I am working my way through Ken Browns book on the cohomology of groups, and in particular chapters 8 and 9 on finiteness conditions, and Euler characteristics.
Most of the concepts in chapter 9 (such as the Hattori-Stallings rank) are defined (initially) in terms of finitely generated projective $\mathbb{Z} G$-moudles, and one thing that I am struggling with in general is what these (non-free) projective $\mathbb{Z} G$-modules look like for different groups $G$.
For instance, in the case of a finite group the projective $\mathbb{Z} G$-modules were classified by Swan: any projective $\mathbb{Z} G$-module is the direct sum of a free module and an ideal $I\le \mathbb{Z} G$.
However, the groups I am interested in are not finite: the case that would be of initial interest is $G=\mathrm{SL}_{2}(\mathbb{Z})$.
So what can be said about projective $\mathbb{Z} [\mathrm{SL}_{2}(\mathbb{Z})]$-modules?
I am also interested in the general case when $G\le\mathrm{SL}_{n}(\mathbb{Z})$ is a finite index subgroup (say a principal congruence subgroup), but here I am guessing that the projective modules over $G$ will be related to those over $\mathrm{SL}_{n}(\mathbb{Z})$ using some standard homological algebra.
I have seen some other papers about resolutions over linear group rings (1), and projective modules over group rings for some groups (2, 3) but I haven't found anything explicitly for the case $G\le\mathrm{SL}_{n}(\mathbb{Z})$ which is what I was hoping for.
Any help, comments, or suggestions are greatly appreciated.