Units of group rings

Abstract

Let G be an abelian group and let ZG be its integral group ring. In special cases (e.g., G finite [4]) it has been noted that G is a direct summand of (ZG)*, the group of units of ZG. However, no explicit construction for the splitting maps is given in the literature. In Section 1 it will be shown that for any abelian group G, there is a canonical splitting of the inclusion G→ZG*. Upon attempting to generalize this result to other coefficient rings, we are led to the concept of a semimodule. We show in Section 2 that G→(AG)* splits if G admits a semimodule structure over A. The final section contains a number of partial results and examples showing the difficulty of deciding whether or not splittings exist in the general case. This paper originated from the construction of maps in algebraic K-theory using the Hochschild homology of a bimodule [6, Chap.X]. Let G→GLnZG be given by gdiag (g, 1,..., 1). It will be shown elsewhere that for an arbitrary group G, there is a canonical splitting of the induced map Hi(G;Z)→Hi(GLn(ZG);Z) (ordinary homology of groups with trivial action on Z for all i ≥ 0, n ≥ 1. As the case i = n = 1 is of independent interest, we present it here rather than obscure it in a paper dealing with algebraic K-theory. For a given integer i, the splitting is constructed via properties of Hochschild homology groups Hi(R, R) for certain rings R. For i = 1 and R a commutative ring, H1(R, R) is canonically isomorphic to Ω1 R Z, the module of Kahler differentials. In Section 1 we use Ω1 R Z and its properties to construct splittings as it will be more familiar than Hochschild homology groups to most readers. © 1976.