Definition 98.4.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks.

1. We say $f$ is DM if $\Delta _ f$ is unramified1.

2. We say $f$ is quasi-DM if $\Delta _ f$ is locally quasi-finite2.

3. We say $f$ is separated if $\Delta _ f$ is proper.

4. We say $f$ is quasi-separated if $\Delta _ f$ is quasi-compact and quasi-separated.

[1] The letters DM stand for Deligne-Mumford. If $f$ is DM then given any scheme $T$ and any morphism $T \to \mathcal{Y}$ the fibre product $\mathcal{X}_ T = \mathcal{X} \times _\mathcal {Y} T$ is an algebraic stack over $T$ whose diagonal is unramified, i.e., $\mathcal{X}_ T$ is DM. This implies $\mathcal{X}_ T$ is a Deligne-Mumford stack, see Theorem 98.21.6. In other words a DM morphism is one whose “fibres” are Deligne-Mumford stacks. This hopefully at least motivates the terminology.
[2] If $f$ is quasi-DM, then the “fibres” $\mathcal{X}_ T$ of $\mathcal{X} \to \mathcal{Y}$ are quasi-DM. An algebraic stack $\mathcal{X}$ is quasi-DM exactly if there exists a scheme $U$ and a surjective flat morphism $U \to \mathcal{X}$ of finite presentation which is locally quasi-finite, see Theorem 98.21.3. Note the similarity to being Deligne-Mumford, which is defined in terms of having an étale covering by a scheme.

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