  In this paper we examine certain filtrations of topological Hochschild
homology and topological cyclic homology. 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(Z/p^m) for i < p-2.
