Some finiteness results in the higher K-theory of orders and group-rings

AbstractLet R be the ring of integers in a number field F, A any R-order in a semi-simple F-algebra Σ. It is well known that for n = 0, 1, Kn(A) is a finitely generated Abelian group and SKn(A) is finite and that a similar result holds for Gn(A) and SGn(A) for n = 0, 1.We now prove that for all n ⩾ 1, Kn(A), Gn(A) are finitely generated Abelian groups and that SKn(A), SGn(A) are finite groups while SG2n (A) = 0. If p is a prime of R, Ap, (Âp) the localisation (completion) of A at p, then we also obtain similar finiteness results for G2n−1(A), SKn(Ap), SKn(Âp), SGn(Ap), SGn(Âp). These results apply in particular to A=Rπ,Ap=Rpπ and Âp=R̂pπ where π is any finite group. We then show that for all n⩾1, K4n+3(Zπ), G4n+3(Zπ) and G4n+3(Z(p)π) are finite groups