Chern classes for twisted \(K\)-theory
type: Item
instance of: scholarly article
title: Chern classes for twisted \(K\)-theory
zbMATH Open document ID: 1097.14005
author: Mark E. Walker
published in: Journal of Pure and Applied Algebra
publication date: 2006-05-29T00:00:00Z
review text: The spectrum level version of the total Chern class map is the map of infinite loop spaces from the \(K\)-theory space of \(X\) taking values in an infinite loop space \({\mathcal H}_{\text{mult}}(X)\) such that its homotopy groups give the units of the motivic cohomology. Such constructions were performed in [\textit{C. P. Boyer} et al., Invent. Math. 113, No.~2, 373--388 (1993; Zbl 0797.55006)] and by \textit{E. Friedlander} and the author [Topology 41, No.~3, 591--644 (2002; Zbl 1003.19003)]. Let \(k\) be a field, \(A\) a finite dimensional central simple \(k\)-algebra, and \(X\) a smooth \(k\)-variety. The author studies the question of Chern classes for coherent \({\mathcal O}_X\otimes A\)-modules which are locally free \({\mathcal O}_X\)-modules. One might try to define the total Chern class map in this context. The basic problem here is to construct the appropriate target for such a map i.e. the twisted form of the motivic cohomology. The author provides such a construction, which is different from Kahn-Levine twisted Chow groups.
reviewed by: Piotr Krasoń
Mathematics Subject Classification ID: 14C17
Mathematics Subject Classification ID: 14F42
Mathematics Subject Classification ID: 19E15
zbMATH DE Number: 5028790
zbMATH Keywords: Chern class map
zbMATH Keywords: motivic cohomology
zbMATH Keywords: central simple algebra
MaRDI profile type: Publication
full work available at URL: j.jpaa.2005.06.020
OpenAlex ID: W2078811223
cites work: Algebraic cycles and infinite loop spaces
cites work: Locally trivial principal homogeneous spaces
cites work: Graded Brauer groups and K-theory with local coefficients
cites work: A theory of algebraic cocycles
cites work: Moving algebraic cycles of bounded degree
cites work: The spectral sequence relating algebraic K-theory to motivic cohomology
cites work: Q4513079
cites work: Semi-topological \(K\)-theory using function complexes
cites work: Rational isomorphisms between \(K\)-theories and cohomology theories
cites work: Q3328651
cites work: Weight filtrations via commuting automorphisms
cites work: Geometric models for algebraic \(K\)-theory
cites work: Q3877755
cites work: K-theory of twisted Grassmannians
cites work: Q3129437
cites work: Q3582378
cites work: Critères de platitude et de projectivité. Techniques de ''platification'' d'un module. (Criterial of flatness and projectivity. Technics of ''flatification of a module.)
cites work: Quaternionic algebraic cycles and reality
cites work: Q4681884
cites work: Q4513077
cites work: Q4513080
DOI: 10.1016/J.JPAA.2005.06.020
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description: scientific article