## Motivic twisted K-theory (2010)

### BibTeX

@MISC{Spitzweck10motivictwisted,

author = {Markus Spitzweck and Paul Arne Østvær},

title = {Motivic twisted K-theory},

year = {2010}

}

### OpenURL

### 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 BGm-bundle for the classifying space of the multiplicative group scheme. We show a Künneth isomorphism for homological motivic twisted K-groups computing the latter as a tensor product of K-groups over the K-theory of BGm. The proof employs an Adams Hopf algebroid and a tri-graded Tor-spectral sequence for motivic twisted K-theory. By adopting the notion of an E∞-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.