K-motives of algebraic varieties
Grigory Garkusha, Ivan Panin

TL;DR
This paper develops a motivic algebra framework for algebraic varieties to define and analyze K-motives, bivariant K-theory, and motivic cohomology, establishing spectral sequences and representations within a triangulated category.
Contribution
It introduces K-motives of algebraic varieties using spectral categories, connecting bivariant K-theory and motivic cohomology through a new motivic spectral sequence.
Findings
Construction of K-motives in a spectral categorical setting
Definition and study of bivariant algebraic K-theory and motivic cohomology
Realization of the Grayson motivic spectral sequence within the K-motive framework
Abstract
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 as well as bivariant motivic kohomology groups are defined and studied. We use Grayson's machinery to produce the Grayson motivic spectral sequence connecting bivariant K-theory to bivariant motivic kohomology. 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.
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