The Stacks project

Lemma 98.11.3. 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 98.11.2 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 65.23.5). 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 72.8.3). 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 65.28.2 and 65.37.5. It is also flat and surjective by Lemma 98.11.1. Hence (2) holds. $\square$


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