The Stacks project

Lemma 65.34.6. 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. Then $f$ is locally quasi-finite if and only if $f$ has relative dimension $0$ at each $x \in |X|$.

Proof. Choose a diagram

\[ \xymatrix{ U \ar[r]_ h \ar[d]_ a & V \ar[d] \\ X \ar[r] & Y } \]

where $U$ and $V$ are schemes and the vertical arrows are surjective and ├ętale, see Spaces, Lemma 63.11.6. The definitions imply that $h$ is locally quasi-finite if and only if $f$ is locally quasi-finite, and that $f$ has relative dimension $0$ at all $x \in |X|$ if and only if $h$ has relative dimension $0$ at all $u \in U$. Hence the result follows from the result for $h$ which is Morphisms, Lemma 29.29.5. $\square$


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