Motivic twisted K–theory

Spitzweck, Markus; Østvær, Paul Arne

doi:10.2140/agt.2012.12.565

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Motivic twisted $K$–theory

Markus Spitzweck and Paul Arne Østvær

Algebraic & Geometric Topology 12 (2012) 565–599

DOI: 10.2140/agt.2012.12.565

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 $B G_{m}$ –bundle for the classifying space of the multiplicative group scheme $G_{m}$ . 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 $B G_{m}$ . The proof employs an Adams Hopf algebroid and a trigraded Tor-spectral sequence for motivic twisted $K$ –theory. By adapting the notion of an $E_{∞}$ –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

References

Publication

Received: 7 April 2011

Revised: 29 November 2011

Accepted: 19 December 2011

Published: 29 March 2012

Authors

Markus Spitzweck
Fakultät für Mathematik Universität Regensburg D-93040 Regensburg Germany
http://www.uni-math.gwdg.de/spitz
Paul Arne Østvær
Department of Mathematics University of Oslo 0316 Oslo Norway

Volume 12, issue 1 (2012)