Lemma 61.12.4 (098C)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 61: Pro-étale Cohomology
Section 61.12: The pro-étale site
Lemma 61.12.4 (cite)

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Lemma 61.12.4. Let $T$ be a scheme.

If $T' \to T$ is an isomorphism then $\{ T' \to T\}$ is a pro-étale covering of $T$ .
If $\{ T_ i \to T\} _{i\in I}$ is a pro-étale covering and for each $i$ we have a pro-étale covering $\{ T_{ij} \to T_ i\} _{j\in J_ i}$ , then $\{ T_{ij} \to T\} _{i \in I, j\in J_ i}$ is a pro-étale covering.
If $\{ T_ i \to T\} _{i\in I}$ is a pro-étale covering and $T' \to T$ is a morphism of schemes then $\{ T' \times _ T T_ i \to T'\} _{i\in I}$ is a pro-étale covering.

Proof. This follows from the fact that composition and base changes of weakly étale morphisms are weakly étale (More on Morphisms, Lemmas 37.64.5 and 37.64.6), Lemma 61.12.2, and the corresponding results for fpqc coverings, see Topologies, Lemma 34.9.8. $\square$

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