Lemma 70.3.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is proper and $Y$ is locally Noetherian. Let $y \in Y$ be a point of codimension $\leq 1$ in $Y$. Let $X^0 \subset |X|$ be the set of points of codimension $0$ on $X$. Assume in addition one of the following conditions is satisfied

for every $x \in X^0$ the transcendence degree of $x/f(x)$ is $0$,

for every $x \in X^0$ with $f(x) \leadsto y$ the transcendence degree of $x/f(x)$ is $0$,

$f$ is quasi-finite at every $x \in X^0$,

$f$ is quasi-finite at a dense set of points of $|X|$,

add more here.

Then there exists an open subspace $Y' \subset Y$ containing $y$ such that $Y' \times _ Y X \to Y'$ is finite.

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