Lemma 35.23.24 (02VI)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 35: Descent
Section 35.23: Properties of morphisms local in the fpqc topology on the target
Lemma 35.23.24 (cite)

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Lemma 35.23.24. 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. Let $f : X \to S$ be a morphism of schemes, and let $\{ S_ i \to S\} $ be an fpqc covering such that each base change $f_ i : X_ i \to S_ i$ is locally quasi-finite. We have already seen (Lemma 35.23.10) that “locally of finite type” is fpqc local on the base, and hence we see that $f$ is locally of finite type. Then it follows from Morphisms, Lemma 29.20.13 that $f$ is locally quasi-finite. The quasi-finite case follows as we have already seen that “quasi-compact” is fpqc local on the base (Lemma 35.23.1). $\square$

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