Lemma 61.18.1 (0992)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 61: Pro-étale Cohomology
Section 61.18: Points of the pro-étale site
Lemma 61.18.1 (cite)

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Lemma 61.18.1. Let $S$ be a scheme. The pro-étale sites $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ , $S_{pro\text{-}\acute{e}tale}$ , $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ , $S_{affine, {pro\text{-}\acute{e}tale}}$ , and $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}$ have enough points.

Proof. The big pro-étale topos of $S$ is equivalent to the topos defined by $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}$ , see Lemma 61.12.11. The topos of sheaves on $S_{pro\text{-}\acute{e}tale}$ is equivalent to the topos associated to $S_{affine, {pro\text{-}\acute{e}tale}}$ , see Lemma 61.12.20. The result for the sites $(\textit{Aff}/S)_{pro\text{-}\acute{e}tale}$ and $S_{affine, {pro\text{-}\acute{e}tale}}$ follows immediately from Deligne's result Sites, Lemma 7.39.4. The case $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ is handled because it is equal to $(\mathit{Sch}/\mathop{\mathrm{Spec}}(\mathbf{Z}))_{pro\text{-}\acute{e}tale}$ . $\square$

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