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# Homology, Homotopy and Applications

## Volume 14 (2012)

### Number 2

### $K$-motives of algebraic varieties

Pages: 211 – 264

DOI: https://dx.doi.org/10.4310/HHA.2012.v14.n2.a13

#### Authors

#### 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.

#### Keywords

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

#### 2010 Mathematics Subject Classification

14F42, 19E08, 55U35

Published 4 December 2012