Mixed motives / Marc Levine.

Format:

Book

Author/Creator:

Levine, Marc, 1952- author.

Series:

Mathematical surveys and monographs ; no. 57.

Mathematical surveys and monographs, 0076-5376 ; volume 57

Language:

English

Subjects (All):

Motives (Mathematics).

Physical Description:

1 online resource (529 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [1998]

Language Note:

English

Summary:

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.

Contents:

""Contents""; ""Preface""; ""Part I. Motives""; ""Introduction: Part I""; ""Chapter I. The Motivic Category""; ""1. The motivic DG category""; ""2. The triangulated motivic category""; ""3. Structure of the motivic categories""; ""Chapter II. Motivic Cohomology and Higher Chow Groups""; ""1. Hypercohomology in the motivic category""; ""2. Higher Chow groups""; ""3. The motivic cycle map""; ""Chapter III. K-Theory and Motives""; ""1. Chern classes""; ""2. Push-forward""; ""3. Riemann-Roch""; ""Chapter IV. Homology, Cohomology, and Duality""; ""1. Duality""; ""2. Classical constructions""

""3. Motives over a perfect field""""Chapter V. Realization of the Motivic Category""; ""1. Realization for geometric cohomology""; ""2. Concrete realizations""; ""Chapter VI. Motivic Constructions and Comparisons""; ""1. Motivic constructions""; ""2. Comparison with the category DM[sub(gm)](k)""; ""Appendix A. Equi-dimensional Cycles""; ""1. Cycles over a normal scheme""; ""2. Cycles over a reduced scheme""; ""Appendix B. K-Theory""; ""1. K-theory of rings and schemes""; ""2. K-theory and homology""; ""Part II. Categorical Algebra""; ""Introduction: Part II""

""Chapter I. Symmetric Monoidal Structures""""1. Foundational material""; ""2. Constructions and computations""; ""Chapter II. DG Categories and Triangulated Categories""; ""1. Differential graded categories""; ""2. Complexes and triangulated categories""; ""3. Constructions""; ""Chapter III. Simplicial and Cosimplicial Constructions""; ""1. Complexes arising from simplicial and cosimplicial objects""; ""2. Categorical cochain operations""; ""3. Homotopy limits""; ""Chapter IV. Canonical Models for Cohomology""; ""1. Sheaves, sites, and topoi""; ""2. Canonical resolutions""; ""Bibliography""

""Subject Index""""A""; ""B""; ""C""; ""D""; ""E""; ""F""; ""G""; ""H""; ""I""; ""K""; ""L""; ""M""; ""N""; ""O""; ""P""; ""Q""; ""R""; ""S""; ""T""; ""U""; ""V""; ""W""; ""Z""; ""Index of Notation""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 501-506) and indexes.

Description based on print version record.

ISBN:

1-4704-8019-0