The Stacks project

Lemma 98.27.6. 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 98.5.3. 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 4.30.12. 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 91.16.1 agrees with the group law on $G$ (proof omitted). Thus Algebraic Stacks, Lemma 91.16.1 shows we obtain a canonical fully faithful $1$-morphism

\[ x_{can} : [U/G] \longrightarrow \mathcal{X} \]

of stacks in groupoids over $(\mathit{Sch}/S)_{fppf}$. 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 $U$, see property (2)(a) of Stacks, Lemma 8.11.3.

By Criteria for Representability, Lemma 94.18.3 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 94.17.1. $\square$


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