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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