The Stacks project

Lemma 35.31.2. The property $\mathcal{P}(f)=$“$f$ is locally quasi-finite” is étale local on the source.

Proof. We are going to use Lemma 35.26.4. By Morphisms, Lemma 29.20.11 the property of being locally quasi-finite is local for Zariski on source and target. By Morphisms, Lemmas 29.20.12 and 29.36.6 we see the precomposition of a locally quasi-finite morphism by an étale morphism is locally quasi-finite. Finally, suppose that $X \to Y$ is a morphism of affine schemes and that $X' \to X$ is a surjective étale morphism of affine schemes such that $X' \to Y$ is locally quasi-finite. Then $X' \to Y$ is of finite type, and by Lemma 35.14.2 we see that $X \to Y$ is of finite type also. Moreover, by assumption $X' \to Y$ has finite fibres, and hence $X \to Y$ has finite fibres also. We conclude that $X \to Y$ is quasi-finite by Morphisms, Lemma 29.20.10. This proves the last assumption of Lemma 35.26.4 and finishes the proof. $\square$

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