Inverse Problems for deformation rings

TL;DR

This paper investigates which complete local rings can be realized as universal deformation rings of group representations, showing that certain non-complete intersection rings arise universally and exploring their inverse problems.

Contribution

It demonstrates that specific rings, including non-complete intersections, can be universal deformation rings, and addresses the inverse inverse problem for these rings.

Findings

The ring $ ext{W}[[t]]/(p^n t,t^2)$ appears as a universal deformation ring.

This ring is not a complete intersection if $p^n ext{W} eq ext{0}$.

The paper characterizes all pairs $( ext{G},V)$ with deformation rings isomorphic to this ring.

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 $p^n\mathcal{W}\neq\{0\}$, so we obtain an answer to a question of M. Flach in all characteristics. We also study the `inverse inverse problem' for the ring $\mathcal{W}[[t]]/(p^n t,t^2)$; this is to determine all pairs $(\Gamma, V)$ such that $R_{\mathcal{W}}(\Gamma,V)$ is isomorphic to this ring.