Universal deformation rings and dihedral blocks with two simple modules

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.