A proper morphism is finite in a neighbourhood of a finite fiber.

Lemma 30.21.2. (For a more general version see More on Morphisms, Lemma 37.39.2.) Let $f : X \to S$ be a morphism of schemes. Let $s \in S$. Assume

1. $S$ is locally Noetherian,

2. $f$ is proper, and

3. $f^{-1}(\{ s\} )$ is a finite set.

Then there exists an open neighbourhood $V \subset S$ of $s$ such that $f|_{f^{-1}(V)} : f^{-1}(V) \to V$ is finite.

Proof. The morphism $f$ is quasi-finite at all the points of $f^{-1}(\{ s\} )$ by Morphisms, Lemma 29.19.7. By Morphisms, Lemma 29.53.2 the set of points at which $f$ is quasi-finite is an open $U \subset X$. Let $Z = X \setminus U$. Then $s \not\in f(Z)$. Since $f$ is proper the set $f(Z) \subset S$ is closed. Choose any open neighbourhood $V \subset S$ of $s$ with $Z \cap V = \emptyset$. Then $f^{-1}(V) \to V$ is locally quasi-finite and proper. Hence it is quasi-finite (Morphisms, Lemma 29.19.9), hence has finite fibres (Morphisms, Lemma 29.19.10), hence is finite by Lemma 30.21.1. $\square$

Comment #854 by Olivier BENOIST on

Suggested slogan: A proper morphism is finite in a neighbourhood of a finite fiber.

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