## Homology, Homotopy and Applications

### $K$-motives of algebraic varieties

#### 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 $K(X; Y)$ 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.

#### Article information

Source
Homology Homotopy Appl., Volume 14, Number 2 (2012), 211-264.

Dates
First available in Project Euclid: 12 December 2012

Garkusha, Grigory; Panin, Ivan. $K$-motives of algebraic varieties. Homology Homotopy Appl. 14 (2012), no. 2, 211--264. https://projecteuclid.org/euclid.hha/1355321489