Abstract
This paper sets out basic properties of motivic twisted K-theory with respect to degree three motivic cohomology classes of weight one. Motivic twisted K-theory is defined in terms of such motivic cohomology classes by taking pullbacks along the universal principal BG(m)-bundle for the classifying space of the multiplicative group scheme G(m). We show a Kunneth isomorphism for homological motivic ...
Abstract
This paper sets out basic properties of motivic twisted K-theory with respect to degree three motivic cohomology classes of weight one. Motivic twisted K-theory is defined in terms of such motivic cohomology classes by taking pullbacks along the universal principal BG(m)-bundle for the classifying space of the multiplicative group scheme G(m). We show a Kunneth isomorphism for homological motivic twisted K-groups computing the latter as a tensor product of K-groups over the K-theory of BG(m). The proof employs an Adams Hopf algebroid and a trigraded Tor-spectral sequence for motivic twisted K-theory. By adapting the notion of an E-infinity-ring spectrum to the motivic homotopy theoretic setting, we construct spectral sequences relating motivic (co)homology groups to twisted K-groups. It generalizes various spectral sequences computing the algebraic K-groups of schemes over fields. Moreover, we construct a Chern character between motivic twisted K-theory and twisted periodized rational motivic cohomology, and show that it is a rational isomorphism. The paper includes a discussion of some open problems.
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