Motivic twisted \(K\)-theory
type: Item
instance of: scholarly article
title: Motivic twisted \(K\)-theory
zbMATH Open document ID: 1282.14040
author: Markus Spitzweck
author: Paul Arne Østvær
published in: Algebraic \& Geometric Topology
publication date: 2012-05-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/1008.4915
reviewed by: Piotr Krasoń
Mathematics Subject Classification ID: 14F42
Mathematics Subject Classification ID: 19L50
Mathematics Subject Classification ID: 55P43
Mathematics Subject Classification ID: 14F99
Mathematics Subject Classification ID: 19D99
zbMATH DE Number: 6035488
zbMATH Keywords: motivic homotopy theory
zbMATH Keywords: twisted \(K\)-theory
zbMATH Keywords: motivic cohomology
zbMATH Keywords: principal bundle
zbMATH Keywords: Adams Hopf algebroid
MaRDI profile type: Publication
OpenAlex ID: W2963312431
arXiv ID: 1008.4915
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DOI: 10.2140/AGT.2012.12.565
review text: Twisted \(K\)-theory introduced by \textit{P. Donovan} and \textit{M. Karoubi} [Publ. Math., Inst. Hautes Étud. Sci. 38, 5--25 (1970; Zbl 0207.22003)] and developed further by \textit{J. Rosenberg} [J. Aust. Math. Soc., Ser. A 47, No. 3, 368--381 (1989; Zbl 0695.46031)] in the direction of analysis is a good example of fruitful application of topological methods in physics. This is evident in the work of \textit{E. Witten} [J. High Energy Phys. 1998, No. 12, Paper No. 19, 41 p. (1998; Zbl 0959.81070)] and it continues today.NEWLINENEWLINEIn the paper the authors define motivic twisted \(K\)-theory with respect to degree three motivic cohomology classes of weight one. Their approach is in the context of motivic homotopy theory (cf. [\textit{B. Dundas} et al., Motivic homotopy theory. Lectures at a summer school in Nordfjordeid, Norway, August 2002. Berlin: Springer (2007; Zbl 1118.14001)] and uses pullbacks along the universal principal \(B{\mathbf{G}}_{m}\)-bundle for the classifying space of the multiplicative group scheme \({\mathbf{G}}_{m}.\) Then basic properties such as Künneth isomorphism for homological motivic twisted \(K\)-groups are proved. The constructions of spectral sequences relating motivic (co)homology groups to twisted \(K\)-groups and a Chern character between motivic twisted \(K\)-theory and twisted periodized rational motivic cohomology are also given. The authors show that the Chern character is a rational isomorphism. There is also more algebraic definition of twisted \(K\)-theory due to \textit{M. E. Walker} [J. Pure Appl. Algebra 206, No. 1--2, 153--188 (2006; Zbl 1097.14005)], but as authors state it is not evident how to compare both.
description: scientific article; zbMATH DE number 6035488