Lemma 61.18.3 (0F6C)—The Stacks project

Loading [MathJax]/extensions/tex2jax.js

Table of contents
Part 3: Topics in Scheme Theory
Chapter 61: Pro-étale Cohomology
Section 61.18: Points of the pro-étale site
Lemma 61.18.3 (cite)

previous lemma
next lemma

numberstags

history
statistics

Lemma 61.18.3. Let $S$ be a scheme. Let $\overline{s}$ be a geometric point of $S$. Let $\mathcal{U} = \{ \varphi _ i : S_ i \to S\} _{i\in I}$ be a pro-étale covering. Then there exist $i \in I$ and geometric point $\overline{s}_ i$ of $S_ i$ mapping to $\overline{s}$.

Proof. Immediate from the fact that $\coprod \varphi _ i$ is surjective and that residue field extensions induced by weakly étale morphisms are separable algebraic (see for example More on Morphisms, Lemma 37.64.11. $\square$

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