K-theory of group-rings of finite groups over maximal orders in division algebras
Let \(\pi\) be a finite group, R the ring of integers in a p-adic field F and A a maximal R-order in a central division algebra D over F. By a result of \textit{R. G. Swan} [Topology 2, 85-110 (1963; Zbl 0119.029)] there is an isomorphism \(G_ 0(R\pi)\to G_ 0(F\pi).\) The author shows that this result does not extend to the non-commutative setting, i.e. the map \(G_ 0(A\pi)\to G_ 0(D\pi)\) is not always injective. This observation then leads to various finiteness theorems about the kernel of \(G_ n(A\pi)\to G_ n(D\pi)\) for \(n\geq 0\) both in the local and in the global case. We only mention the following more general result, which settles a question posed by Swan: Let R be the ring of integers in a number field F and \(\Lambda\) any R-order in a separable F-algebra \(\Sigma\). For all \(n\geq 0\) the group \(G_ n(\Lambda)\) is finitely generated and the kernel of \(G_ n(\Lambda)\to G_ n(\Sigma)\) is a finite group, which is trivial for even n.
scientific article; zbMATH DE number 3873528
scientific article