Definition 101.4.2. Let $\mathcal{X}$ be an algebraic stack over the base scheme $S$. Denote $p : \mathcal{X} \to S$ the structure morphism.

We say $\mathcal{X}$ is

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

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

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

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

*DM*if $\mathcal{X}$ is DM^{1}over $\mathop{\mathrm{Spec}}(\mathbf{Z})$.We say $\mathcal{X}$ is

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

*separated*if $\mathcal{X}$ is separated over $\mathop{\mathrm{Spec}}(\mathbf{Z})$.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 94.19.2.

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