The Stacks project

Lemma 99.11.10. A reduced, locally Noetherian algebraic stack $\mathcal{Z}$ such that $|\mathcal{Z}|$ is a singleton is regular.

Proof. Let $W \to \mathcal{Z}$ be a surjective smooth morphism where $W$ is a scheme. Let $k$ be a field and let $\mathop{\mathrm{Spec}}(k) \to \mathcal{Z}$ be surjective, flat, and locally of finite presentation (see Lemma 99.11.3). The algebraic space $T = W \times _\mathcal {Z} \mathop{\mathrm{Spec}}(k)$ is smooth over $k$ in particular regular, see Spaces over Fields, Lemma 71.16.1. Since $T \to W$ is locally of finite presentation, flat, and surjective it follows that $W$ is regular, see Descent on Spaces, Lemma 73.9.4. By definition this means that $\mathcal{Z}$ is regular. $\square$


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