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Universal deformation rings and dihedral defect groups

Author: Frauke M. Bleher
Journal: Trans. Amer. Math. Soc. 361 (2009), 3661-3705
MSC (2000): Primary 20C20; Secondary 20C15, 16G10
Published electronically: February 10, 2009
MathSciNet review: 2491895
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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 dihedral defect group $ D$, which is Morita equivalent to the principal $ 2$-modular block of a finite simple group. We determine the universal deformation ring $ R(G,V)$ for every $ kG$-module $ V$ which belongs to $ B$ and has stable endomorphism ring $ k$. It follows that $ R(G,V)$ is always isomorphic to a subquotient ring of $ WD$. Moreover, we obtain an infinite series of examples of universal deformation rings which are not complete intersections.

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Additional Information

Frauke M. Bleher
Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242-1419

Keywords: Universal deformation rings, dihedral defect groups, special biserial algebras, stable endomorphism rings
Received by editor(s): July 26, 2006
Received by editor(s) in revised form: April 27, 2007
Published electronically: February 10, 2009
Additional Notes: The author was supported in part by NSF Grant DMS01-39737 and NSA Grant H98230-06-1-0021.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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