The Stacks project

Lemma 100.4.13. Let $\mathcal{X}$ be an algebraic stack over the base scheme $S$.

  1. $\mathcal{X}$ is DM $\Leftrightarrow $ $\mathcal{X}$ is DM over $S$.

  2. $\mathcal{X}$ is quasi-DM $\Leftrightarrow $ $\mathcal{X}$ is quasi-DM over $S$.

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

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

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

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

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

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

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

Proof. Parts (5), (6), (7), and (8) follow immediately from Lemma 100.4.12 and Spaces, Definition 64.13.2. To prove (3) and (4) think of $X$ and $Y$ as algebraic stacks over $\mathop{\mathrm{Spec}}(\mathbf{Z})$ and apply Lemma 100.4.12. Similarly, to prove (1) and (2), think of $\mathcal{X}$ as an algebraic stack over $\mathop{\mathrm{Spec}}(\mathbf{Z})$ consider the morphisms

\[ \mathcal{X} \longrightarrow \mathcal{X} \times _ S \mathcal{X} \longrightarrow \mathcal{X} \times _{\mathop{\mathrm{Spec}}(\mathbf{Z})} \mathcal{X} \]

Both arrows are representable by algebraic spaces. The second arrow is unramified and locally quasi-finite as the base change of the immersion $\Delta _{S/\mathbf{Z}}$. Hence the composition is unramified (resp. locally quasi-finite) if and only if the first arrow is unramified (resp. locally quasi-finite), see Morphisms of Spaces, Lemmas 66.38.3 and 66.38.11 (resp. Morphisms of Spaces, Lemmas 66.27.3 and 66.27.8). $\square$

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