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On the structure of certain injective modules over group algebras of soluble groups of finite rank

Journal of Algebra (Impact Factor: 0.6). 11/1983; 85(1):51–75. DOI: 10.1016/0021-8693(83)90118-7
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    ABSTRACT: The author presents some easily checked conditions for a module to be injective, and applies them in the situation where the module is a module algebra over a Hopf algebra H, where H has polynormal augmentation ideal and every finite-dimensional simple H-module is one dimensional. The quantum group O q (SL(2)) satisfies these conditions, and the author constructs a module algebra for it. He also uses these ideas to describe the Hopf dual for O q (SL(2)) and related Hopf algebras.
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    ABSTRACT: Let k be an uncountable field of characteristic zero and G a polycyclic-by-finite group. If P is a prime ideal of kG and the largest Ore set of elements of kG which are regular mod P, it is shown that the injective hull of any simple kG-module is artinian. The technique used is to construct normal elements in completions of kG at the augmentation ideals of certain abelian normal subgroups. We also show that if N is a normal subgroup of G such that N is torsion free nilpotent of finite rank and is polycyclic-by-finite, then the completion is Noetherian.
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    ABSTRACT: It is proved that if k is a field of characteristic p>0 and G a soluble-by-finite group of finite abelian section p-rank (i.e. all abelian sections have finite p-rank) then the largest locally finite dimensional submodule of the injective hull of any finite dimensional kG- module is Artinian. This result was proved for polycyclic-by-finite G by I. M. Musson [Q. J. Math., Oxf. II. Ser. 31, 429-448 (1980; Zbl 0413.16012)] and there is a corresponding result to Musson’s for fields of characteristic 0 due to S. Donkin [Proc. Lond. Math. Soc., III. Ser. 44, 333-348 (1982; Zbl 0483.20004)]. In a later paper I. M. Musson [J. Algebra 85, 51-75 (1983; Zbl 0525.16007)] extended Donkin’s theorem to finite extensions of torsion-free soluble groups of finite torsion-free rank. This present paper is written in the spirit of these earlier ones, the general idea being to use affine group schemes to derive the fact that the completion of kG with respect to its augmentation ideal is a Noetherian ring (when k is algebraically closed) and the augmentation ideal has a restricted form of the weak Artin-Rees property.
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