The Stacks project

Lemma 101.17.5. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. If $f$ is locally of finite type and $\mathcal{Y}$ is locally Noetherian, then $\mathcal{X}$ is locally Noetherian.

Proof. Let

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

be a commutative diagram where $U$, $V$ are schemes, $V \to \mathcal{Y}$ is surjective and smooth, and $U \to V \times _\mathcal {Y} \mathcal{X}$ is surjective and smooth. Then $U \to V$ is locally of finite type. If $\mathcal{Y}$ is locally Noetherian, then $V$ is locally Noetherian. By Morphisms, Lemma 29.15.6 we see that $U$ is locally Noetherian, which means that $\mathcal{X}$ is locally Noetherian. $\square$

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