Filtered topological cyclic homology and relative K–theory of nilpotent ideals

Abstract

In this paper certain filtrations of topological Hochschild homology and topological cyclic homology are examined. As an example we show how the filtration with respect to a nilpotent ideal gives rise to an analog of a theorem of Goodwillie saying that rationally relative $K$–theory and relative cyclic homology agree. Our variation says that the $p$–torsion parts agree in a range of degrees. We use it to compute $K_i\left(\mathbb{Z}∕p^n\right)$ for $i\le p-3$.