The Stacks project

Lemma 74.11.24. The properties $\mathcal{P}(f) =$“$f$ is locally quasi-finite” and $\mathcal{P}(f) =$“$f$ is quasi-finite” are fpqc local on the base.

Proof. We have already seen that “quasi-compact” is fpqc local on the base, see Lemma 74.11.1. Hence it is enough to prove the lemma for “locally quasi-finite”. We will use Lemma 74.10.4 to prove this. Assumptions (1) and (2) of that lemma follow from Morphisms of Spaces, Lemma 67.27.6. 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 quasi-finite. We have to show that $f$ is locally quasi-finite. Let $U$ be a scheme and let $U \to X$ be surjective and étale. By Morphisms of Spaces, Lemma 67.27.6 again, it is enough to show that $U \to Z$ is locally quasi-finite. Since $f'$ is locally quasi-finite, 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 quasi-finite. As $\{ Z' \to Z\} $ is an fpqc covering we conclude that $U \to Z$ is locally quasi-finite by Descent, Lemma 35.23.24 as desired. $\square$

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