2010/08/29 by Markus Spitzweck, Spitzweck, Markus, Paul Arne Østvær +1
Mathematics · #14F42 #19E08 #55P43 #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics #K-Theory and Homology (math.KT) #math.AG #math.AT #math.KT #msc:14F42 #msc:19E08 #msc:55P43
paper · pdf · doi:10.48550/arxiv.1008.4915
35 pages
arxiv created 2010/08/29 · arxiv updated 2010/08/31
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. 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 BGm. 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.