The Stacks project

Lemma 98.11.2. Let $\mathcal{Z}$ be an algebraic stack. The following are equivalent

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

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

  3. there exists a locally of finite type, surjective, flat morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{Z}$ where $k$ is a field.

Proof. Assume (1). Let $W$ be a scheme and let $W \to \mathcal{Z}$ be a surjective smooth morphism. Then $W$ is a reduced scheme. Let $\eta \in W$ be a generic point of an irreducible component of $W$. Since $W$ is reduced we have $\mathcal{O}_{W, \eta } = \kappa (\eta )$. It follows that the canonical morphism $\eta = \mathop{\mathrm{Spec}}(\kappa (\eta )) \to W$ is flat. We see that the composition $\eta \to \mathcal{Z}$ is flat (see Morphisms of Spaces, Lemma 65.30.3). It is also surjective as $|\mathcal{Z}|$ is a singleton. In other words (2) holds.

Assume (2). Let $W$ be a scheme and let $W \to \mathcal{Z}$ be a surjective smooth morphism. Choose a field $k$ and a 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 regular (see Spaces over Fields, Lemma 70.16.1) and in particular reduced. Since $W \times _\mathcal {Z} \mathop{\mathrm{Spec}}(k) \to W$ is surjective and flat we conclude that $W$ is reduced (Descent on Spaces, Lemma 72.8.2). In other words (1) holds.

It is clear that (3) implies (2). Finally, assume (2). Pick a nonempty affine scheme $W$ and a smooth morphism $W \to \mathcal{Z}$. Pick a closed point $w \in W$ and set $k = \kappa (w)$. The composition

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

is locally of finite type by Morphisms of Spaces, Lemmas 65.23.2 and 65.37.6. It is also flat and surjective by Lemma 98.11.1. Hence (3) holds. $\square$


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