The Stacks project

Lemma 59.72.2. Let $f: X \to Y$ be a morphism of schemes which is quasi-compact, quasi-separated, and locally of finite type. If $\eta $ is a generic point of an irreducible component of $Y$ such that $f^{-1}(\eta )$ is finite, then there exists an open $V \subset Y$ containing $\eta $ such that $f^{-1}(V) \to V$ is finite.

Proof. This is Morphisms, Lemma 29.51.1. $\square$

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