The Stacks project

Lemma 101.4.11. 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.