The Stacks project

Lemma 64.28.7. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.

  1. If $Y$ is locally Noetherian and $f$ locally of finite type then $f$ is locally of finite presentation.

  2. If $Y$ is locally Noetherian and $f$ of finite type and quasi-separated then $f$ is of finite presentation.

Proof. Assume $f : X \to Y$ locally of finite type and $Y$ locally Noetherian. This means there exists a diagram as in Lemma 64.22.1 with $h$ locally of finite type and surjective vertical arrow $a$. By Morphisms, Lemma 29.20.9 $h$ is locally of finite presentation. Hence $X \to Y$ is locally of finite presentation by definition. This proves (1). If $f$ is of finite type and quasi-separated then it is also quasi-compact and quasi-separated and (2) follows immediately. $\square$


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