Universal deformation rings and dihedral 2-groups

Details Universal deformation rings and dihedral 2-groups Universal deformation rings and dihedral 2-groups

2007-05-07

arxiv.org/abs/0705.0834v2

J. London Math. Soc. 79 (2009), 225-237

Mathematics

Representation Theory

16 pages, 1 table

Scientific paper

10.1112/jlms/jdn071

Let $k$ be an algebraically closed field of characteristic 2, and let $W$ be the ring of infinite Witt vectors over $k$. Suppose $D$ is a dihedral 2-group. We prove that the universal deformation ring $R(D,V)$ of an endo-trivial $kD$-module $V$ is always isomorphic to $W[\mathbb{Z}/2\times\mathbb{Z}/2]$. As a consequence we obtain a similar result for modules $V$ with stable endomorphism ring $k$ belonging to an arbitrary nilpotent block with defect group $D$. This confirms for such $V$ conjectures on the ring structure of the universal deformation ring of $V$ which had previously been shown for $V$ belonging to cyclic blocks or to blocks with Klein four defect groups.

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