The Stacks project

Lemma 28.2.1. Let $X$ be a scheme. A subset $E$ of $X$ is locally constructible in $X$ if and only if $E \cap U$ is constructible in $U$ for every affine open $U$ of $X$.

Proof. Assume $E$ is locally constructible. Then there exists an open covering $X = \bigcup U_ i$ such that $E \cap U_ i$ is constructible in $U_ i$ for each $i$. Let $V \subset X$ be any affine open. We can find a finite open affine covering $V = V_1 \cup \ldots \cup V_ m$ such that for each $j$ we have $V_ j \subset U_ i$ for some $i = i(j)$. By Topology, Lemma 5.15.4 we see that each $E \cap V_ j$ is constructible in $V_ j$. Since the inclusions $V_ j \to V$ are quasi-compact (see Schemes, Lemma 26.19.2) we conclude that $E \cap V$ is constructible in $V$ by Topology, Lemma 5.15.6. The converse implication is immediate. $\square$

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