Relative \(K\)-theory and topological cyclic homology
type: Item
instance of: scholarly article
title: Relative \(K\)-theory and topological cyclic homology
zbMATH Open document ID: 0913.19002
DOI: 10.1007/BF02392744
author: Bjørn Ian Dundas
published in: Acta Mathematica
publication date: 1998-10-04T00:00:00Z
review text: This is a combined review with the article above [\textit{R. McCarthy}, Acta Math. 179, No. 2, 197-222 (1997; Zbl 0913.19001)]. From the introduction of McCarthy's paper: ``In recent years, the study of the algebraic \(K\)-theory space \(K(R)\) of a ring \(R\) has been approached by the introduction of spaces with a more homological flavor. One collection of such spaces is connected to \(K\)-theory by various trace maps and is particularly effective in measuring the relative \(K\)-theory \(K(f)\) associated to a surjective ring homomorphism \(f:R\to S\) whose kernel is nilpotent. \textit{T. G. Goodwillie}'s main theorem [in: Ann. Math., II. Ser. 124, 347-402 (1986; Zbl 0627.18004)]\ shows that the rational homotopy type of \(K(f)\) can be recovered from cyclic homology. The main theorem of this paper shows that, for any prime \(p\), the \(p\)-adic homotopy type of \(K(f)\) can be recovered from topological homology, \(\text{TC}(f)\).'' More precisely, if \(f:R\to S\) is a map of simplicial rings such that \(\pi_0 f\) is surjective with nilpotent kernel then \[ \text{trc}: K(f)^\wedge_p \overset \simeq \leftrightarrow \text{TC}(f)^\wedge_p \] where \(\text{trc}: K\to \text{TC}\) denotes the cyclic trace of \textit{M. Bökstedt, W. C. Hsiang} and \textit{I. Madsen} [``The cyclotomic trace and algebraic \(K\)-theory of spaces'', Invent. Math. 111, No. 3, 465-539 (1993; Zbl 0804.55004)]. The approach here is based on Goodwillie's calculus of functors. In outline: By an argument of Goodwillie it suffices to prove the theorem for maps of the form \[ f:A\times M\to A \] where \(A\) is a ring, \(M\) is an \(A\)-bimodule and \(A\times M\) is the twisted product with multiplication given by \((a,m)(a', m')=(aa', am'+ ma')\). Now define functors from simplicial sets to spectra by \[ F(X):= K(A\times \widetilde M(X))^\wedge_p \quad \text{ and } \quad G(X):= \text{TC}(A\times \widetilde M(X))^\wedge_p \] where \(\widetilde M(X)\) is the reduced free \(M\)-group generated by the s…
Mathematics Subject Classification ID: 19D55
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 19E20
Mathematics Subject Classification ID: 18G60
Mathematics Subject Classification ID: 55N20
Mathematics Subject Classification ID: 55N15
zbMATH DE Number: 1172064
zbMATH Keywords: simplicial rings
zbMATH Keywords: cyclic trace
zbMATH Keywords: simplicial sets
zbMATH Keywords: spectra
zbMATH Keywords: Eilenberg-MacLane spaces
zbMATH Keywords: Waldhausen's \(A\)-theory
zbMATH Keywords: free infinite loop space
zbMATH Keywords: stable \(K\)-theory
zbMATH Keywords: rational homotopy type
zbMATH Keywords: cyclic homology
zbMATH Keywords: \(p\)-adic homotopy type
zbMATH Keywords: topological cyclic homology
reviewed by: U. Tillmann
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1172064