Modules for group algebras
D. J. Benson · Cambridge University Press eBooks · 1991
In Chapter 1 we gave a brief summary of some standard material on rings and modules. In this chapter we investigate what more we can say if the ring is the group algebra RG of a finite group G over a ring of coefficients R . The major new feature we find here is that we may give the tensor product over R of two RG -modules the structure of an RG -module. Of course, we also try to relate the subgroup structure of the group with the representation theory. Throughout this chapter, R will denote a commutative ring of coefficients, and κ will denote a field of coefficients. All RG -modules and κ G -modules will be finitely generated . Operations on RG -modules D efinition 3.1.1. If G is a finite group and R is a commutative ring, we may form the group ring RG whose elements are the formal linear combinations with r i ∈ R and g i ∈ G . Addition and multiplication are given by Thus RG is an R -algebra, which as an R -module is free of rank | G |. Of course, this definition also makes sense for infinite groups, provided we restrict our attention to finite sums. The group ring RG is an augmented algebra with augmentation ε : RG → R given by (cf. Section 2.4). Thus it makes sense to talk of the trivial RG -module R . We write H n ( G, M ) and H n ( G, M ) for the homology and cohomology groups with coefficients in M , namely the groups H n ( RG, M ) and H n ( RG, M ) defined in Section 2.6. Note that in the former case we should regard the left RG -module M as a right module via mg = g −1 m .