# The Stacks Project

## Tag: 06QG

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Lemma 70.19.6. Let $\pi : \mathcal{X} \to U$ be a morphism from an algebraic stack to an algebraic space and let $x : U \to \mathcal{X}$ be a section of $\pi$. Set $G = \mathit{Isom}_\mathcal{X}(x, x)$, see Definition 70.5.3. If $\mathcal{X}$ is a gerbe over $U$, then
1. there is a canonical equivalence of stacks in groupoids $$x_{can} : [U/G] \longrightarrow \mathcal{X}.$$ where $[U/G]$ is the quotient stack for the trivial action of $G$ on $U$,
2. $G \to U$ is flat and locally of finite presentation, and
3. $U \to \mathcal{X}$ is surjective, flat, and locally of finite presentation.

Proof. Set $R = U \times_{x, \mathcal{X}, x} U$. The morphism $R \to U \times U$ factors through the diagonal $\Delta_U : U \to U \times U$ as it factors through $U \times_U U = U$. Hence $R = G$ because \begin{align*} G & = \mathit{Isom}_\mathcal{X}(x, x) \\ & = U \times_{x, \mathcal{X}} \mathcal{I}_\mathcal{X} \\ & = U \times_{x, \mathcal{X}} (\mathcal{X} \times_{\Delta, \mathcal{X} \times_S \mathcal{X}, \Delta} \mathcal{X}) \\ & = (U \times_{x, \mathcal{X}, x} U) \times_{U \times U, \Delta_U} U \\ & = R \times_{U \times U, \Delta_U} U \\ & = R \end{align*} for the fourth equality use Categories, Lemma 4.28.12. Let $t, s : R \to U$ be the projections. The composition law $c : R \times_{s, U, t} R \to R$ constructed on $R$ in Algebraic Stacks, Lemma 63.16.1 agrees with the group law on $G$ (proof omitted). Thus Algebraic Stacks, Lemma 63.16.1 shows we obtain a canonical fully faithful $1$-morphism $$x_{can} : [U/G] \longrightarrow \mathcal{X}$$ of stacks in groupoids over $(\textit{Sch}/S)_{fppf}$. To see that it is an equivalence it suffices to show that it is essentially surjective. To do this it suffices to show that any object of $\mathcal{X}$ over a scheme $T$ comes fppf locally from $x$ via a morphism $T \to U$, see Stacks, Lemma 8.4.8. However, this follows the condition that $\pi$ turns $\mathcal{X}$ into a gerbe over $X$, see property (2)(a) of Stacks, Lemma 8.11.3.

By Criteria for Representability, Lemma 66.18.3 we conclude that $G \to U$ is flat and locally of finite presentation. Finally, $U \to \mathcal{X}$ is surjective, flat, and locally of finite presentation by Criteria for Representability, Lemma 66.17.1. $\square$

\begin{lemma}
\label{lemma-gerbe-with-section}
Let $\pi : \mathcal{X} \to U$ be a morphism from an algebraic stack to
an algebraic space and let $x : U \to \mathcal{X}$ be a section of $\pi$.
Set $G = \mathit{Isom}_\mathcal{X}(x, x)$, see
Definition \ref{definition-isom}.
If $\mathcal{X}$ is a gerbe over $U$, then
\begin{enumerate}
\item there is a canonical equivalence of stacks in groupoids
$$x_{can} : [U/G] \longrightarrow \mathcal{X}.$$
where $[U/G]$ is the quotient stack for the trivial
action of $G$ on $U$,
\item $G \to U$ is flat and locally of finite presentation, and
\item $U \to \mathcal{X}$ is surjective, flat, and locally of finite
presentation.
\end{enumerate}
\end{lemma}

\begin{proof}
Set $R = U \times_{x, \mathcal{X}, x} U$. The morphism $R \to U \times U$
factors through the diagonal $\Delta_U : U \to U \times U$ as it factors
through $U \times_U U = U$. Hence $R = G$ because
\begin{align*}
G & = \mathit{Isom}_\mathcal{X}(x, x) \\
& = U \times_{x, \mathcal{X}} \mathcal{I}_\mathcal{X} \\
& = U \times_{x, \mathcal{X}}
(\mathcal{X}
\times_{\Delta, \mathcal{X} \times_S \mathcal{X}, \Delta}
\mathcal{X}) \\
& = (U \times_{x, \mathcal{X}, x} U) \times_{U \times U, \Delta_U} U \\
& = R \times_{U \times U, \Delta_U} U \\
& = R
\end{align*}
for the fourth equality use
Categories, Lemma \ref{categories-lemma-diagonal-2}.
Let $t, s : R \to U$ be the projections.
The composition law $c : R \times_{s, U, t} R \to R$ constructed on $R$ in
Algebraic Stacks, Lemma \ref{algebraic-lemma-map-space-into-stack}
agrees with the group law on $G$ (proof omitted). Thus
Algebraic Stacks, Lemma \ref{algebraic-lemma-map-space-into-stack}
shows we obtain a canonical fully faithful $1$-morphism
$$x_{can} : [U/G] \longrightarrow \mathcal{X}$$
of stacks in groupoids over $(\Sch/S)_{fppf}$. To see that it is
an equivalence it suffices to show that it is essentially surjective.
To do this it suffices to show that any object of $\mathcal{X}$ over
a scheme $T$ comes fppf locally from $x$ via a morphism $T \to U$, see
Stacks, Lemma \ref{stacks-lemma-characterize-essentially-surjective-when-ff}.
However, this follows the condition that $\pi$ turns $\mathcal{X}$
into a gerbe over $X$, see property (2)(a) of
Stacks, Lemma \ref{stacks-lemma-when-gerbe}.

\medskip\noindent
By
Criteria for Representability, Lemma \ref{criteria-lemma-BG-algebraic}
we conclude that $G \to U$ is flat and locally of finite presentation.
Finally, $U \to \mathcal{X}$ is surjective, flat, and locally of finite
presentation by
Criteria for Representability, Lemma
\ref{criteria-lemma-flat-quotient-flat-presentation}.
\end{proof}


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