Motivic twisted<i>K</i>–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 BG m -bundle for the classifying space of the multiplicative group scheme G m . We show a Knneth 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 1 -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. 14F42, 55P43, 19L50; 14F99, 19D99