On the K-theory and cyclic homology of a square-zero ideal I
type: Item
instance of: scholarly article
title: On the K-theory and cyclic homology of a square-zero ideal I
zbMATH Open document ID: 0659.18013
DOI: 10.1016/0022-4049(87)90095-8
author: Crichton Ogle
published in: Journal of Pure and Applied Algebra
publication date: 1987-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 18G99
Mathematics Subject Classification ID: 18F25
zbMATH DE Number: 4077520
zbMATH Keywords: relative K-theory
zbMATH Keywords: cyclic homology
zbMATH Keywords: square-zero ideal
zbMATH Keywords: stable K-groups
MaRDI profile type: Publication
full work available at URL: 0022-4049(87)90095-8
OpenAlex ID: W1973054693
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description: scientific article; zbMATH DE number 4077520