The norm residue theorem in motivic cohomology / Christian Haesemeyer, Charles A. Weibel.

Format:

Book

Author/Creator:

Haesemeyer, Christian, author.

Weibel, Charles A., 1950- author.

Series:

Annals of mathematics studies ; no. 200.

Annals of mathematics studies ; no. 200

Language:

English

Subjects (All):

Homology theory.

Physical Description:

xiii, 299 pages : illustrations ; 25 cm.

Place of Publication:

Princeton, New Jersey : Princeton University Press, 2019.

Summary:

This book presents the complete proof of the Bloch-Kato conjecture and several related conjectures of Beilinson and Lichtenbaum in algebraic geometry. Brought together here for the first time, these conjectures describe the structure of étale cohomology and its relation to motivic cohomology and Chow groups.Although the proof relies on the work of several people, it is credited primarily to Vladimir Voevodsky. The authors draw on a multitude of published and unpublished sources to explain the large-scale structure of Voevodsky's proof and introduce the key figures behind its development. They go on to describe the highly innovative geometric constructions of Markus Rost, including the construction of norm varieties, which play a crucial role in the proof. The book then addresses symmetric powers of motives and motivic cohomology operations.Comprehensive and self-contained, The Norm Residue Theorem in Motivic Cohomology unites various components of the proof that until now were scattered across many sources of varying accessibility, often with differing hypotheses, definitions, and language.

Contents:

Part I p. 1

1 An Overview of the Proof p. 3

1.1 First Reductions p. 3

1.2 The Quick Proof p. 6

1.3 Norm Varieties and Rost Varieties p. 9

1.4 The Beilinson-Lichtenbaum Conditions p. 14

1.5 Simplicial Schemes p. 16

1.6 Motivic Cohomology Operations p. 19

2 Relation to Beilinson-Lichtenbaum p. 23

2.1 BL(n) Implies BL(n - 1) p. 24

2.2 H90(n) Implies H90(n - 1) p. 26

2.3 Cohomology of Singular Varieties p. 28

2.4 Cohomology with Supports p. 31

2.5 Rationally Contractible Presheaves p. 34

2.6 Bloch-Kato Implies Beilinson-Lichtenbaum p. 37

2.7 Condition H90(n) Implies BL(n) p. 39

3 Hilbert 90 for Knm p. 42

3.1 Hilbert 90 for Knm p. 42

3.2 A Galois Cohomology Sequence p. 45

3.3 Hilbert 90 for especial Fields p. 48

3.4 Cohomology Elements p. 50

4 Rost Motives and H90 p. 54

4.1 Chow Motives p. 54

4.2 x-Duality p. 56

4.3 Rost Motives p. 59

4.4 Rost Motives Imply Hilbert 90 p. 61

5 Existence of Rost Motives p. 65

5.1 A Candidate for the Rost Motive p. 66

5.2 Axioms (ii) and (iii) p. 67

5.3 End(M) Is a Local Ring p. 70

5.4 Existence of a Rost Motive p. 71

6 Motives over S p. 76

6.1 Motives over a Scheme p. 76

6.2 Motives over a Simplicial Scheme p. 77

6.3 Motives over a Smooth Simplicial Scheme p. 79

6.4 The Slice Filtration p. 82

6.5 Embedded Schemes p. 84

6.6 The Operations Φi p. 86

6.7 The Operation ΦV p. 90

7 The Motivic Group Hbm₋₁, ₋₁ p. 95

7.1 Properties of H₋₁, ₋₁ p. 95

7.2 The Case of Norm Varieties p. 100

Part II p. 103

8 Degree Formulas p. 105

8.1 Algebraic Cobordism p. 105

8.2 The General Degree Formula p. 107

8.3 Other Degree Formulas p. 109

8.4 An Equivariant Degree Formula p. 112

8.5 The η-invariant p. 114

9 Rost's Chain Lemma p. 119

9.1 Forms on Vector Bundles p. 120

9.2 The Chain Lemma when n = 2 p. 122

9.3 The Symbol Chain p. 126

9.4 The Tower of Varieties Pr and Qr p. 129

9.5 Models for Moves of Type Cn p. 133

9.6 Proof of the Chain Lemma p. 135

9.7 Nice G-actions p. 137

9.8 Chain Lemma, Revisited p. 140

10 Existence of Norm Varieties p. 144

10.1 Properties of Norm Varieties p. 144

10.2 Two Vn-1-varieties p. 147

10.3 Norm Varieties Are Vn-1-varieties p. 151

10.4 Existence of Norm Varieties p. 153

11 Existence of Rost Varieties p. 158

11.1 The Multiplication Principle p. 159

11.2 The Norm Principle p. 161

11.3 Weil Restriction p. 162

11.4 Another Splitting Variety p. 163

11.5 Expressing Norms p. 168

Part III p. 173

12 Model Structures for the A¹-homotopy Category p. 175

12.1 The Projective Model Structure p. 176

12.2 Radditive Presheaves p. 182

12.3 The Radditive Projective Model Structure p. 186

12.4 Δ-closed Classes and Weak Equivalences p. 190

12.5 Bousfield Localization p. 194

12.6 Bousfield Localization and Δ-closed Classes p. 196

12.7 Nisnevich-Local Projective Model Structure p. 199

12.8 Model Categories of Sheaves p. 205

12.9 A¹-local Model Structure p. 207

13 Cohomology Operations p. 213

13.1 Motivic Cohomology Operations p. 213

13.2 Steenrod Operations p. 217

13.3 Construction of Steenrod Operations p. 219

13.4 The Milnor Operations Qi p. 220

13.5 Qn of the Degree Map p. 223

13.6 Margolis Homology p. 225

13.7 A Motivic Degree Theorem p. 228

14 Symmetric Powers of Motives p. 232

14.1 Symmetric Powers of Varieties p. 232

14.2 Symmetric Powers of Correspondences p. 235

14.3 Weak Equivalences and Symmetric Powers p. 238

14.4 SG of Quotients X/U p. 241

14.5 Nisnevich G-local Equivalences p. 245

14.6 Symmetric Powers and Shifts p. 248

15 Motivic Classifying Spaces p. 253

15.1 Symmetric Powers and Operations p. 253

15.2 Operations on H¹,¹ p. 256

15.3 Scalar Weight p. 258

15.4 The Motive of (V-0)/C with VC = 0 p. 259

15.5 The Motive Sltr(Ln) p. 264

15.6 A Künneth Formula p. 268

15.7 Operations of Pure Scalar Weight p. 269

15.8 Uniqueness of ßPn p. 271.

Notes:

Includes bibliographical references (pages [283]-292) and index.

ISBN:

0691191042

9780691191041

0691181829

9780691181820

OCLC:

1059268154