Lemma 65.28.9. Let $S$ be a scheme. Let $f : X \to Y$ and $Y \to Z$ be morphisms of algebraic spaces over $S$. If $X$ is locally of finite presentation over $Z$, and $Y$ is locally of finite type over $Z$, then $f$ is locally of finite presentation.

Proof. Choose a scheme $W$ and a surjective étale morphism $W \to Z$. Then choose a scheme $V$ and a surjective étale morphism $V \to W \times _ Z Y$. Finally choose a scheme $U$ and a surjective étale morphism $U \to V \times _ Y X$. By definition $U$ is locally of finite presentation over $W$ and $V$ is locally of finite type over $W$. By Morphisms, Lemma 29.21.11 the morphism $U \to V$ is locally of finite presentation. Hence $f$ is locally of finite presentation. $\square$

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