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The Stacks project

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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