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The Stacks project

Definition 101.4.1. 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 101.21.6. 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 101.21.3. Note the similarity to being Deligne-Mumford, which is defined in terms of having an étale covering by a scheme.

Comments (2)

Comment #5789 by Leo on

It may be worth pointing out that Laumon-Moret-Bailly seem to define quasiseparated to have separated diagonal (Definition 4.1). Some other places in the literature follow this convention, for example in Olsson's Log Geometry and Algebraic Stacks Remark 3.17, he explicitly states is not quasiseparated even though his Theorem 3.2 shows that its diagonal is quasicompact and quasiseparated (a consequence of his notion of "locally separated," which is the stacks project's notion + quasiseparated).

Comment #5804 by on

OK, I think this isn't necessary because we mention in the introduction 94.1 to this chapter that we are working without any separation assumptions. Moreover, just the fact that you are referring to Definition 4.1 of their book (which is their definition of algebraic stacks and not the definition of a quasi-separated morphism of algebraic stacks) seems to suggest that things are quite a bit different.

There are also:

  • 2 comment(s) on Section 101.4: Separation axioms

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View Definition 101.4.1 as pdf