Chern classes for twisted \(K\)-theory
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.
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