# The Stacks Project

## Tag: 06QJ

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Proposition 69.19.9. Let $\mathcal{X}$ be an algebraic stack. The following are equivalent
1. $\mathcal{X}$ is a gerbe, and
2. $\mathcal{I}_\mathcal{X} \to \mathcal{X}$ is flat and locally of finite presentation.

Proof. Assume (1). Choose a morphism $\mathcal{X} \to X$ into an algebraic space $X$ which turns $\mathcal{X}$ into a gerbe over $X$. Let $X' \to X$ is a surjective, flat, locally finitely presented morphism and set $\mathcal{X}' = X' \times_X \mathcal{X}$. Note that $\mathcal{X}'$ is a gerbe over $X'$ by Lemma 69.19.3. Then both squares in $$\xymatrix{ \mathcal{I}_{\mathcal{X}'} \ar[r] \ar[d] & \mathcal{X}' \ar[r] \ar[d] & X' \ar[d] \\ \mathcal{I}_\mathcal{X} \ar[r] & \mathcal{X} \ar[r] & X }$$ are fibre product squares, see Lemma 69.5.4. Hence to prove $\mathcal{I}_\mathcal{X} \to \mathcal{X}$ is flat and locally of finite presentation it suffices to do so after such a base change by Lemmas 69.17.4 and 69.18.7. Thus we can apply Lemma 69.19.7 to assume that $\mathcal{X} = [U/G]$. By Lemma 69.19.6 we see $G$ is flat and locally of finite presentation over $U$ and that $x : U \to [U/G]$ is surjective, flat, and locally of finite presentation. Moreover, the pullback of $\mathcal{I}_\mathcal{X}$ by $x$ is $G$ and we conclude that (2) holds by descent again, i.e., by Lemmas 69.17.4 and 69.18.7.

Conversely, assume (2). Choose a smooth presentation $\mathcal{X} = [U/R]$, see Algebraic Stacks, Section 62.16. Denote $G \to U$ the stabilizer group algebraic space of the groupoid $(U, R, s, t, c, e, i)$, see Groupoids in Spaces, Definition 55.15.2. By Lemma 69.5.6 we see that $G \to U$ is flat and locally of finite presentation as a base change of $\mathcal{I}_\mathcal{X} \to \mathcal{X}$, see Lemmas 69.17.3 and 69.18.3. Consider the following action $$a : G \times_{U, t} R \to R, \quad (g, r) \mapsto c(g, r)$$ of $G$ on $R$. This action is free on $T$-valued points for any scheme $T$ as $R$ is a groupoid. Hence $R' = R/G$ is an algebraic space and the quotient morphism $\pi : R \to R'$ is surjective, flat, and locally of finite presentation by Bootstrap, Lemma 57.11.5. The projections $s, t : R \to U$ are $G$-invariant, hence we obtain morphisms $s' , t' : R' \to U$ such that $s = s' \circ \pi$ and $t = t' \circ \pi$. Since $s, t : R \to U$ are flat and locally of finite presentation we conclude that $s', t'$ are flat and locally of finite presentation, see Morphisms of Spaces, Lemmas 45.28.5 and Descent on Spaces, Lemma 51.14.1. Consider the morphism $$j' = (t', s') : R' \longrightarrow U \times U.$$ We claim this is a monomorphism. Namely, suppose that $T$ is a scheme and that $a, b : T \to R'$ are morphisms which have the same image in $U \times U$. By definition of the quotient $R' = R/G$ there exists an fppf covering $\{h_j : T_j \to T\}$ such that $a \circ h_j = \pi \circ a_j$ and $b \circ h_j = \pi \circ b_j$ for some morphisms $a_j, b_j : T_j \to R$. Since $a_j, b_j$ have the same image in $U \times U$ we see that $g_j = c(a_j, i(b_j))$ is a $T_j$-valued point of $G$ such that $c(g_j, b_j) = a_j$. In other words, $a_j$ and $b_j$ have the same image in $R'$ and the claim is proved. Since $j : R \to U \times U$ is a pre-equivalence relation (see Groupoids in Spaces, Lemma 55.11.2) and $R \to R'$ is surjective (as a map of sheaves) we see that $j' : R' \to U \times U$ is an equivalence relation. Hence Bootstrap, Theorem 57.10.1 shows that $X = U/R'$ is an algebraic space. Finally, we claim that the morphism $$\mathcal{X} = [U/R] \longrightarrow X = U/R'$$ turns $\mathcal{X}$ into a gerbe over $X$. This follows from Groupoids in Spaces, Lemma 55.26.1 as $R \to R'$ is surjective, flat, and locally of finite presentation (if needed use Bootstrap, Lemma 57.4.5 to see this implies the required hypothesis). $\square$

\begin{proposition}
\label{proposition-when-gerbe}
Let $\mathcal{X}$ be an algebraic stack. The following are equivalent
\begin{enumerate}
\item $\mathcal{X}$ is a gerbe, and
\item $\mathcal{I}_\mathcal{X} \to \mathcal{X}$ is flat and locally of
finite presentation.
\end{enumerate}
\end{proposition}

