Lemma 61.25.6 (09BM)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 61: Pro-étale Cohomology
Section 61.25: Closed immersions and pro-étale sites
Lemma 61.25.6 (cite)

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Lemma 61.25.6. Let $i : Z \to X$ be a closed immersion of schemes. If $X \setminus i(Z)$ is a retrocompact open of $X$ , then $i_{{pro\text{-}\acute{e}tale}, *}$ is exact.

Proof. The question is local on $X$ hence we may assume $X$ is affine. Say $X = \mathop{\mathrm{Spec}}(A)$ and $Z = \mathop{\mathrm{Spec}}(A/I)$ . There exist $f_1, \ldots , f_ r \in I$ such that $Z = V(f_1, \ldots , f_ r)$ set theoretically, see Algebra, Lemma 10.29.1. By Lemma 61.25.4 we may assume that $Z = \mathop{\mathrm{Spec}}(A/(f_1, \ldots , f_ r))$ . In this case the functor $i_{{pro\text{-}\acute{e}tale}, *}$ is exact by Lemma 61.24.1. $\square$

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