Lemma 72.10.10. The property $\mathcal{P}(f) =$“$f$ is locally of finite presentation” is fpqc local on the base.

Proof. We will use Lemma 72.9.4 to prove this. Assumptions (1) and (2) of that lemma follow from Morphisms of Spaces, Lemma 65.28.4. 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 presentation. We have to show that $f$ is locally of finite presentation. Let $U$ be a scheme and let $U \to X$ be surjective and étale. By Morphisms of Spaces, Lemma 65.28.4 again, it is enough to show that $U \to Z$ is locally of finite presentation. Since $f'$ is locally of finite presentation, 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 presentation. As $\{ Z' \to Z\}$ is an fpqc covering we conclude that $U \to Z$ is locally of finite presentation by Descent, Lemma 35.20.11 as desired. $\square$

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