The Stacks project

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.

  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 94.19.2.

[1] Theorem 101.21.6 shows that this is equivalent to $\mathcal{X}$ being a Deligne-Mumford stack.

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