The Stacks project

Lemma 67.23.6. Let $S$ be a scheme. Let $f : X \to Y$, $g : Y \to Z$ be morphisms of algebraic spaces over $S$. If $g \circ f : X \to Z$ is locally of finite type, then $f : X \to Y$ is locally of finite type.

Proof. We can find a diagram

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

where $U$, $V$, $W$ are schemes, the vertical arrows are ├ętale and surjective, see Spaces, Lemma 65.11.6. At this point we can use Lemma 67.23.4 and Morphisms, Lemma 29.15.8 to conclude. $\square$

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