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In this paper we examine certain filtrations of topological Hochschildhomology and topological cyclic homology. As an example we show how thefiltration with respect to a nilpotent ideal gives rise to an analog of atheorem of Goodwillie saying that rationally relative K-theory and relativecyclic homology agree. Our variation says that the p-torsion parts agree in arange of degrees. We use it to compute K_i(Z/p^m) for i < p-2.

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