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

Abstract

Let Г be a polycyclic-by-finite group, R a commutative Noetherian ring, G 0( RГ) the Grothendieck group of finitely generated RГ-modules, and G 0( RГ, F) the subgroup generated by the classes of modules induced from finite subgroups of Г. It is expected that G 0 (RГ)=G 0 (RГ, F ) . As partial evidence for this, we show that G 0 (RГ) G 0 (RГ, F ) is torsion, with an explicit bound for the exponent, in the case where Г is abelian-by-finite and R a regular Noetherian Hilbert ring of finite global dimension.