The Stacks project

Lemma 67.23.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally of finite type and $Y$ is locally Noetherian, then $X$ is locally Noetherian.

Proof. Let

\[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } \]

be a commutative diagram where $U$, $V$ are schemes and the vertical arrows are surjective ├ętale. If $f$ is locally of finite type, then $U \to V$ is locally of finite type. If $Y$ is locally Noetherian, then $V$ is locally Noetherian. By Morphisms, Lemma 29.15.6 we see that $U$ is locally Noetherian, which means that $X$ is locally Noetherian. $\square$

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