The Stacks project

Lemma 72.3.2. 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

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

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

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

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

  5. add more here.

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

Proof. By Lemma 72.3.1 the morphism $f$ is quasi-finite at every point lying over $y$. Let $\overline{y} : \mathop{\mathrm{Spec}}(k) \to Y$ be a geometric point lying over $y$. Then $|X_{\overline{y}}|$ is a discrete space (Decent Spaces, Lemma 68.18.10). Since $X_{\overline{y}}$ is quasi-compact as $f$ is proper we conclude that $|X_{\overline{y}}|$ is finite. Thus we can apply Cohomology of Spaces, Lemma 69.23.2 to conclude. $\square$

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