Lemma 67.34.8. Let $S$ be a scheme. Consider a cartesian diagram

where $X \to Y$ is a morphism of algebraic spaces over $S$ which is locally of finite type and where $k$ is a field over $S$. Let $z \in |F|$ be such that $\dim _ z(F) = 0$. Then, after replacing $X$ by an open subspace containing $p(z)$, the morphism

is locally quasi-finite.

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