On the K · Theory and Cyclic Homology of a Square

The purpose of this paper is to establish an integral connection between the relative K-theory and cyclic homology of a square-zero ideal IeR, 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, 1) is graded in analogy to relative K-theory and fits into a a long-exact sequence··· -> HCn(R) -> HCn(R/I)----> HCn _ 1 (R, I) -> HCn(R) -> .... In Section 3, we construct double-brackets symbols (after Loday), and prove their relevant properties. In Section 4, we prove that the summand of HCn _I (R, I) generated by these symbols splits off of Kn(R, 1) after inverting n. By severely restricting the quotient rings R/I under consideration, we get some information on . the remaining piece of Kn(R, 1). We also give an application to the stable K-groups KZ'(R, A) of Hatcher, Igusa and Waldhausen. It is a t~eorem of Goodwillie's that rationally there is an isomorphism K*(R.,I.) ® Q~ HC*_I (R*,! *)® Q when I.eR. is a nilpotent ideal (generalizing earlier results of Burghelea and Staffeldt). Therefore the splitting mentioned is only of interest when taken integrally. The author would like to thank Chuck Weibel for his careful reading of an earlier version of this paper and for the many helpful suggestions which followed. We now state the main results.