#### Title

Universal Deformation Rings and Dihedral Blocks with Two Simple Modules

#### Document Type

Article

#### Publication Date

11-2011

#### Department

Mathematics

#### Language

English

#### Publication Title

Journal of Algebra

#### 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.

#### DOI

10.1016/j.jalgebra.2011.08.010

#### Recommended Citation

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

## Comments

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