On the K-theory and cyclic homology of a square-zero ideal I
The purpose of this paper is to establish an integral connection between the relative K-theory and cyclic homology of a square-zero ideal \(I\subset R\), R a ring with unit. We restrict attention to the case when R is a split extension of R/I by I. The outline of the paper is as follows: in Section 2, we compute a direct summand of \(HC_*(R,I)\), where \(HC_*(R,I)\) is graded in analogy to relative K-theory and fits into a long-exact sequence \[ ...\quad \to \quad HC_ n(R)\quad \to \quad HC_ n(R/I)\to^{\partial}HC_{n-1}(R,I)\quad \to \quad HC_ n(R)\quad \to \quad...\quad. \] In Section 3, we construct double- brackets symbols (after Loday), and prove their relevant properties. In Section 4, we prove that the summand of \(HC_{n-1}(R,I)\) generated by these symbols splits off of \(K_ n(R,I)\) after inverting n. By severely restricting the quotient rings R/I under consideration, we get some information on the remaining piece of \(K_ n(R,I)\). We also give an application to the stable K-groups \(K^ m_*(R,A)\) of Hatcher, Igusa and Waldhausen. It is a theorem of Goodwillie's that rationally there is an isomorphism \(K_*(R_.,I_.)\otimes Q\to^{\cong}HC_{*- 1}(R_*I_*)\otimes Q\) when \(I_.\subset R_.\) is a nilpotent ideal (generalizing earlier results of Burghelea and Staffeldt). Therefore the splitting mentioned is only of interest when taken integrally.
scientific article; zbMATH DE number 4077520
scientific article