# The Stacks Project

## Tag: 06QI

This tag has label stacks-morphisms-lemma-gerbe-fppf and it points to

The corresponding content:

Lemma 69.19.8. 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 68.5.4 and Lemmas 69.17.4 and 69.18.7 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 69.19.7 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 69.19.6. This implies that $\pi$ is surjective, flat, and locally of finite presentation by Properties of Stacks, Lemma 68.5.5 and Lemmas 69.17.5 and 69.18.8. $\square$

\begin{lemma}
\label{lemma-gerbe-fppf}
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.
\end{lemma}

\begin{proof}
By
Properties of Stacks, Lemma
\ref{stacks-properties-lemma-descent-surjective}
and
Lemmas \ref{lemma-descent-flat} and
\ref{lemma-descent-finite-presentation}
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 \ref{lemma-local-structure-gerbe}
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 \ref{lemma-gerbe-with-section}.
This implies that $\pi$ is surjective, flat, and locally
of finite presentation by
Properties of Stacks,
Lemma \ref{stacks-properties-lemma-surjective-permanence}
and
Lemmas \ref{lemma-flat-permanence} and
\ref{lemma-flat-finite-presentation-permanence}.
\end{proof}


To cite this tag (see How to reference tags), use:

\cite[\href{http://stacks.math.columbia.edu/tag/06QI}{Tag 06QI}]{stacks-project}


In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).