Lemma 76.35.2 (0A4Y)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 76: More on Morphisms of Spaces
Section 76.35: Applications of Zariski's Main Theorem, I
Lemma 76.35.2 (cite)

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Lemma 76.35.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ . Let $y \in |Y|$ . Assume

$f$ is proper, and
$f$ is quasi-finite at all $x \in |X|$ lying over $y$ (Decent Spaces, Lemma 68.18.10).

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

Proof. By Morphisms of Spaces, Lemma 67.34.7 the set of points at which $f$ is quasi-finite is an open $U \subset X$ . Let $Z = X \setminus U$ . Then $y \not\in f(Z)$ . Since $f$ is proper the set $f(Z) \subset Y$ is closed. Choose any open neighbourhood $V \subset Y$ of $y$ with $Z \cap V = \emptyset$ . Then $f^{-1}(V) \to V$ is locally quasi-finite and proper. Hence $f^{-1}(V) \to V$ is finite by Lemma 76.35.1. $\square$

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