Universal Deformation Rings and Dihedral Blocks with Two Simple Modules
Journal of Algebra
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.
Bleher, Frauke M., Giovanna Llosent, and Jennifer B. Schaefer. "Universal Deformation Rings and Dihedral Blocks with Two Simple Modules." Journal of Algebra 345, no. 1 (2011), 49-71. doi: 10.1016/j.jalgebra.2011.08.010