The Stacks project

Lemma 98.27.12. Let $\mathcal{Z}$ be a reduced, locally Noetherian algebraic stack such that $|\mathcal{Z}|$ is a singleton. Then $\mathcal{Z}$ is a gerbe over a reduced, locally Noetherian algebraic space $Z$ with $|Z|$ a singleton.

Proof. By Properties of Stacks, Lemma 97.11.3 there exists a surjective, flat, locally finitely presented morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{Z}$ where $k$ is a field. Then $\mathcal{I}_ Z \times _\mathcal {Z} \mathop{\mathrm{Spec}}(k) \to \mathop{\mathrm{Spec}}(k)$ is representable by algebraic spaces and locally of finite type (as a base change of $\mathcal{I}_\mathcal {Z} \to \mathcal{Z}$, see Lemmas 98.5.1 and 98.17.3). Therefore it is locally of finite presentation, see Morphisms of Spaces, Lemma 64.28.7. Of course it is also flat as $k$ is a field. Hence we may apply Lemmas 98.24.4 and 98.26.11 to see that $\mathcal{I}_\mathcal {Z} \to \mathcal{Z}$ is flat and locally of finite presentation. We conclude that $\mathcal{Z}$ is a gerbe by Proposition 98.27.9. Let $\pi : \mathcal{Z} \to Z$ be a morphism to an algebraic space such that $\mathcal{Z}$ is a gerbe over $Z$. Then $\pi $ is surjective, flat, and locally of finite presentation by Lemma 98.27.8. Hence $\mathop{\mathrm{Spec}}(k) \to Z$ is surjective, flat, and locally of finite presentation as a composition, see Properties of Stacks, Lemma 97.5.2 and Lemmas 98.24.2 and 98.26.2. Hence by Properties of Stacks, Lemma 97.11.3 we see that $|Z|$ is a singleton and that $Z$ is locally Noetherian and reduced. $\square$


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