The Stacks project

Proposition 101.29.1. Let $\mathcal{X}$ be a reduced algebraic stack such that $\mathcal{I}_\mathcal {X} \to \mathcal{X}$ is quasi-compact. Then there exists a dense open substack $\mathcal{U} \subset \mathcal{X}$ which is a gerbe.

Proof. According to Proposition 101.28.9 it is enough to find a dense open substack $\mathcal{U}$ such that $\mathcal{I}_\mathcal {U} \to \mathcal{U}$ is flat and locally of finite presentation. Note that $\mathcal{I}_\mathcal {U} = \mathcal{I}_\mathcal {X} \times _\mathcal {X} \mathcal{U}$, see Lemma 101.5.5.

Choose a presentation $\mathcal{X} = [U/R]$. Let $G \to U$ be the stabilizer group algebraic space of the groupoid $R$. By Lemma 101.5.7 we see that $G \to U$ is the base change of $\mathcal{I}_\mathcal {X} \to \mathcal{X}$ hence quasi-compact (by assumption) and locally of finite type (by Lemma 101.5.1). Let $W \subset U$ be the largest open (possibly empty) subscheme such that the restriction $G_ W \to W$ is flat and locally of finite presentation (we omit the proof that $W$ exists; hint: use that the properties are local). By Morphisms of Spaces, Proposition 67.32.1 we see that $W \subset U$ is dense. Note that $W \subset U$ is $R$-invariant by More on Groupoids in Spaces, Lemma 79.6.2. Hence $W$ corresponds to an open substack $\mathcal{U} \subset \mathcal{X}$ by Properties of Stacks, Lemma 100.9.11. Since $|U| \to |\mathcal{X}|$ is open and $|W| \subset |U|$ is dense we conclude that $\mathcal{U}$ is dense in $\mathcal{X}$. Finally, the morphism $\mathcal{I}_\mathcal {U} \to \mathcal{U}$ is flat and locally of finite presentation because the base change by the surjective smooth morphism $W \to \mathcal{U}$ is the morphism $G_ W \to W$ which is flat and locally of finite presentation by construction. See Lemmas 101.25.4 and 101.27.11. $\square$

Comments (0)

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Your email address will not be published.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 06RC. Beware of the difference between the letter 'O' and the digit '0'.