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 DM1 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.
Comments (0)
There are also: