[Paper Review] Deformation functors of local actions

Jakub Byszewski|arXiv (Cornell University)|Dec 1, 2011

Homotopy and Cohomology in Algebraic Topology7 references3 citations

TL;DR

This paper investigates the deformation theory of local group actions on curves in positive characteristic, introducing restriction and induction maps between deformation functors of a group, its normal subgroup, and quotient group. It proposes a decomposition of the deformation functor of a group as a smooth extension of a fibered product involving its subgroup and quotient group, providing cohomological evidence via obstruction theory and a main result on the obstruction space for the morphism of deformation functors.

ABSTRACT

We study the behaviour of infinitesimal deformation functors of local group actions with regard to passing to subgroups and quotient groups. Inspired by the cohomological information, we conjecture the existence of a decomposition of a deformation functor of a local $G$-action as a smooth extension of a fibered product of functors related to a subgroup and a quotient group of $G$.

Motivation & Objective

• To understand the behavior of infinitesimal deformation functors of local group actions under passage to subgroups and quotient groups.
• To generalize cohomological operations (restriction and induction) to the level of deformation functors.
• To investigate whether the deformation functor of a group $ G $ decomposes as a smooth extension of a fibered product of functors associated to a normal subgroup $ N $ and the quotient $ G/N $.
• To provide cohomological evidence for such a decomposition via obstruction theory.

Proposed method

• Define restriction and induction maps between deformation functors $ D_G $, $ D_N $, and $ D_{G/N} $, lifting standard group cohomology operations.
• Construct a morphism $ (\operatorname{res}, \operatorname{ind}) : D_G \to D_N^{G/N} \times D_{G/N} $, analyzing its deformation-theoretic properties.
• Use spectral sequence techniques and the inhomogeneous complex to compute cohomological data, particularly the tangent maps and obstruction classes.
• Establish a complete obstruction space for the morphism $ (\operatorname{res}, \operatorname{ind}) $ under pro-representability assumptions, using the total complex of a bicomplex.
• Apply the Hochschild–Serre spectral sequence to relate cohomology of $ G $, $ N $, and $ G/N $, and compute the induced map on cohomology.
• Verify that the obstruction space computation aligns with cohomological expectations, supporting the conjectured decomposition.

Experimental results

Research questions

• RQ1 Does there exist a pro-representable functor $ F $ such that the map $ D_G \to D_N^{G/N} \times_F D_{G/N} $ is smooth?
• RQ2 How do the restriction and induction maps on deformation functors relate to the corresponding maps in group cohomology?
• RQ3 Can the deformation functor $ D_G $ be described as a smooth extension of a fibered product of $ D_N^{G/N} $ and $ D_{G/N} $, reflecting subgroup and quotient group structures?
• RQ4 What is the complete obstruction space for the morphism $ (\operatorname{res}, \operatorname{ind}) : D_G \to D_N^{G/N} \times D_{G/N} $, and how does it relate to cohomology?
• RQ5 To what extent can this decomposition technique be used to inductively solve lifting problems for group actions on curves to characteristic zero?

Key findings

• The tangent map to the restriction map $ \operatorname{res} $ coincides with the standard restriction map in group cohomology $ \operatorname{H}^1(G, \Theta) \to \operatorname{H}^1(N, \Theta)^{G/N} $.
• A cohomological description of the tangent map to the induction map $ \operatorname{ind} $ is established in Theorem 3.4.
• At the cohomological level, the morphism $ (\operatorname{res}, \operatorname{ind}) $ is compatible with the fibered product structure, supporting the conjectured decomposition (Corollary 4.3).
• Under pro-representability assumptions, Theorem 4.9 constructs a complete obstruction space for the morphism $ (\operatorname{res}, \operatorname{ind}) $ using cohomological data from a bicomplex.
• The obstruction space computation provides strong evidence for a positive answer to the conjectured smooth decomposition of $ D_G $ as an extension of $ D_N^{G/N} \times_F D_{G/N} $.
• The result suggests that versal deformation rings of $ D_G $ could be explicitly computed from those of $ D_N $ and $ D_{G/N} $, as noted in Remark 4.5.