Inverse problems for deformation rings

Abstract:

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 $n \ge 1$, the ring $W[[t]]/(p^n t,t^2)$ arises as a universal deformation ring. This ring is not a complete intersection if $p^nW\neq \{0\}$, so we obtain an answer to a question of M. Flach in all characteristics. We also study the ‘inverse inverse problem’ for the ring $W[[t]]/(p^n t,t^2)$; this is to determine all pairs $(\Gamma , V)$ such that $R_{W}(\Gamma ,V)$ is isomorphic to this ring.