Filtrations, stable Clifford theory and group-graded rings and modules
Let \({\mathcal R}\) be a commutative ring, N a normal subgroup of a finite group G, M an \({\mathcal R}N\)-module which is stable in G. E. Cline's ``Stable Clifford theory'' looked at the ``decomposition'' of an \({\mathcal R}G\)-module I whose restriction \(I_ N\) to N was a direct sum of copies of M as a tensor product. This was given a categorical formulation by E. C. Dade. Both have as a background group-graded rings and modules, principally graded by the elements of the group \(\bar G=G/N.\) More recently, \textit{J. L. Alperin}, \textit{M. J. Collins} and \textit{D. A. Sibley} [Bull. Lond. Math. Soc. 16, 416-420 (1984; Zbl 0533.20003)], used filtrations (induced by the Jacobson radical and the augmentation ideal of \(kN\) (k a field of characteristic p)) to relate the projective covers Q and \(\bar Q\) of a simple \(k\bar G\)-module, the covers being considered as \(kG\)- and \(k\bar G\)-modules respectively. Q and \(\bar Q\otimes L\) have the same composition factors, for a certain kG-module L constructed via these filtrations. The author has in mind various generalizations of these results and sets down the necessary theory of group-graded rings and modules, compounded by filtrations, to discuss these generalizations.
scientific article; zbMATH DE number 4040152
scientific article