The Stacks project

Lemma 61.12.9. Let $S$ be a scheme. Let $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site containing $S$. Both $S_{pro\text{-}\acute{e}tale}$ and $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}$ are sites.

Proof. Let us show that $S_{pro\text{-}\acute{e}tale}$ is a site. It is a category with a given set of families of morphisms with fixed target. Thus we have to show properties (1), (2) and (3) of Sites, Definition 7.6.2. Since $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ is a site, it suffices to prove that given any covering $\{ U_ i \to U\} $ of $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ with $U \in \mathop{\mathrm{Ob}}\nolimits (S_{pro\text{-}\acute{e}tale})$ we also have $U_ i \in \mathop{\mathrm{Ob}}\nolimits (S_{pro\text{-}\acute{e}tale})$. This follows from the definitions as the composition of weakly étale morphisms is weakly étale.

To show that $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}$ is a site, reasoning as above, it suffices to show that the collection of standard pro-étale coverings of affines satisfies properties (1), (2) and (3) of Sites, Definition 7.6.2. This follows from Lemma 61.12.2 and the corresponding result for standard fpqc coverings (Topologies, Lemma 34.9.10). $\square$

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