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The Stacks project

Lemma 100.11.3. Let \mathcal{Z} be an algebraic stack. The following are equivalent

  1. \mathcal{Z} is reduced, locally Noetherian, and |\mathcal{Z}| is a singleton, and

  2. there exists a locally finitely presented, surjective, flat morphism \mathop{\mathrm{Spec}}(k) \to \mathcal{Z} where k is a field.

Proof. Assume (2) holds. By Lemma 100.11.2 we see that \mathcal{Z} is reduced and |\mathcal{Z}| is a singleton. Let W be a scheme and let W \to \mathcal{Z} be a surjective smooth morphism. Choose a field k and a locally finitely presented, surjective, flat morphism \mathop{\mathrm{Spec}}(k) \to \mathcal{Z}. Then W \times _\mathcal {Z} \mathop{\mathrm{Spec}}(k) is an algebraic space smooth over k, hence locally Noetherian (see Morphisms of Spaces, Lemma 67.23.5). Since W \times _\mathcal {Z} \mathop{\mathrm{Spec}}(k) \to W is flat, surjective, and locally of finite presentation, we see that \{ W \times _\mathcal {Z} \mathop{\mathrm{Spec}}(k) \to W\} is an fppf covering and we conclude that W is locally Noetherian (Descent on Spaces, Lemma 74.9.3). In other words (1) holds.

Assume (1). Pick a nonempty affine scheme W and a smooth morphism W \to \mathcal{Z}. Pick a closed point w \in W and set k = \kappa (w). Because W is locally Noetherian the morphism w : \mathop{\mathrm{Spec}}(k) \to W is of finite presentation, see Morphisms, Lemma 29.21.7. Hence the composition

\mathop{\mathrm{Spec}}(k) \xrightarrow {w} W \longrightarrow \mathcal{Z}

is locally of finite presentation by Morphisms of Spaces, Lemmas 67.28.2 and 67.37.5. It is also flat and surjective by Lemma 100.11.1. Hence (2) holds. \square


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