Modules in the sources of Green's exact sequences for cyclic blocks
The paper under review is a contribution to the theory of blocks with cyclic defect groups initiated by R. Brauer and E. C. Dade and further developed by W. Feit, J. A. Green and others. The new idea is to combine these classical results with new methods introduced by L. Puig. The aim is to determine the source algebra of a block with cyclic defect groups, an algebra which is in general smaller than the block itself but larger than its basic algebra and which determines, for example, the generalized decomposition numbers of the block. This aim is almost achieved. In order to describe the results, let R be a suitable complete discrete valuation ring with residue field F, b a block (idempotent) of RG with cyclic defect group \(P\neq 1\), H the normalizer of the unique minimal subgroup of P and c the Brauer correspondent of b in RH. Then the source algebra of RHc has the form \(S\otimes_ RR[PE]\) where S is the R-algebra of R-endomorphisms of an indecomposable endo-permutation RP-module with vertex P and where PE denotes the semidirect product of P with the inertial factor group E of b (i.e. the order of E divides p-1). The major part of the paper is concerned with the structure of an R- algebra A which differs, roughly speaking, from the source algebra of b by S and which contains R[PE] as a unitary subalgebra. The author shows that the restriction of a simple A-module to RP remains indecomposable and that \(F\otimes_ RA\) is uniquely determined by the Brauer tree of b. Moreover, A can be computed from an exact sequence of projective A- modules defined by Green. The author determines the restrictions of the modules in this sequence to RP. A consequence of his results is the fact that the sources of all simple RGb-modules are direct summands of tensor products of indecomposable endopermutation FP-modules with vertex P with indecomposable FP-modules of dimension at most \(| E|\). What remains open is the precise form of the embedding of P into A although this is determined by the homomorphisms appearing in Green's exact sequence.
indecomposable endo- permutation RP-module
scientific article; zbMATH DE number 4114861