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K (A, I)→ K (A)→ K (A/I)→ K (A, I)[− 1] and an induced natural long-exact sequence of K-groups→ Kq (A, I)→ Kq (A)→ Kq (A/I)→ Kq-1 (A, I)→ If the ideal ICA is nilpotent, the relative K-theory K (A, I) can be expressed completely in terms of the cyclic homology of Connes [14] and the topological cyclic homology of Bökstedt-Hsiang-Madsen [7]. Indeed, on the one hand, Goodwillie [21] has shown that rationally