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Product Code:  ULECT/61 
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Electronic ISBN:  9781470410391 
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Book DetailsUniversity Lecture SeriesVolume: 61; 2013; 149 ppMSC: Primary 14; 19;
The theory of motives was created by Grothendieck in the 1960s as he searched for a universal cohomology theory for algebraic varieties. The theory of pure motives is well established as far as the construction is concerned. Pure motives are expected to have a number of additional properties predicted by Grothendieck's standard conjectures, but these conjectures remain wide open. The theory for mixed motives is still incomplete.
This book deals primarily with the theory of pure motives. The exposition begins with the fundamentals: Grothendieck's construction of the category of pure motives and examples. Next, the standard conjectures and the famous theorem of Jannsen on the category of the numerical motives are discussed. Following this, the important theory of finite dimensionality is covered. The concept of ChowKünneth decomposition is introduced, with discussion of the known results and the related conjectures, in particular the conjectures of BlochBeilinson type. We finish with a chapter on relative motives and a chapter giving a short introduction to Voevodsky's theory of mixed motives.ReadershipGraduate students and research mathematicians interested in algebraic cycles and motives.

Table of Contents

Chapters

Chapter 1. Algebraic cycles and equivalence relations

Appendix A. Survey of some of the main results on Chow groups

Appendix B. Proof of the theorem of VoisinVoevodsky

Chapter 2. Motives: Construction and first properties

Chapter 3. On Grothendieck’s standard conjectures

Chapter 4. Finite dimensionality of motives

Chapter 5. Properties of finite dimensional motives

Chapter 6. ChowKünneth decomposition; The Picard and Albanese motive

Appendix C. ChowKünneth decomposition in a special case

Chapter 7. On the conjectural BlochBeilinson filtration

Chapter 8. Relative ChowKünneth decomposition

Appendix D. Surfaces fibered over a curve

Chapter 9. Beyond pure motives

Appendix E. The category of motivic complexes


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The theory of motives was created by Grothendieck in the 1960s as he searched for a universal cohomology theory for algebraic varieties. The theory of pure motives is well established as far as the construction is concerned. Pure motives are expected to have a number of additional properties predicted by Grothendieck's standard conjectures, but these conjectures remain wide open. The theory for mixed motives is still incomplete.
This book deals primarily with the theory of pure motives. The exposition begins with the fundamentals: Grothendieck's construction of the category of pure motives and examples. Next, the standard conjectures and the famous theorem of Jannsen on the category of the numerical motives are discussed. Following this, the important theory of finite dimensionality is covered. The concept of ChowKünneth decomposition is introduced, with discussion of the known results and the related conjectures, in particular the conjectures of BlochBeilinson type. We finish with a chapter on relative motives and a chapter giving a short introduction to Voevodsky's theory of mixed motives.
Graduate students and research mathematicians interested in algebraic cycles and motives.

Chapters

Chapter 1. Algebraic cycles and equivalence relations

Appendix A. Survey of some of the main results on Chow groups

Appendix B. Proof of the theorem of VoisinVoevodsky

Chapter 2. Motives: Construction and first properties

Chapter 3. On Grothendieck’s standard conjectures

Chapter 4. Finite dimensionality of motives

Chapter 5. Properties of finite dimensional motives

Chapter 6. ChowKünneth decomposition; The Picard and Albanese motive

Appendix C. ChowKünneth decomposition in a special case

Chapter 7. On the conjectural BlochBeilinson filtration

Chapter 8. Relative ChowKünneth decomposition

Appendix D. Surfaces fibered over a curve

Chapter 9. Beyond pure motives

Appendix E. The category of motivic complexes