Open Access
Translator Disclaimer
2012 Motivic twisted $K$–theory
Markus Spitzweck, Paul Arne Østvær
Algebr. Geom. Topol. 12(1): 565-599 (2012). DOI: 10.2140/agt.2012.12.565


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.


Download Citation

Markus Spitzweck. Paul Arne Østvær. "Motivic twisted $K$–theory." Algebr. Geom. Topol. 12 (1) 565 - 599, 2012.


Received: 7 April 2011; Revised: 29 November 2011; Accepted: 19 December 2011; Published: 2012
First available in Project Euclid: 19 December 2017

zbMATH: 1282.14040
MathSciNet: MR2916287
Digital Object Identifier: 10.2140/agt.2012.12.565

Primary: 14F42 , 19L50 , 55P43
Secondary: 14F99 , 19D99

Keywords: Adams Hopf algebroid , bundle , motivic cohomology , motivic homotopy theory , twisted $K$–theory

Rights: Copyright © 2012 Mathematical Sciences Publishers


Vol.12 • No. 1 • 2012
Back to Top