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Decomposition of augmentation ideals and relation modules. (English) Zbl 0531.20002

Let R be a commutative ring, let G be a group, and let \(...\to P_ n\to^{\partial_ n}...\to^{\partial_ 1}P_ 0\to^{\partial_ 0}R\to 0\) be a projective resolution of the trivial RG-module R. For \(i\geq 0\), write \(k_ i=im \partial_ i.\) If d is a non-negative integer, an RG-module K is a d-th kernel for G over R if there exists such a projective resolution with \(K_ d\cong K\). The basic problem discussed in this paper is: When are d-th kernels indecomposable? Particular interest centres on the cases \(R={\mathbb{Z}}\) and \(d=0\) or \(d=1\), which yield the augmentation ideal of \({\mathbb{Z}}G\), and the relation modules of \({\mathbb{Z}}G\), respectively. Let S be any ring. An S-module M is called a Heller module if whenever \(M\oplus F\cong U\oplus V\) as S- modules, where F is a free module of finite rank, then U or V is projective; M is strongly Heller if the same conclusion holds when F is allowed to have arbitrary rank. The strategy employed here is to obtain conditions on G ensuring that d-th kernels are (strong) Heller modules, then apply these to study the existence of decompositions of augmentation ideals and relation modules. This pattern (and many of the other ideas which appear here) were first used to handle the case where G is finite [K. W. Gruenberg and K. W. Roggenkamp, Proc. Lond. Math. Soc., III. Ser. 31, 149-166 (1975; Zbl 0313.20004)]. There are many interesing results in this paper, but their statements and proofs are too technical to elaborate on in this review.
Reviewer: K.A.Brown

MSC:

20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
20C12 Integral representations of infinite groups

Citations:

Zbl 0313.20004
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