#### Volume 12, issue 1 (2012)

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Motivic twisted $K$–theory

### Markus Spitzweck and Paul Arne Østvær

Algebraic & Geometric Topology 12 (2012) 565–599
##### 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 $\mathsf{B}{\mathbf{G}}_{m}$–bundle for the classifying space of the multiplicative group scheme ${\mathbf{G}}_{m}$. 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 $\mathsf{B}{\mathbf{G}}_{m}$. The proof employs an Adams Hopf algebroid and a trigraded Tor-spectral sequence for motivic twisted $K$–theory. By adapting the notion of an ${E}_{\infty }$–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.

##### Keywords
motivic homotopy theory, twisted $K$–theory, motivic cohomology, bundle, Adams Hopf algebroid
##### Mathematical Subject Classification 2010
Primary: 14F42, 55P43, 19L50
Secondary: 14F99, 19D99