Universal deformation rings and dihedral blocks with two simple modules

TL;DR

This paper classifies universal deformation rings for modules over dihedral blocks with two simple modules in characteristic 2, revealing conditions under which these rings are subquotients of the Witt ring and relating to Morita equivalence.

Contribution

It explicitly determines the universal deformation rings for modules in dihedral blocks with two simples and links the parameter c to Morita equivalence to principal blocks.

Findings

R(G,V) is a subquotient of WD if and only if c=0.

c=0 precisely when B is Morita equivalent to a principal block.

Provides explicit descriptions of deformation rings for modules in these 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 G is a finite group and B is a block of kG with a dihedral defect group D such that there are precisely two isomorphism classes of simple B-modules. We determine the universal deformation ring R(G,V) for every finitely generated kG-module V which belongs to B and whose stable endomorphism ring is isomorphic to k. The description by Erdmann of the quiver and relations of the basic algebra of B is usually only determined up to a certain parameter c which is either 0 or 1. We show that R(G,V) is isomorphic to a subquotient ring of WD for all V as above if and only if c=0, giving an answer to a question raised by the first author and Chinburg in this case. Moreover, we prove that c=0 if and only if B is Morita equivalent to a principal block.