Lemma 28.2.5 (054E)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 28: Properties of Schemes
Section 28.2: Constructible sets
Lemma 28.2.5 (cite)

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Lemma 28.2.5. Let $X$ be a quasi-compact and quasi-separated scheme. Any locally constructible subset of $X$ is constructible.

Proof. As $X$ is quasi-compact we can choose a finite affine open covering $X = V_1 \cup \ldots \cup V_ m$ . As $X$ is quasi-separated each $V_ i$ is retrocompact in $X$ by Lemma 28.2.3. Hence by Topology, Lemma 5.15.6 we see that $E \subset X$ is constructible in $X$ if and only if $E \cap V_ j$ is constructible in $V_ j$ . Thus we win by Lemma 28.2.1. $\square$

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