The Stacks project

Lemma 33.17.2. Let $f : X \to Y$ be a proper morphism. Let $y \in Y$ be a point such that $\mathcal{O}_{Y, y}$ is Noetherian of dimension $\leq 1$. Assume in addition one of the following conditions is satisfied

  1. for every generic point $\eta $ of an irreducible component of $X$ the field extension $\kappa (\eta )/\kappa (f(\eta ))$ is finite (or algebraic),

  2. for every generic point $\eta $ of an irreducible component of $X$ such that $f(\eta ) \leadsto y$ the field extension $\kappa (\eta )/\kappa (f(\eta ))$ is finite (or algebraic),

  3. $f$ is quasi-finite at every generic point of $X$,

  4. $Y$ is locally Noetherian and $f$ is quasi-finite at a dense set of points of $X$,

  5. add more here.

Then there exists an open neighbourhood $V \subset Y$ of $y$ such that $f^{-1}(V) \to V$ is finite.

Proof. By Lemma 33.17.1 the morphism $f$ is quasi-finite at every point of the fibre $X_ y$. Hence $X_ y$ is a discrete topological space (Morphisms, Lemma 29.20.6). As $f$ is proper the fibre $X_ y$ is quasi-compact, i.e., finite. Thus we can apply Cohomology of Schemes, Lemma 30.21.2 to conclude. $\square$

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