On Modules over Infinite Group Rings

Let $R$ be a commutative ring and $\Gamma$ be an infinite discrete group. The algebraic $K$-theory of the group ring $R[\Gamma]$ is an important object of computation in geometric topology and number theory. When the group ring is Noetherian there is a companion $G$-theory of $R[\Gamma]$ which is often easier to compute. However, it is exceptionally rare that the group ring is Noetherian for an infinite group. In this paper, we define a version of $G$-theory for any finitely generated discrete group. This construction is based on the coarse geometry of the group. Therefore it has some expected properties such as independence from the choice of a word metric. We prove that, whenever $R$ is a regular Noetherian ring of finite global homological dimension and $\Gamma$ has finite asymptotic dimension and a finite model for the classifying space $B\Gamma$, the natural Cartan map from the $K$-theory of $R[\Gamma]$ to $G$-theory is an equivalence. On the other hand, our $G$-theory is indeed better suited for computation as we show in a separate paper. Some results and constructions in this paper might be of independent interest as we learn to construct projective resolutions of finite type for certain modules over group rings.