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 of our results, 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 that appears in the Farrell–Jones conjecture in terms of group homology and the $K$-groups of finite cyclic subgroups.

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. $S^{-1}\mathrm{Nil}=\mathrm{Nil}{S}^{-1}$. An important corollary is that the Nil-groups appearing in the decomposition of the $K$-groups of virtually cyclic groups are torsion groups.