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Softcover ISBN:  9781470423971 
Product Code:  ULECT/63 
List Price:  $69.00 
MAA Member Price:  $62.10 
AMS Member Price:  $55.20 
eBook ISBN:  9781470426279 
Product Code:  ULECT/63.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $52.00 
Softcover ISBN:  9781470423971 
eBook ISBN:  9781470426279 
Product Code:  ULECT/63.B 
List Price:  $134.00 $101.50 
MAA Member Price:  $120.60 $91.35 
AMS Member Price:  $107.20 $81.20 

Book DetailsUniversity Lecture SeriesVolume: 63; 2015; 114 ppMSC: Primary 14; 18; Secondary 19
The theory of motives began in the early 1960s when Grothendieck envisioned the existence of a “universal cohomology theory of algebraic varieties”. The theory of noncommutative motives is more recent. It began in the 1980s when the Moscow school (Beilinson, Bondal, Kapranov, Manin, and others) began the study of algebraic varieties via their derived categories of coherent sheaves, and continued in the 2000s when Kontsevich conjectured the existence of a “universal invariant of noncommutative algebraic varieties”.
This book, prefaced by Yuri I. Manin, gives a rigorous overview of some of the main advances in the theory of noncommutative motives. It is divided into three main parts. The first part, which is of independent interest, is devoted to the study of DG categories from a homotopical viewpoint. The second part, written with an emphasis on examples and applications, covers the theory of noncommutative pure motives, noncommutative standard conjectures, noncommutative motivic Galois groups, and also the relations between these notions and their commutative counterparts. The last part is devoted to the theory of noncommutative mixed motives. The rigorous formalization of this latter theory requires the language of Grothendieck derivators, which, for the reader's convenience, is revised in a brief appendix.
ReadershipGraduate students and research mathematicians interested in algebraic geometry, including noncommutative algebraic geometry.

Table of Contents

Chapters

Introduction

Chapter 1. Differential graded categories

Chapter 2. Additive invariants

Chapter 3. Background on pure motives

Chapter 4. Noncommutative pure motives

Chapter 5. Noncommutative (standard) conjugates

Chapter 6. Noncommutative motivic Galois groups

Chapter 7. Jacobians of noncommutative Chow motives

Chapter 8. Localizing invariants

Chapter 9. Noncommutative mixed motives

Chapter 10. Noncommutative motivic Hopf dg algebras

Appendix A. Grothendieck derivators


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The theory of motives began in the early 1960s when Grothendieck envisioned the existence of a “universal cohomology theory of algebraic varieties”. The theory of noncommutative motives is more recent. It began in the 1980s when the Moscow school (Beilinson, Bondal, Kapranov, Manin, and others) began the study of algebraic varieties via their derived categories of coherent sheaves, and continued in the 2000s when Kontsevich conjectured the existence of a “universal invariant of noncommutative algebraic varieties”.
This book, prefaced by Yuri I. Manin, gives a rigorous overview of some of the main advances in the theory of noncommutative motives. It is divided into three main parts. The first part, which is of independent interest, is devoted to the study of DG categories from a homotopical viewpoint. The second part, written with an emphasis on examples and applications, covers the theory of noncommutative pure motives, noncommutative standard conjectures, noncommutative motivic Galois groups, and also the relations between these notions and their commutative counterparts. The last part is devoted to the theory of noncommutative mixed motives. The rigorous formalization of this latter theory requires the language of Grothendieck derivators, which, for the reader's convenience, is revised in a brief appendix.
Graduate students and research mathematicians interested in algebraic geometry, including noncommutative algebraic geometry.

Chapters

Introduction

Chapter 1. Differential graded categories

Chapter 2. Additive invariants

Chapter 3. Background on pure motives

Chapter 4. Noncommutative pure motives

Chapter 5. Noncommutative (standard) conjugates

Chapter 6. Noncommutative motivic Galois groups

Chapter 7. Jacobians of noncommutative Chow motives

Chapter 8. Localizing invariants

Chapter 9. Noncommutative mixed motives

Chapter 10. Noncommutative motivic Hopf dg algebras

Appendix A. Grothendieck derivators