This book combines foundational constructions in the theory of motives and results relating motivic cohomology to more explicit constructions. Prerequisite for understanding the work is a basic background in algebraic geometry.

The author constructs and describes a triangulated category of mixed motives over an arbitrary base scheme. Most of the classical constructions of cohomology are described in the motivic setting, including Chern classes from higher $K$-theory, push-forward for proper maps, Riemann-Roch, duality, as well as an associated motivic homology, Borel-Moore homology and cohomology with compact supports.

Readership

Graduate students and research mathematicians interested in algebraic geometry and $K$-theory.

Reviews

"All in all, everyone interested in mixed motives and willing to take a serious look at the topic, should try his/her hand on this impressive work."

-- Zentralblatt MATH

"We must go out of our way to ensure that our libraries acquire books like this, and then we should `encourage' our best PhD students to read them!"

-- Bulletin of the London Mathematical Society

Table of Contents

• Introduction: Part I
• The motivic category
• Motivic cohomology and higher Chow groups
• K-theory and motives
• Homology, cohomology and duality
• Realization of the motivic category
• Motivic constructions and comparisons
• Equi-dimensional cycles
• K-theory

• Introduction: Part II
• Symmetric monoidal structures
• DG categories and triangulated categories
• Simplicial and cosimplicial constructions
• Canonical models for cohomology
• Bibliography
• Subject index
• Index of notation