The Stacks project

Proof. We will use Lemma 74.10.4 to prove this. Assumptions (1) and (2) of that lemma follow from Morphisms of Spaces, Lemma 67.23.4. Let $Z' \to Z$ be a surjective flat morphism of affine schemes over $S$. Let $f : X \to Z$ be a morphism of algebraic spaces, and assume that the base change $f' : Z' \times _ Z X \to Z'$ is locally of finite type. We have to show that $f$ is locally of finite type. Let $U$ be a scheme and let $U \to X$ be surjective and étale. By Morphisms of Spaces, Lemma 67.23.4 again, it is enough to show that $U \to Z$ is locally of finite type. Since $f'$ is locally of finite type, and since $Z' \times _ Z U$ is a scheme étale over $Z' \times _ Z X$ we conclude (by the same lemma again) that $Z' \times _ Z U \to Z'$ is locally of finite type. As $\{ Z' \to Z\} $ is an fpqc covering we conclude that $U \to Z$ is locally of finite type by Descent, Lemma 35.23.10 as desired. $\square$