The Stacks project

Lemma 68.23.2. (For a more general version see More on Morphisms of Spaces, Lemma 75.35.2). Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\overline{y}$ be a geometric point of $Y$. Assume

  1. $Y$ is locally Noetherian,

  2. $f$ is proper, and

  3. $|X_{\overline{y}}|$ is finite.

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

Proof. The morphism $f$ is quasi-finite at all the geometric points of $X$ lying over $\overline{y}$ by Morphisms of Spaces, Lemma 66.34.8. By Morphisms of Spaces, Lemma 66.34.7 the set of points at which $f$ is quasi-finite is an open subspace $U \subset X$. Let $Z = X \setminus U$. Then $\overline{y} \not\in f(Z)$. Since $f$ is proper the set $f(Z) \subset Y$ is closed. Choose any open neighbourhood $V \subset Y$ of $\overline{y}$ with $Z \cap V = \emptyset $. Then $f^{-1}(V) \to V$ is locally quasi-finite and proper. Hence $f^{-1}(V) \to V$ has discrete fibres $X_ k$ (Morphisms of Spaces, Lemma 66.27.5) which are quasi-compact hence finite. Thus $f^{-1}(V) \to V$ is finite by Lemma 68.23.1. $\square$

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