Universal deformation rings and dihedral blocks with two simple modules

Details Universal deformation rings and dihedral blocks with two simple modules Universal deformation rings and dihedral blocks with two simple modules

2010-12-08

arxiv.org/abs/1012.1668v1

J. Algebra 345 (2011), 49-71

Mathematics

Group Theory

22 pages, 4 figures

Scientific paper

10.1016/j.jalgebra.2011.08.010

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.

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