Lemma 100.11.10. 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 100.11.3). The algebraic space T = W \times _\mathcal {Z} \mathop{\mathrm{Spec}}(k) is smooth over k in particular regular, see Spaces over Fields, Lemma 72.16.1. Since T \to W is locally of finite presentation, flat, and surjective it follows that W is regular, see Descent on Spaces, Lemma 74.9.4. By definition this means that \mathcal{Z} is regular. \square
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