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Motivic twisted \(K\)-theory. (English) Zbl 1282.14040
Twisted \(K\)-theory introduced by P. Donovan and M. Karoubi [Publ. Math., Inst. Hautes Étud. Sci. 38, 5–25 (1970; Zbl 0207.22003)] and developed further by 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 E. Witten [J. High Energy Phys. 1998, No. 12, Paper No. 19, 41 p. (1998; Zbl 0959.81070)] and it continues today.
In 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. [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 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.

MSC:
14F42 Motivic cohomology; motivic homotopy theory
19L50 Twisted \(K\)-theory; differential \(K\)-theory
55P43 Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)
14F99 (Co)homology theory in algebraic geometry
19D99 Higher algebraic \(K\)-theory
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