Chern maps for \(U\)-theory
type: Item
instance of: scholarly article
title: Chern maps for \(U\)-theory
zbMATH Open document ID: 0867.19004
DOI: 10.1016/S0022-4049(95)00174-3
author: Jack M. Shapiro
author: Dongyuan Yao
published in: Journal of Pure and Applied Algebra
publication date: 1997-07-20T00:00:00Z
review text: Let \({\mathcal A}\) be an exact \(k\)-linear category with a duality functor. In this paper the authors define the homologies and their variants for \({\mathcal A}\) using a Quillen's \(Q\)-construction and show that such a construction yields a generalization of the Dennis trace map and the Jones-Goodwillie Chern map into the \(U\)-theory of \({\mathcal A}\). This is done as follows. They first define the Dennis trace map from \(K_*({\mathcal A})\) to \(HH_*({\mathcal A})\) and the Jones-Goodwillie Chern map from \(K_*({\mathcal A})\) to \(HN_*({\mathcal A})\), which are equivalent to the maps defined by \textit{R. McCarthy} using the \(S\)-construction. These maps are extended to \(_\varepsilon U_*({\mathcal A})\) by composing these maps with the canonical map from \(_\varepsilon U_*({\mathcal A})\) to \(K_*({\mathcal A})\) which comes from forgetting the \(\varepsilon\)-hermitian form. It is proven that when 2 is invertible in the ground ring \(k\), these generalized maps factor through the corresponding involutive homologies. And also it is shown that sliding \(2p\) dimensions to the positive direction in particular the second one can be transformed into a map from \(_\varepsilon U_*({\mathcal A})\) to the dihedral homology \(HD_{*+2p} ({\mathcal A})\) which is the involutive cyclic homology. Furthermore the authors note that if all short exact sequences in \({\mathcal A}\) split, then the \(L\)-theory \(_\varepsilon L_*({\mathcal A})\) of \({\mathcal A}\) can be defined analogously and statements similar to the above can be made. This provides another proof of the early results of \textit{G. Cortinas} and \textit{J. Lodder}.
reviewed by: Haruo Minami
Mathematics Subject Classification ID: 19D55
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 19D06
zbMATH DE Number: 986705
zbMATH Keywords: exact category
zbMATH Keywords: \(U\)-theory
zbMATH Keywords: Chern map
zbMATH Keywords: dihedral homology
MaRDI profile type: Publication
cites work: On a theorem of Giffen
cites work: $L$-theory and dihedral homology.
cites work: \(L\)-theory and dihedral homology. II
cites work: Relative algebraic K-theory and cyclic homology
cites work: Some algebraic properties of cyclic homology groups
cites work: Le théorème fondamental de la K-théorie hermitienne
cites work: Q5288307
cites work: Homologies diédrale et quaternionique. (Dihedral and quaternionic homology)
cites work: The cyclic homology of an exact category
cites work: Configuration-spaces and iterated loop-spaces
cites work: Hermitian \(\mathcal{U}\)-theory of exact categories with duality functors
cites work: Q3934683
cites work: Q3701607
description: scientific article; zbMATH DE number 986705