Lemma 99.4.11. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks over the base scheme $S$.

1. If $\mathcal{Y}$ is DM over $S$ and $f$ is DM, then $\mathcal{X}$ is DM over $S$.

2. If $\mathcal{Y}$ is quasi-DM over $S$ and $f$ is quasi-DM, then $\mathcal{X}$ is quasi-DM over $S$.

3. If $\mathcal{Y}$ is separated over $S$ and $f$ is separated, then $\mathcal{X}$ is separated over $S$.

4. If $\mathcal{Y}$ is quasi-separated over $S$ and $f$ is quasi-separated, then $\mathcal{X}$ is quasi-separated over $S$.

5. If $\mathcal{Y}$ is DM and $f$ is DM, then $\mathcal{X}$ is DM.

6. If $\mathcal{Y}$ is quasi-DM and $f$ is quasi-DM, then $\mathcal{X}$ is quasi-DM.

7. If $\mathcal{Y}$ is separated and $f$ is separated, then $\mathcal{X}$ is separated.

8. If $\mathcal{Y}$ is quasi-separated and $f$ is quasi-separated, then $\mathcal{X}$ is quasi-separated.

Proof. Parts (1), (2), (3), and (4) follow immediately from Lemma 99.4.10 and Definition 99.4.2. For (5), (6), (7), and (8) think of $\mathcal{X}$ and $\mathcal{Y}$ as algebraic stacks over $\mathop{\mathrm{Spec}}(\mathbf{Z})$ and apply Lemma 99.4.10. Details omitted. $\square$

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