The Stacks project

Theorem 66.29.2 (Chevalley's Theorem). Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is quasi-compact and locally of finite presentation. Then the image of every étale locally constructible subset of $|X|$ is an étale locally constructible subset of $|Y|$.

Proof. Let $E \subset |X|$ be étale locally constructible. Let $V \to Y$ be an étale morphism with $V$ affine. It suffices to show that the inverse image of $f(E)$ in $V$ is constructible, see Properties of Spaces, Definition 65.8.2. Since $f$ is quasi-compact $V \times _ Y X$ is a quasi-compact algebraic space. Choose an affine scheme $U$ and a surjective étale morphism $U \to V \times _ Y X$ (Properties of Spaces, Lemma 65.6.3). By Properties of Spaces, Lemma 65.4.3 the inverse image of $f(E)$ in $V$ is the image under $U \to V$ of the inverse image of $E$ in $U$. Thus the result follows from the case of schemes, see Morphisms, Lemma 29.22.2. $\square$


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