Universal deformation rings and dihedral 2-groups

TL;DR

This paper proves that the universal deformation ring of endo-trivial modules over dihedral 2-groups is isomorphic to a specific group ring, confirming conjectures for a broad class of modules and blocks.

Contribution

It establishes the universal deformation ring structure for endo-trivial modules over dihedral 2-groups, extending known results to more general blocks and modules.

Findings

Universal deformation ring isomorphic to W[Z/2 x Z/2]

Results confirmed for modules with stable endomorphism ring k

Extends previous conjectures to broader classes of modules and blocks

Abstract

Let $k$ be an algebraically closed field of characteristic 2, and let $W$ be the ring of infinite Witt vectors over $k$. Suppose $D$ is a dihedral 2-group. We prove that the universal deformation ring $R(D,V)$ of an endo-trivial $kD$-module $V$ is always isomorphic to $W[\mathbb{Z}/2\times\mathbb{Z}/2]$. As a consequence we obtain a similar result for modules $V$ with stable endomorphism ring $k$ belonging to an arbitrary nilpotent block with defect group $D$. This confirms for such $V$ conjectures on the ring structure of the universal deformation ring of $V$ which had previously been shown for $V$ belonging to cyclic blocks or to blocks with Klein four defect groups.