The Stacks project

Lemma 98.27.8. Let $\pi : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. If $\mathcal{X}$ is a gerbe over $\mathcal{Y}$, then $\pi $ is surjective, flat, and locally of finite presentation.

Proof. By Properties of Stacks, Lemma 97.5.4 and Lemmas 98.24.4 and 98.26.11 it suffices to prove to the lemma after replacing $\pi $ by a base change with a surjective, flat, locally finitely presented morphism $\mathcal{Y}' \to \mathcal{Y}$. By Lemma 98.27.7 we may assume $\mathcal{Y} = U$ is an algebraic space and $\mathcal{X} = [U/G]$ over $U$. Then $U \to [U/G]$ is surjective, flat, and locally of finite presentation, see Lemma 98.27.6. This implies that $\pi $ is surjective, flat, and locally of finite presentation by Properties of Stacks, Lemma 97.5.5 and Lemmas 98.24.5 and 98.26.12. $\square$


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