$K$-motives of algebraic varieties

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 $H^{p,q}(X,Y,\mathbb{Z})$ 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.

Keywords

motivic homotopy theory, algebraic $K$-theory, spectral category

2010 Mathematics Subject Classification

14F42, 19E08, 55U35

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Published 4 December 2012