Lemma 100.4.3. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks.

1. If $f$ is separated, then $f$ is quasi-separated.

2. If $f$ is DM, then $f$ is quasi-DM.

3. If $f$ is representable by algebraic spaces, then $f$ is DM.

Proof. To see (1) note that a proper morphism of algebraic spaces is quasi-compact and quasi-separated, see Morphisms of Spaces, Definition 66.40.1. To see (2) note that an unramified morphism of algebraic spaces is locally quasi-finite, see Morphisms of Spaces, Lemma 66.38.7. Finally (3) follows from Lemma 100.3.4. $\square$

## Comments (0)

There are also:

• 2 comment(s) on Section 100.4: 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Your email address will not be published.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 050E. Beware of the difference between the letter 'O' and the digit '0'.