The Stacks project

Lemma 72.8.3. Let $f : X \to Y$ be a morphism of algebraic spaces. If $f$ is locally of finite presentation, flat, and surjective and $X$ is locally Noetherian, then $Y$ is locally Noetherian.

Proof. Choose a scheme $V$ and a surjective ├ętale morphism $V \to Y$. Choose a scheme $U$ and a surjective ├ętale morphism $U \to X \times _ Y V$. As $f$ is surjective, flat, and locally of finite presentation the morphism of schemes $U \to V$ is surjective, flat, and locally of finite presentation. In this way we reduce the problem to the case of schemes (as being locally Noetherian for $X$ and $Y$ is defined in terms of being locally Noetherian of $U$ and $V$, see Properties of Spaces, Section 64.7). In the case of schemes the result follows from Descent, Lemma 35.13.1. $\square$


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