The Stacks project

Lemma 64.28.5. A morphism which is locally of finite presentation is locally of finite type. A morphism of finite presentation is of finite type.

Proof. Let $f : X \to Y$ be a morphism of algebraic spaces which is locally of finite presentation. This means there exists a diagram as in Lemma 64.22.1 with $h$ locally of finite presentation and surjective vertical arrow $a$. By Morphisms, Lemma 29.20.8 $h$ is locally of finite type. Hence $X \to Y$ is locally of finite type by definition. If $f$ is of finite presentation then it is quasi-compact and it follows that $f$ is of finite type. $\square$


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