Lemma 61.18.2 (0F6B)—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 61.18.2 (cite)

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Lemma 61.18.2. Let $S$ be a scheme and let $\overline{s} : \mathop{\mathrm{Spec}}(k) \to S$ be a geometric point. The category of pro-étale neighbourhoods of $\overline{s}$ is cofiltered.

Proof. The proof is identitical to the proof of Étale Cohomology, Lemma 59.29.4 but using the corresponding facts about weakly étale morphisms proven in More on Morphisms, Lemmas 37.64.5, 37.64.6, and 37.64.13. $\square$

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