Lemma 61.25.3 (09AA)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 61: Pro-étale Cohomology
Section 61.25: Closed immersions and pro-étale sites
Lemma 61.25.3 (cite)

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Lemma 61.25.3. Let $i : Z \to X$ be a closed immersion of schemes. Then

$i_{pro\text{-}\acute{e}tale}^{-1}$ commutes with limits,
$i_{{pro\text{-}\acute{e}tale}, *}$ is fully faithful, and
$i_{pro\text{-}\acute{e}tale}^{-1}i_{{pro\text{-}\acute{e}tale}, *} \cong \text{id}_{\mathop{\mathit{Sh}}\nolimits (Z_{pro\text{-}\acute{e}tale})}$.

Proof. Assertions (2) and (3) are equivalent by Sites, Lemma 7.41.1. Parts (1) and (3) are (Zariski) local on $X$, hence we may assume that $X$ is affine. In this case the result follows from Lemma 61.25.2. $\square$

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