Review of Galois Deformations

We'll review the study of Galois deformations. Here's the setup. Let G be a pro nite group, and let ρ : G→ GLn(k) be a representation of G over a nite eld k of characteristic p. Let Λ be a complete DVR with residue eld k, and let CΛ denote the category whose objects are artinian local Λ-algebras with residue eld k, and whose morphisms are local homomorphisms. Let ĈΛ denote the category of complete noetherian local Λ-algebras with residue eld k, which is the completion of CΛ under limits. Frequently, Λ will be W (k), the ring of Witt vectors over k. We now de ne two deformation functors associated to ρ. The rst is the deformation functor Def(ρ) : ĈΛ → Sets given by Def(ρ)(A) = {(ρ,M, φ)}/ ∼=, where M is a free A-module of rank n, ρ : G → AutA(M), and φ : ρ ⊗A k → ρ is an isomorphism. The second is the framed deformation functor Def (ρ) : ĈΛ → Sets given by Def (ρ)(A) = {(ρ,M, φ,B) | (ρ,M, φ) ∈ Def(ρ)(A)}/ ∼=, where B is a basis of M which is sent to the standard basis for k under φ. We can compute both Def and Def at the level of its artinian quotients: if m is the maximal ideal of A, then