K-motives of algebraic varieties

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.