About this Title
Gonçalo Tabuada, Massachusetts Institute of Technology, Cambridge, MA
Publication: University Lecture Series
Publication Year 2015: Volume 63
ISBNs: 978-1-4704-2397-1 (print); 978-1-4704-2627-9 (online)
MathSciNet review: MR3379910
MSC: Primary 14A22; Secondary 14C15
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 non-commutative algebraic geometry.
Table of Contents
- 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