Table of Contents
Preface xi
Acknowledgments xv
Part I 1
1 An Overview of the Proof 3
1.1 First Reductions 3
1.2 The Quick Proof 6
1.3 Norm Varieties and Rost Varieties 9
1.4 The Beilinson-Lichtenbaum Conditions 14
1.5 Simplicial Schemes 16
1.6 Motivic Cohomology Operations 19
1.7 Historical Notes 21
2 Relation to Beilinson-Lichtenbaum 23
2.1 BL(n) Implies BL(n - 1) 24
2.2 H90(n) Implies H90(n - 1) 26
2.3 Cohomology of Singular Varieties 28
2.4 Cohomology with Supports 31
2.5 Rationally Contractible Presheaves 34
2.6 Bloch-Kato Implies Beilinson-Lichtenbaum 37
2.7 Condition H90(n) Implies BL(n) 39
2.8 Historical Notes 41
3 Hilbert 90 for $$$ 42
3.1 Hilbert 90 for $$$ 42
3.2 A Galois Cohomology Sequence 45
3.3 Hilbert 90 for especial Fields 48
3.4 Cohomology Elements 50
3.5 Historical Notes 53
4 Rost Motives and H90 54
4.1 Chow Motives 54
4.2 x-Duality 56
4.3 Rost Motives 59
4.4 Rost Motives Imply Hilbert 90 61
4.5 Historical Notes 64
5 Existence of Rost Motives 65
5.1 A Candidate for the Rost Motive 66
5.2 Axioms (ii) and (iii) 67
5.3 End(M) Is a Local Ring 70
5.4 Existence of a Rost Motive 71
5.5 Historical Notes 74
6 Motives over S 76
6.1 Motives over a Scheme 76
6.2 Motives over a Simplicial Scheme 77
6.3 Motives over a Smooth Simplicial Scheme 79
6.4 The Slice Filtration 82
6.5 Embedded Schemes 84
6.6 The Operations φi 86
6.7 The Operation φV 90
6.8 Historical Notes 94
7 The Motivic Group $$$ 95
7.1 Properties of H-1, -1 95
7.2 The Case of Norm Varieties 100
7.3 Historical Notes 101
Part II 103
8 Degree Formulas 105
8.1 Algebraic Cobordism 105
8.2 The General Degree Formula 107
8.3 Other Degree Formulas 109
8.4 An Equivariant Degree Formula 112
8.5 The η-invariant 114
8.6 Historical Notes 117
9 Rost's Chain Lemma 119
9.1 Forms on Vector Bundles 120
9.2 The Chain Lemma when n = 2 122
9.3 The Symbol Chain 126
9.4 The Tower of Varieties Pr and Qr 129
9.5 Models for Moves of Type Cn 133
9.6 Proof of the Chain Lemma 135
9.7 Nice G-actions 137
9.8 Chain Lemma, Revisited 140
9.9 Historical Notes 143
10 Existence of Norm Varieties 144
10.1 Properties of Norm Varieties 144
10.2 Two νn-1-varieties 147
10.3 Norm Varieties Are νn-1-varieties 151
10.4 Existence of Norm Varieties 153
10.5 Historical Notes 156
11 Existence of Rost Varieties 158
11.1 The Multiplication Principle 159
11.2 The Norm Principle 161
11.3 Weil Restriction 162
11.4 Another Splitting Variety 163
11.5 Expressing Norms 168
11.6 Historical Notes 171
Part III 173
12 Model Structures for the A1-homotopy Category 175
12.1 The Projective Model Structure 176
12.2 Radditive Presheaves 182
12.3 The Radditive Projective Model Structure 186
12.4 Δ-closed Classes and Weak Equivalences 190
12.5 Bousfield Localization 194
12.6 Bousfield Localization and Δ-closed Classes 196
12.7 Nisnevich-Local Projective Model Structure 199
12.8 Model Categories of Sheaves 205
12.9 A1-local Model Structure 207
12.10 Historical Notes 211
13 Cohomology Operations 213
13.1 Motivic Cohomology Operations 213
13.2 Steenrod Operations 217
13.3 Construction of Steenrod Operations 219
13.4 The Milnor Operations Qi 220
13.5 Qn of the Degree Map 223
13.6 Margolis Homology 225
13.7 A Motivic Degree Theorem 228
13.8 Historical Notes 230
14 Symmetric Powers of Motives 232
14.1 Symmetric Powers of Varieties 232
14.2 Symmetric Powers of Correspondences 235
14.3 Weak Equivalences and Symmetric Powers 238
14.4 SG of Quotients X/U 241
14.5 Nisnevich G-local Equivalences 245
14.6 Symmetric Powers and Shifts 248
14.7 Historical Notes 252
15 Motivic Classifying Spaces 253
15.1 Symmetric Powers and Operations 253
15.2 Operations on H1,1 256
15.3 Scalar Weight 258
15.4 The Motive of (V-0)/C with VC = 0 259
15.5 The Motive $$$ 264
15.6 A Künneth Formula 268
15.7 Operations of Pure Scalar Weight 269
15.8 Uniqueness of βPn 271
15.9 Historical Notes 276
Glossary 277
Bibliography 283
Index 293
