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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.
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Mathematical Surveys and Monographs Volume 57 Mixed Motives Marc Levine FO UN 8 DED 1 SOCIETY ΑΓΕΩ ΜΕ ΤΡΗΤΟΣ ΜΗ ΕΙΣΙΤΩ R AME ICAN L HEMATIC AT A M 88 American Mathematical Society Editorial Board Georgia Benkart Howard A. Masur Tudor Stefan Ratiu, Chair Michael Renardy 1991 Mathematics Subject Classification. Primary 19E15, 14C25; Secondary 14C15, 14C17, 14C40, 19D45, 19E08, 19E20. Research supported in part by the National Science Foundation and the Deutsche Forschungsgemeinschaft. Abstract. The author constructs and describes a triangulated category of mixed motives over an arbitrary base scheme. The resulting cohomology theory satisfies the Bloch-Ogus axioms; if the base scheme is a smooth scheme of dimension at most one over a field, this cohomology theory agrees with Bloch’s higher Chow groups. Most of the classical constructions of cohomology can be made 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. The motivic category admits a real