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Universal deformation rings and Klein four defect groups. (English) Zbl 1047.20006

Let \(k\) be a field of finite characteristic \(p\), \(W\) the ring of infinite Witt vectors over \(k\), \(R\) a local commutative \(W\)-algebra which is isomorphic to the projective limit of its discrete Artinian quotients, with residue field \(k\), and \(G\) a profinite group. Then, for a finite dimensional representation \(V\) of \(G\) over \(k\) we say that \(V\) has a lift to \(R\) if there is an \(R\)-free \(RG\)-module \(M\) so that the \(kG\)-modules \(k\otimes_RM\) and \(V\) are isomorphic. The isomorphism class of lifts is a deformation, and we call for any \(V\) the functor sending an algebra \(R\) to the set of deformations the deformation functor. The representation \(V\) has a universal deformation ring if this deformation functor is representable, and then the representing object is the universal deformation ring \(R(G,V)\).
The author gives the ring structure of the ring \(R(G,V)\) for \(V\) being a representation of the group \(G\) belonging to a block with Klein four defect group \(D\), and \(k\) being algebraically closed of characteristic \(2\). In previous joint work with T. Chinburg the author showed the existence of such universal deformation rings in case the stable endomorphism ring of \(V\) is at most one-dimensional. Moreover, then the universal deformation ring coincides with the one of the first syzygy of \(V\). The case of one-dimensional \(V\) is solved in previous work of Mazur. The author shows that universal deformation rings of Morita equivalent blocks coincide, for corresponding modules, and so by the classification of tame blocks of group rings, the problem is reduced to the cases of alternating groups of degree \(4\), degree \(5\) and the Klein four group. In each case the modules with one-dimensional stable endomorphism ring are rare, and explicit computations in each of the finite number of cases gives that the universal deformation ring is either \(WD\), \(k\) or \(W\).
The paper ends with a detailed and tricky study of the symmetric group of degree \(4\), where the universal deformation rings for two specific two-dimensional modules are computed.

MSC:

20C05 Group rings of finite groups and their modules (group-theoretic aspects)
11S20 Galois theory
20C20 Modular representations and characters
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