The Behavior of Nil-Groups under Localization and the Relative Assembly Map

We study the behavior of the Nil-subgroups of K-groups under localization. As a consequence we obtain that the relative assembly map from the family of finite subgroups to the family of virtually cyclic subgroups is rationally an isomorphism. Combined with the equivariant Chern character we obtain a complete computation of the rationalized source of the K-theoretic assembly map in terms of group homology. Specifically we prove that under mild assumptions we can always write the Nil-groups and End-groups of the localized ring as a certain colimit over the Nil-groups and End-groups of the ring, generalizing a result of Vorst. We define Frobenius and Verschiebung operations on certain Nil-groups. These operations provide the tool to prove that Nil-groups are modules over the ring of Witt-vectors and are either trivial or not finitely generated as abelian groups. Combining the localization results with the Witt-vector module structure we obtain that Nil and localization at an appropriate multiplicatively closed set S commute, i.e. SNil = NilS. An important corollary is that the Nil-groups appearing in the decomposition of the K-groups of virtually cyclic groups are torsion groups.