The Stacks project

Lemma 68.21.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is locally of finite type. Let $X^0 \subset |X|$, resp. $Y^0 \subset |Y|$ denote the set of codimension $0$ points of $X$, resp. $Y$. Let $y \in Y^0$. The following are equivalent

  1. $f^{-1}(\{ y\} ) \subset X^0$,

  2. $f$ is quasi-finite at all points lying over $y$,

  3. $f$ is quasi-finite at all $x \in X^0$ lying over $y$.

Proof. Let $V$ be a scheme and let $V \to Y$ be a surjective ├ętale morphism. Let $U$ be a scheme and let $U \to V \times _ Y X$ be a surjective ├ętale morphism. Then $f$ is quasi-finite at the image $x$ of a point $u \in U$ if and only if $U \to V$ is quasi-finite at $u$. Moreover, $x \in X^0$ if and only if $u$ is the generic point of an irreducible component of $U$ (Properties of Spaces, Lemma 66.11.1). Thus the lemma reduces to the case of the morphism $U \to V$, i.e., to Morphisms, Lemma 29.51.4. $\square$

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