The Stacks project

Lemma 66.8.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E \subset |X|$ be a subset. The following are equivalent

  1. for every étale morphism $U \to X$ where $U$ is a scheme the inverse image of $E$ in $U$ is a locally constructible subset of $U$,

  2. for every étale morphism $U \to X$ where $U$ is an affine scheme the inverse image of $E$ in $U$ is a constructible subset of $U$,

  3. for some surjective étale morphism $U \to X$ where $U$ is a scheme the inverse image of $E$ in $U$ is a locally constructible subset of $U$.

Proof. By Properties, Lemma 28.2.1 we see that (1) and (2) are equivalent. It is immediate that (1) implies (3). Thus we assume we have a surjective étale morphism $\varphi : U \to X$ where $U$ is a scheme such that $\varphi ^{-1}(E)$ is locally constructible. Let $\varphi ' : U' \to X$ be another étale morphism where $U'$ is a scheme. Then we have

\[ E'' = \text{pr}_1^{-1}(\varphi ^{-1}(E)) = \text{pr}_2^{-1}((\varphi ')^{-1}(E)) \]

where $\text{pr}_1 : U \times _ X U' \to U$ and $\text{pr}_2 : U \times _ X U' \to U'$ are the projections. By Morphisms, Lemma 29.22.1 we see that $E''$ is locally constructible in $U \times _ X U'$. Let $W' \subset U'$ be an affine open. Since $\text{pr}_2$ is étale and hence open, we can choose a quasi-compact open $W'' \subset U \times _ X U'$ with $\text{pr}_2(W'') = W'$. Then $\text{pr}_2|_{W''} : W'' \to W'$ is quasi-compact. We have $W' \cap (\varphi ')^{-1}(E) = \text{pr}_2(E'' \cap W'')$ as $\varphi $ is surjective, see Lemma 66.4.3. Thus $W' \cap (\varphi ')^{-1}(E) = \text{pr}_2(E'' \cap W'')$ is locally constructible by Morphisms, Theorem 29.22.3 as desired. $\square$

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