Motivic twisted K-theory

TL;DR

This paper develops the foundational theory of motivic twisted K-theory, establishing key properties, spectral sequences, and a Chern character, thereby advancing the understanding of twisted K-theory in the motivic setting.

Contribution

It introduces a motivic twisted K-theory framework, proves a Kuenneth isomorphism, constructs spectral sequences, and establishes a rational Chern character, extending algebraic K-theory computations.

Findings

Kuenneth isomorphism for motivic twisted K-groups

Spectral sequences relating motivic (co)homology to twisted K-theory

Chern character as a rational isomorphism

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. We show a Kuenneth 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 tri-graded Tor-spectral sequence for motivic twisted K-theory. By adopting 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.