K-motives of algebraic varieties

Grigory Garkusha and Ivan Panin

A kind of motivic algebra of spectral categories and modules over them is developed to introduce K-motives of algebraic varieties. As an application, bivariant algebraic K-theory K(X,Y) as well as bivariant motivic cohomology groups Hp,q(X,Y,Z) are defined and studied. We use Grayson's machinery to produce the Grayson motivic spectral sequence connecting bivariant K-theory to bivariant motivic cohomology. It is shown that the spectral sequence is naturally realized in the triangulated category of K-motives constructed in the paper. It is also shown that ordinary algebraic K-theory is represented by the K-motive of the point.

Homology, Homotopy and Applications, Vol. 14 (2012), No. 2, pp.211-264.

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