\begin{proof}
Assume (1). Choose a morphism $\mathcal{X} \to X$ into an algebraic space $X$
which turns $\mathcal{X}$ into a gerbe over $X$. Let $X' \to X$ is a
surjective, flat, locally finitely presented morphism and
set $\mathcal{X}' = X' \times_X \mathcal{X}$. Note that $\mathcal{X}'$
is a gerbe over $X'$ by
Lemma \ref{lemma-base-change-gerbe}.
Then both squares in
$$\xymatrix{ \mathcal{I}_{\mathcal{X}'} \ar[r] \ar[d] & \mathcal{X}' \ar[r] \ar[d] & X' \ar[d] \\ \mathcal{I}_\mathcal{X} \ar[r] & \mathcal{X} \ar[r] & X }$$
are fibre product squares, see
Lemma \ref{lemma-cartesian-square-intertia}.
Hence to prove $\mathcal{I}_\mathcal{X} \to \mathcal{X}$ is flat and
locally of finite presentation it suffices to do so after such a base
change by
Lemmas \ref{lemma-descent-flat} and
\ref{lemma-descent-finite-presentation}.
Thus we can apply
Lemma \ref{lemma-local-structure-gerbe}
to assume that $\mathcal{X} = [U/G]$.
By
Lemma \ref{lemma-gerbe-with-section}
we see $G$ is flat and locally of finite presentation over $U$ and
that $x : U \to [U/G]$ is surjective, flat, and locally of finite
presentation. Moreover, the pullback of $\mathcal{I}_\mathcal{X}$
by $x$ is $G$ and we conclude that (2) holds by descent again, i.e., by
Lemmas \ref{lemma-descent-flat} and
\ref{lemma-descent-finite-presentation}.

\medskip\noindent
Conversely, assume (2). Choose a smooth presentation $\mathcal{X} = [U/R]$, see
Algebraic Stacks, Section \ref{algebraic-section-stack-to-presentation}.
Denote $G \to U$ the stabilizer group algebraic space of the groupoid
$(U, R, s, t, c, e, i)$, see
Groupoids in Spaces, Definition
\ref{spaces-groupoids-definition-stabilizer-groupoid}.
By
Lemma \ref{lemma-presentation-inertia}
we see that $G \to U$ is flat and locally of finite presentation as
a base change of $\mathcal{I}_\mathcal{X} \to \mathcal{X}$, see
Lemmas \ref{lemma-base-change-flat} and
\ref{lemma-base-change-finite-presentation}.
Consider the following action
$$a : G \times_{U, t} R \to R, \quad (g, r) \mapsto c(g, r)$$
of $G$ on $R$. This action is free on $T$-valued points for any
scheme $T$ as $R$ is a groupoid. Hence $R' = R/G$ is an algebraic
space and the quotient morphism $\pi : R \to R'$ is surjective,
flat, and locally of finite presentation by
Bootstrap, Lemma \ref{bootstrap-lemma-quotient-free-action}.
The projections $s, t : R \to U$ are $G$-invariant, hence
we obtain morphisms $s' , t' : R' \to U$ such that $s = s' \circ \pi$
and $t = t' \circ \pi$.
Since $s, t : R \to U$ are flat and locally of finite presentation
we conclude that $s', t'$ are flat and locally of finite presentation, see
Morphisms of Spaces, Lemmas
\ref{spaces-morphisms-lemma-flat-permanence} and
Descent on Spaces, Lemma
\ref{spaces-descent-lemma-locally-finite-presentation-fppf-local-source}.
Consider the morphism
$$j' = (t', s') : R' \longrightarrow U \times U.$$
We claim this is a monomorphism. Namely, suppose that $T$ is a scheme
and that $a, b : T \to R'$ are morphisms which have the same image
in $U \times U$. By definition of the quotient $R' = R/G$ there
exists an fppf covering $\{h_j : T_j \to T\}$ such
that $a \circ h_j = \pi \circ a_j$ and $b \circ h_j = \pi \circ b_j$
for some morphisms $a_j, b_j : T_j \to R$. Since $a_j, b_j$ have the same
image in $U \times U$ we see that $g_j = c(a_j, i(b_j))$ is a $T_j$-valued
point of $G$ such that $c(g_j, b_j) = a_j$. In other words, $a_j$ and
$b_j$ have the same image in $R'$ and the claim is proved.
Since $j : R \to U \times U$ is a pre-equivalence relation (see
Groupoids in Spaces, Lemma
\ref{spaces-groupoids-lemma-groupoid-pre-equivalence})
and $R \to R'$ is surjective (as a map of sheaves) we see that
$j' : R' \to U \times U$ is an equivalence relation.
Hence
Bootstrap, Theorem \ref{bootstrap-theorem-final-bootstrap}
shows that $X = U/R'$ is an algebraic space.
Finally, we claim that the morphism
$$\mathcal{X} = [U/R] \longrightarrow X = U/R'$$
turns $\mathcal{X}$ into a gerbe over $X$. This follows from
Groupoids in Spaces, Lemma \ref{spaces-groupoids-lemma-when-gerbe}
as $R \to R'$ is surjective, flat, and locally of finite presentation
(if needed use
Bootstrap, Lemma
\ref{bootstrap-lemma-surjective-flat-locally-finite-presentation}
to see this implies the required hypothesis).
\end{proof}


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