The Stacks project

Lemma 69.19.2. Notation and assumptions as in Lemma 69.19.1. 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 66.8.4, 66.4.4, 66.28.3, 66.23.3, 66.40.3, 66.45.5. Conversely, let $P$ be a property of morphisms of algebraic spaces which is ├ętale 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 ├ętale covering, to prove that $g$ has $P$ it suffices to show that $h$ has $P$. Thus we conclude using Morphisms of Spaces, Lemmas 66.8.8, 66.4.12, 66.28.4, 66.23.4, 66.40.2, 66.45.3. $\square$


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