Research Catalog

Title

Noncommutative motives / Goncalo Tabuada.

Author

Tabuada, Gonçalo, 1979-

Publication

Providence, Rhode Island : American Mathematical Society, 2015.

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Holdings

Details

Description

114 pages; 26 cm.

Summary

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.

Series Statement

University lecture series ; volume 63

Uniform Title

University lecture series (Providence, R.I.) ; 63.

Subject

• $K$-theory > Higher algebraic $K$-theory > $K$-theory and homology; cyclic homology and cohomology
• Algebraic geometry > Cycles and subschemes > (Equivariant) Chow groups and rings; motives
• Algebraic geometry > Foundations > Noncommutative algebraic geometry
• Algebraic varieties
• Algebraic varieties
• Category theory; homological algebra > Abelian categories > Derived categories, triangulated categories
• Category theory; homological algebra > Categories with structure > Enriched categories (over closed or monoidal categories)
• Category theory; homological algebra > Homological algebra > Homotopical algebra
• Motives (Mathematics)
• Noncommutative algebras
• Noncommutative algebras

Bibliography (note)

• Includes bibliographical references and index.

Processing Action (note)

• committed to retain

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.

ISBN

• 9781470423971 (alk. paper)
• 1470423979 (alk. paper)

LCCN

^^2015018204

Owning Institutions

Harvard Library