Deformation rings which are not local complete intersections

TL;DR

This paper investigates which complete local rings can be realized as versal deformation rings of finite group representations, showing that certain non-complete intersection rings can indeed occur, answering a longstanding question.

Contribution

The authors demonstrate that specific non-complete intersection rings arise as versal deformation rings, extending the known class of such rings in all characteristics.

Findings

Certain rings $ ext{W}[[t]]/(p^n t,t^2)$ occur as deformation rings.

These rings are not local complete intersections when $p^n ext{W} eq ext{0}$.

The results answer a question posed by M. Flach.

Abstract

We study the inverse problem for the versal deformation rings $R(\Gamma,V)$ of finite dimensional representations $V$ of a finite group $\Gamma$ over a field $k$ of positive characteristic $p$. This problem is to determine which complete local commutative Noetherian rings with residue field $k$ can arise up to isomorphism as such $R(\Gamma,V)$. We show that for all integers $n \ge 1$ and all complete local commutative Noetherian rings $\mathcal{W}$ with residue field $k$, the ring $\mathcal{W}[[t]]/(p^n t,t^2)$ arises in this way. This ring is not a local complete intersection if $p^n\mathcal{W}\neq\{0\}$, so we obtain an answer to a question of M. Flach in all characteristics.