Volume 12, issue 1 (2012)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 6, 3213–3852
Issue 5, 2565–3212
Issue 4, 1917–2564
Issue 3, 1283–1916
Issue 2, 645–1281
Issue 1, 1–643

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Subscriptions
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Author Index
To Appear
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Motivic twisted $K$–theory

Markus Spitzweck and Paul Arne Østvær

Algebraic & Geometric Topology 12 (2012) 565–599
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 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.

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