Induced resolutions and grothendieck groups of polycyclic-by-finite-groups

Let G{cyrillic} be a polycyclic-by-finite group, R a commutative Noetherian ring, G₀(RG{cyrillic}) the Grothendieck group of finitely generated RG{cyrillic}-modules, and G₀(RG{cyrillic},F) the subgroup generated by the classes of modules induced from finite subgroups of G{cyrillic}. It is expected that G₀(RG{cyrillic})=G₀(RG{cyrillic},F). As partial evidence for this, we show that G₀(RG{cyrillic}) G₀(RG{cyrillic},F) is torsion, with an explicit bound for the exponent, in the case where G{cyrillic} is abelian-by-finite and R a regular Noetherian Hilbert ring of finite global dimension. © 1988.