Mixed motives

Abstract

The author constructs and describes a triangulated category of mixed motives over anarbitrary base scheme. The resulting cohomology theory satisfies the Bloch-Ogus axioms; if thebase scheme is a smooth scheme of dimension at most one over a field, this cohomology theoryagrees with Bloch’s higher Chow groups. Most of the classical constructions of cohomology canbe made in the motivic setting, including Chern classes from higherK-theory, push-forward forproper maps, Riemann-Roch, duality, as well as an associated motivic homology, Borel-Moorehomology and cohomology with compact supports. The motivic category admits a realizationfunctor for each Bloch-Ogus cohomology theory which satisfies certain axioms; as examples theauthor constructs Betti, etale, and Hodge realizations over smooth base schemes.This book is a combination of foundational constructions in the theory of motives, togetherwith results relating motivic cohomology with more explicit constructions, such as Bloch’s higherChow groups. It is aimed at research mathematicians interested in algebraic cycles, motives andK-theory, starting at the graduate level. It presupposes a basic background in algebraic geometryand commutative algebra.