The Stacks project

Proposition 98.28.1. 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 98.27.9 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 98.5.5.

Choose a presentation $\mathcal{X} = [U/R]$. Let $G \to U$ be the stabilizer group algebraic space of the groupoid $R$. By Lemma 98.5.7 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 98.5.1). 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 64.32.1 we see that $W \subset U$ is dense. Note that $W \subset U$ is $R$-invariant by More on Groupoids in Spaces, Lemma 76.6.2. Hence $W$ corresponds to an open substack $\mathcal{U} \subset \mathcal{X}$ by Properties of Stacks, Lemma 97.9.10. 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 98.24.4 and 98.26.11. $\square$


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