Lemma 70.19.2. Notation and assumptions as in Lemma 70.19.1. Let g : X \to W correspond to h : Y \to V via the equivalence. Then g is quasi-compact, quasi-separated, separated, locally of finite presentation, of finite presentation, locally of finite type, of finite type, proper, integral, finite, and add more here if and only if h is so.
Proof. If g is quasi-compact, quasi-separated, separated, locally of finite presentation, of finite presentation, locally of finite type, of finite type, proper, finite, so is h as a base change of g by Morphisms of Spaces, Lemmas 67.8.4, 67.4.4, 67.28.3, 67.23.3, 67.40.3, 67.45.5. Conversely, let P be a property of morphisms of algebraic spaces which is étale local on the base and which holds for the identity morphism of any algebraic space. Since \{ W^0 \to W, V \to W\} is an étale covering, to prove that g has P it suffices to show that h has P. Thus we conclude using Morphisms of Spaces, Lemmas 67.8.8, 67.4.12, 67.28.4, 67.23.4, 67.40.2, 67.45.3. \square
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