Inverse Problems for deformation rings

Abstract

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

17 pages; this paper is closely related to arXiv:1003.3143. In the third version, we added a new Section 5 with further examples

Cited by in corpus (5)

• Universal deformation rings of modules for algebras of dihedral type of polynomial growth
• Brauer's generalized decomposition numbers and universal deformation rings
• The inverse deformation problem
• Deformation rings and images of Galois representations
• Large universal deformation rings