The Stacks project

Lemma 29.20.9. Let $f : X \to S$ be a morphism of schemes. Then $f$ is quasi-finite if and only if $f$ is locally quasi-finite and quasi-compact.

Proof. Assume $f$ is quasi-finite. It is quasi-compact by Definition 29.15.1. Let $x \in X$. We see that $f$ is quasi-finite at $x$ by Lemma 29.20.6. Hence $f$ is quasi-compact and locally quasi-finite.

Assume $f$ is quasi-compact and locally quasi-finite. Then $f$ is of finite type. Let $x \in X$ be a point. By Lemma 29.20.6 we see that $x$ is an isolated point of its fibre. The lemma is proved. $\square$

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