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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