The Stacks project

112.5.7 Rigidification

Rigidification is a process for removing a flat subgroup from the inertia. For example, if $X$ is a projective variety, the morphism from the Picard stack to the Picard scheme is a rigidification of the group of automorphism $\mathbf{G}_ m$.

• Abramovich, Corti, Vistoli: Twisted bundles and admissible covers [acv]

  Let $\mathcal{X}$ be an algebraic stack over $S$ and $H$ be a flat, finitely presented separated group scheme over $S$. Assume that for every object $\xi \in \mathcal{X}(T)$ there is an embedding $H(T) \hookrightarrow \text{Aut}_{\mathcal{X}(T)}(\xi )$ which is compatible under pullbacks in the sense that for every arrow $\phi : \xi \rightarrow \xi '$ over $f: T \rightarrow T'$ and $g \in H(T')$, $g \circ \phi = \phi \circ f^*g$. Then there exists an algebraic stack $\mathcal{X}/H$ and a morphism $\rho : \mathcal{X} \rightarrow \mathcal{X}/H$ which is an fppf gerbe such that for every $\xi \in \mathcal{X}(T)$, the morphism $\text{Aut}_{\mathcal{X}(T)} (\xi ) \rightarrow \text{Aut}_{\mathcal{X}/H (T)} (\xi ) $ is surjective with kernel $H(T)$.

• Romagny: Group actions on stacks and applications [romagny_actions]

  Discusses how group actions behave with respect to rigidifications.

• Abramovich, Graber, Vistoli: Gromov-Witten theory for Deligne-Mumford stacks [agv]

  The appendix gives a summary of rigidification as in [acv] with two alternative interpretations. This paper also contains constructions for gluing algebraic stacks along closed substacks and for taking roots of line bundles.

• Abramovich, Olsson, Vistoli: Tame stacks in positive characteristic ([tame])

  The appendix handles the more complicated situation where the flat subgroup stack of the inertia $H \subset I_\mathcal {X}$ is normal but not necessarily central.