The Stacks project

Lemma 29.20.14. Let $f : X \to S$ be a morphism of schemes of finite type. Let $s \in S$. There are at most finitely many points of $X$ lying over $s$ at which $f$ is quasi-finite.

Proof. The fibre $X_ s$ is a scheme of finite type over a field, hence Noetherian (Lemma 29.15.6). Hence the topology on $X_ s$ is Noetherian (Properties, Lemma 28.5.5) and can have at most a finite number of isolated points (by elementary topology). Thus our lemma follows from Lemma 29.20.6. $\square$

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