The Stacks project

Lemma 67.27.8. Let $S$ be a scheme. Let $X \to Y \to Z$ be morphisms of algebraic spaces over $S$. If $X \to Z$ is locally quasi-finite, then $X \to Y$ is locally quasi-finite.

Proof. Choose a commutative diagram

\[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \ar[r] & W \ar[d] \\ X \ar[r] & Y \ar[r] & Z } \]

with vertical arrows ├ętale and surjective. (See Spaces, Lemma 65.11.6.) Apply Morphisms, Lemma 29.20.17 to the top row. $\square$

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