Contents
Introduction v
Chapter 1. Algebraic Cycles and Equivalence Relations 1
1.1. Algebraic Cycles 1
1.2. Equivalence Relations 3
Appendix A: Survey of Some of the Main Results on Chow Groups 13
A-1. Divisors 13
A-2. Classical Results on the Picard and Albanese Varieties 14
A-3. Higher Codimension 16
Appendix B: Proof of the Theorem of Voisin–Voevodsky 19
Chapter 2. Motives: Construction and First Properties 23
2.1. Correspondences 23
2.2. (Pure) Motives 25
2.3. Examples 27
2.4. Further Remarks and Properties 28
2.5. Chow Groups and Cohomology Groups of Motives 29
2.6. Relations Between the Various Categories of Motives 30
2.7. Motives of Curves 30
2.8. Manin’s Identity Principle 33
Chapter 3. On Grothendieck’s Standard Conjectures 35
3.1. The Standard Conjectures 35
3.2. Jannsen’s Theorem 39
Chapter 4. Finite Dimensionality of Motives 43
4.1. Introduction 43
4.2. Preliminaries on Group Representations 43
4.3. Action of the Symmetric Group on Products 44
4.4. Dimension of Motives 46
4.5. The Sign Conjecture and Finite Dimensionality 47
4.6. Curves 47
Chapter 5. Properties of Finite Dimensional Motives 51
5.1. Sums and Tensor Products 51
5.2. Smash Nilpotence of Morphisms 54
5.3. Morphisms Between Finite Dimensional Motives 55
5.4. Surjective Morphisms and Finite Dimensionality 56
5.5. Finite Dimensionality and Nilpotence 59
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