## Homology, Homotopy and Applications

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

### $K$-motives of algebraic varieties

Grigory Garkusha and Ivan Panin

#### 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

**Permanent link to this document**

https://projecteuclid.org/euclid.hha/1355321489

**Mathematical Reviews number (MathSciNet)**

MR3007094

**Zentralblatt MATH identifier**

1284.14029

**Subjects**

Primary: 14F42: Motivic cohomology; motivic homotopy theory [See also 19E15] 19E08: $K$-theory of schemes [See also 14C35] 55U35: Abstract and axiomatic homotopy theory

**Keywords**

Motivic homotopy theory algebraic $K$-theory spectral category

#### Citation

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