Lemma 61.18.3 (0F6C)—The Stacks project

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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)

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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. $\square$

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