The Stacks project

Lemma 101.8.2. Let $j : \mathcal{X} \to \mathcal{Y}$ be an immersion of algebraic stacks.

  1. If $\mathcal{Y}$ is locally Noetherian, then $\mathcal{X}$ is locally Noetherian and $j$ is quasi-compact.

  2. If $\mathcal{Y}$ is Noetherian, then $\mathcal{X}$ is Noetherian.

Proof. Choose a scheme $V$ and a surjective smooth morphism $V \to \mathcal{Y}$. Then $U = \mathcal{X} \times _\mathcal {Y} V$ is a scheme and $V \to U$ is an immersion, see Properties of Stacks, Definition 100.9.1. Recall that $\mathcal{Y}$ is locally Noetherian if and only if $V$ is locally Noetherian. In this case $U$ is locally Noetherian too (Morphisms, Lemmas 29.15.5 and 29.15.6) and $U \to V$ is quasi-compact (Properties, Lemma 28.5.3). This shows that $j$ is quasi-compact (Lemma 101.7.10) and that $\mathcal{X}$ is locally Noetherian. Finally, if $\mathcal{Y}$ is Noetherian, then we see from the above that $\mathcal{X}$ is quasi-compact and locally Noetherian. To finish the proof observe that $j$ is separated and hence $\mathcal{X}$ is quasi-separated because $\mathcal{Y}$ is so by Lemma 101.4.11. $\square$

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