Inverse problems for deformation rings

Abstract

Let W W be a complete Noetherian local commutative ring with residue field k k of positive characteristic p p . We study the inverse problem for the universal deformation rings R W ( Γ , V ) R_{W}(\Gamma ,V) relative to W W of finite dimensional representations V V of a profinite group Γ \Gamma over k k . We show that for all p p and n ≥ 1 n \ge 1 , the ring W [ [ t ] ] / ( p n t , t 2 ) W[[t]]/(p^n t,t^2) arises as a universal deformation ring. This ring is not a complete intersection if p n W ≠ { 0 } p^nW\neq \{0\} , so we obtain an answer to a question of M. Flach in all characteristics. We also study the ‘inverse inverse problem’ for the ring W [ [ t ] ] / ( p n t , t 2 ) W[[t]]/(p^n t,t^2) ; this is to determine all pairs ( Γ , V ) (\Gamma , V) such that R W ( Γ , V ) R_{W}(\Gamma ,V) is isomorphic to this ring.