The Stacks project

Definition 95.4.1. 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 95.21.6. 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 95.21.3. Note the similarity to being Deligne-Mumford, which is defined in terms of having an étale covering by a scheme.

Comments (0)

There are also:

  • 2 comment(s) on Section 95.4: Separation axioms

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 04YW. Beware of the difference between the letter 'O' and the digit '0'.