The Stacks project

Lemma 65.28.6. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is of finite presentation and $Y$ is Noetherian, then $X$ is Noetherian.

Proof. Assume $f$ is of finite presentation and $Y$ Noetherian. By Lemmas 65.28.5 and 65.23.5 we see that $X$ is locally Noetherian. As $f$ is quasi-compact and $Y$ is quasi-compact we see that $X$ is quasi-compact. As $f$ is of finite presentation it is quasi-separated (see Definition 65.28.1) and as $Y$ is Noetherian it is quasi-separated (see Properties of Spaces, Definition 64.24.1). Hence $X$ is quasi-separated by Lemma 65.4.9. Hence we have checked all three conditions of Properties of Spaces, Definition 64.24.1 and we win. $\square$


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