INVERSE PROBLEMS FOR DEFORMATION RINGS

Let W be a complete Noetherian local commutative ring with residue field k of positive characteristic p. We study the inverse problem for the universal deformation rings R-W(Gamma, V) relative to W of finite dimensional representations V of a profinite group Gamma over k. We show that for all p and n >= 1, the ring W[[t]]/(p(n)t, t(2)) arises as a universal deformation ring. This ring is not a complete intersection if p(n)W not equal {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)); this is to determine all pairs (Gamma, V) such that R-W(Gamma, V) is isomorphic to this ring.