Lemma 67.29.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $E \subset |Y|$ be a subset. If $E$ is étale locally constructible in $Y$, then $f^{-1}(E)$ is étale locally constructible in $X$.
67.29 Constructible sets
This section is the continuation of Properties of Spaces, Section 66.8.
Proof. Choose a scheme $V$ and a surjective étale morphism $\varphi : V \to Y$. Choose a scheme $U$ and a surjective étale morphism $U \to V \times _ Y X$. Then $U \to X$ is surjective étale and the inverse image of $f^{-1}(E)$ in $U$ is the inverse image of $\varphi ^{-1}(E)$ by $U \to V$. Thus the lemma follows from the case of schemes for $U \to V$ (Morphisms, Lemma 29.22.1) and the definition (Properties of Spaces, Definition 66.8.2). $\square$
Theorem 67.29.2 (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 66.8.2. 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 66.6.3). By Properties of Spaces, Lemma 66.4.3 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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