Summary of “cohomological Control of Deformation Theory via A∞-structure”

This is a brief summary of one part of the forthcoming work “Cohomological control of deformation theory via A∞-structure.” Let G be a profinite group. The main result is that a natural A∞-structure on cohomology groups induces presentations of universal deformation rings for G-representations, more general moduli spaces for G-representations, and universal deformation rings for Galois pseudorepresentations. Nothing in this summary is particular to the case that G is a Galois group. Remaining parts of the forthcoming paper (not described here) give applications to number theory. 1. Fine and coarse moduli of Galois representations In this section, we give background for the result, quickly summarizing [WE15]. 1.1. Fine moduli of representations. The most often-applied moduli theory of representations of a profinite group, due to Mazur [Maz89], proceeds as follows: fix a residual representation ρ̄ : G→ GLd(F) and study its deformations, which is often represented by a universal deformation ring Rρ̄. In [WE15], I have studied the moduli of all representations, a space we will call “Rep.” Universal deformations rings Rρ̄ are complete local rings in Rep. Because we must take account of the profinite topology on G, it is natural to restrict the coefficient rings (on which we evaluate Rep) to quotients of completions of Z[x1, . . . , xn] at some ideal containing a rational prime p. To understand Rep, it is helpful to introduce pseudorepresentations, a notion due to Chenevier [Che14]. An A-valued pseudorepresentation D : G → A of dimension d is a collection of characteristic polynomial coefficient functions D = (f1 = Tr, f2, . . . , fd = det) : G→ A satisfying conditions that would be expected if it came from an A-valued representation. We write PsR for the (fine) moduli scheme of pseudorepresentations. There is a natural map ψ : Rep→ PsR associating a representation to its characteristic polynomial. Although not every pseudorepresentation arises from a representation, it is critically important that pseudorepresentations valued in a field are in bijection with semi-simple representations [Che14, Thm. A]. Accordingly, we write D̄ : G → F for a residual pseudorepresentation valued in a finite field F, and write ρ̄ D̄ : G→ GLd(F) for the associated semi-simple representation. Chenevier has shown that each D̄ has a universal deformation ring RD̄, which we call a pseudodeformation ring. Unlike the moduli of representations Rep, PsR is the disjoint union of deformation spaces of residual pseudorepresentations [Che14, Thm. F]. Consequently, we study one connected component of Rep at a time, written ψ : RepD̄ → SpecRD̄. Date: 2015-11-02. 1Chenevier’s definition develops notions due to Wiles [Wil88] and Taylor [Tay91].