Inverse Problems for deformation rings

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