Lemma 66.21.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume that $f$ is quasi-separated and locally of finite type and $Y$ quasi-separated. Let $y \in |Y|$ be a point of codimension $0$ on $Y$. The following are equivalent:

the set $f^{-1}(\{ y\} )$ is finite,

the space $|X_ k|$ is finite where $\mathop{\mathrm{Spec}}(k) \to Y$ represents $y$,

there exist open subspaces $X' \subset X$ and $Y' \subset Y$ with $f(X') \subset Y'$, $y \in |Y'|$, and $f^{-1}(\{ y\} ) \subset |X'|$ such that $f|_{X'} : X' \to Y'$ is finite.

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