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Motivic twisted $K$–theory

Markus Spitzweck and Paul Arne Østvær

Algebraic & Geometric Topology 12 (2012) 565–599

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 Gm. 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 trigraded Tor-spectral sequence for motivic twisted K–theory. By adapting 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.

motivic homotopy theory, twisted $K$–theory, motivic cohomology, bundle, Adams Hopf algebroid
Mathematical Subject Classification 2010
Primary: 14F42, 55P43, 19L50
Secondary: 14F99, 19D99
Received: 7 April 2011
Revised: 29 November 2011
Accepted: 19 December 2011
Published: 29 March 2012
Markus Spitzweck
Fakultät für Mathematik
Universität Regensburg
D-93040 Regensburg
Paul Arne Østvær
Department of Mathematics
University of Oslo
0316 Oslo