Motivic twisted $K$–theory

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.