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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