Hodge decompositions of Loday symbols in \(K\)-theory and cyclic homology
type: Item
instance of: scholarly article
title: Hodge decompositions of Loday symbols in \(K\)-theory and cyclic homology
zbMATH Open document ID: 0824.19002
DOI: 10.1007/BF00961021
author: Susan C. Geller
author: C. Weibel
published in: \(K\)-Theory
publication date: 1995-11-09T00:00:00Z
review text: Let \(A\) be a commutative ring with unit supposed to be of finite type over some field \(\ell\) of characteristic zero. Denote by \(HH_ *(A)\), \(HC_ *(A)\) and \(K_ *(A)\) the Hochschild homology, cyclic homology and the \(K\)-theory of \(A\), respectively. All three of them admit direct sum decompositions (Hodge decompositions) according to the action of the Adams operations \(\psi^ k\) and \(\lambda^ k\), e.g. \(HH_ n(A)= \bigoplus^ n_{i= 0} HH^{(i)}_ n(A)\), such that \(\psi^ k\) and \(\lambda^ k\) act as multiplication by \(k^{i+ 1}\) and \((- 1)^ k k^ i\), respectively, on \(HH^{(i)}_ n(A)\). Similarly for \(HC^{(i)}_ n(A)\). On \(K^{(i)}_ *(A)\), \(\psi^ k\) and \(\lambda^ k\) act as multiplication by \(k^ i\) and \((- 1)^ k k^{i- 1}\), respectively. In its greatest generality the above Hodge decompositions are not well understood. In the paper under consideration the Hodge decomposition for special elements in \(K\)-theory (and also in Hochschild and cyclic homology), the so-called Loday symbols, is explored. Hochschild homology, cyclic homology and \(K\)-theory are not unrelated: one has the SBI sequence for Hochschild homology and cyclic homology, the Dennis trace map from \(K\)-theory to Hochschild homology, and the Chern character from \(K\)-theory to cyclic homology. In particular, if \(A\) is a \(\mathbb{Q}\)-algebra the Dennis trace map factors through \(K_ m(A)\otimes \mathbb{Q}\) and one can compare the Hodge decomposition on \(K\)-theory with the Hodge decompositions on Hochschild and cyclic homologies, e.g. for such algebras the Dennis trace map preserves the Hodge decomposition of Loday symbols. If \(A= \bigoplus_{i\geq 0} A_ i\) is a graded \(\ell\)-algebra and \(H\) is any functor from \(\ell\)-algebras to abelian groups, let \(\widetilde H(A)\) denote the kernel of the augmentation map \(H(A)\to H(A_ 0)\). With this notation the Dennis trace map \(D\) factors as \(\widetilde K_ m(A)@> \nu>> HC_{m- 1}(A)@> B>>\widetilde{HH}_ m(A)\). It is show…
Mathematics Subject Classification ID: 19D55
zbMATH DE Number: 753669
zbMATH Keywords: Hodge decomposition
zbMATH Keywords: cyclic homology
zbMATH Keywords: Loday symbols
zbMATH Keywords: Hochschild homology
reviewed by: W. W. J. Hulsbergen
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00961021
OpenAlex ID: W2067078374
description: scientific article; zbMATH DE number 753669