cohomology
Section 61.18: Points of the pro-étale site
Lemma 61.18.4 (cite)

previous lemma

numberstags

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

Comments (0)

There are also:

7 comment(s) on Section 61.18: Points of the pro-étale site

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$ ). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Comments (0)

Post a comment