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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