Definition 99.4.2. Let $\mathcal{X}$ be an algebraic stack over the base scheme $S$. Denote $p : \mathcal{X} \to S$ the structure morphism.

1. We say $\mathcal{X}$ is DM over $S$ if $p : \mathcal{X} \to S$ is DM.

2. We say $\mathcal{X}$ is quasi-DM over $S$ if $p : \mathcal{X} \to S$ is quasi-DM.

3. We say $\mathcal{X}$ is separated over $S$ if $p : \mathcal{X} \to S$ is separated.

4. We say $\mathcal{X}$ is quasi-separated over $S$ if $p : \mathcal{X} \to S$ is quasi-separated.

5. We say $\mathcal{X}$ is DM if $\mathcal{X}$ is DM1 over $\mathop{\mathrm{Spec}}(\mathbf{Z})$.

6. We say $\mathcal{X}$ is quasi-DM if $\mathcal{X}$ is quasi-DM over $\mathop{\mathrm{Spec}}(\mathbf{Z})$.

7. We say $\mathcal{X}$ is separated if $\mathcal{X}$ is separated over $\mathop{\mathrm{Spec}}(\mathbf{Z})$.

8. We say $\mathcal{X}$ is quasi-separated if $\mathcal{X}$ is quasi-separated over $\mathop{\mathrm{Spec}}(\mathbf{Z})$.

In the last 4 definitions we view $\mathcal{X}$ as an algebraic stack over $\mathop{\mathrm{Spec}}(\mathbf{Z})$ via Algebraic Stacks, Definition 92.19.2.

[1] Theorem 99.21.6 shows that this is equivalent to $\mathcal{X}$ being a Deligne-Mumford stack.

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