Valuations, Orderings, and Milnor K-Theory

Introduction ix

Conventions xiii

Part I. Abelian Groups

Chapter 1. Preliminaries on Abelian Groups 5

§1.1. General facts 5

§1.2. Divisible hulls 7

§1.3. Rational ranks 11

§1.4. Characters 12

Chapter 2. Ordered Abelian Groups 15

§2.1. Basic properties and examples 15

§2.2. Ranks 17

§2.3. Cores 19

§2.4. Cofinality and infinitesimals 20

§2.5. Ordered abelian groups of rank 1 21

§2.6. Push-downs 23

§2.7. Well-ordered sets 24

§2.8. Formal power series 27

§2.9. Generalized rational functions 30

Part II. Valuations and Orderings

Chapter 3. Valuations 37

§3.1. Valuation rings 37

§3.2. Valuations 38

§3.3. Places 42

§3.4. Discrete valuations 43

Chapter 4. Examples of Valuations 47

§4.1. Valuations from unique factorization domains 47

§4.2. Valuations on power series fields 48

§4.3. Gauss valuations 50

Chapter 5. Coarsenings of Valuations 55

§5.1. Coarser and finer 55

§5.2. Quotients and compositions of valuations 56

§5.3. Coarsenings in the mixed characteristic case 60

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vi CONTENTS

Chapter 6. Orderings 63

§6.1. Ordered fields 63

§6.2. Examples of orderings 66

§6.3. Archimedean orderings 67

Chapter 7. The Tree of Localities 69

§7.1. Localities 69

§7.2. Localities on residue fields 70

§7.3. The tree structure 71

Chapter 8. Topologies 75

§8.1. Basic properties 75

§8.2. Continuity of roots 77

§8.3. Bounded sets 79

Chapter 9. Complete Fields 81

§9.1. Metrics 81

§9.2. Examples 82

§9.3. Completions 83

Chapter 10. Approximation Theorems 87

§10.1. Approximation by independent localities 87

§10.2. Approximation by incomparable valuations 90

§10.3. Consequences 93

Chapter 11. Canonical Valuations 95

§11.1. Compatible localities 95

§11.2. S-cores 98

§11.3. Explicit constructions 100

§11.4. Existence of valuations 103

Chapter 12. Valuations of Mixed Characteristics 107

§12.1. Multiplicative representatives 107

§12.2. λ-adic expansions 109

§12.3. p-perfect structures 110

§12.4. Rings of Witt vectors 116

§12.5. Mixed valuations under a finiteness assumption 118

Part III. Galois Theory

Chapter 13. Infinite Galois Theory 125

Chapter 14. Valuations in Field Extensions 127

§14.1. Chevalley’s theorem 127

§14.2. Valuations in algebraic extensions 128

§14.3. The Galois action 130

Chapter 15. Decomposition Groups 133

§15.1. Definition and basic properties 133

§15.2. Immediateness of decomposition fields 134

§15.3. Relatively Henselian fields 136

CONTENTS vii

Chapter 16. Ramification Theory 141

§16.1. Inertia groups 141

§16.2. Ramification groups 143

Chapter 17. The Fundamental Equality 151

§17.1. The fundamental inequality 151

§17.2. Ostrowski’s theorem 153

§17.3. Defectless fields 157

§17.4. Extensions of discrete valuations 158

Chapter 18. Hensel’s Lemma 161

§18.1. The main variants 161

§18.2. nth powers 164

§18.3. Example: complete valued fields 166

§18.4. Example: power series fields 168

§18.5. The Krasner–Ostrowski lemma 170

Chapter 19. Real Closures 175

§19.1. Extensions of orderings 175

§19.2. Relative real closures 177

§19.3. Sturm’s theorem 181

§19.4. Uniqueness of real closures 184

Chapter 20. Coarsening in Algebraic Extensions 187

§20.1. Extensions of localities 187

§20.2. Coarsening and Galois groups 189

§20.3. Local closedness and quotients 190

§20.4. Ramification pairings under coarsening 191

Chapter 21. Intersections of Decomposition Groups 193

§21.1. The case of independent valuations 193

§21.2. The case of incomparable valuations 194

§21.3. Transition properties for Henselity 195

Chapter 22. Sections 199

§22.1. Complements of inertia groups 199

§22.2. Complements of ramification groups 203

Part IV. K-Rings

Chapter 23. κ-Structures 209

§23.1. Basic notions 209

§23.2. Constructions of κ-structures 210

§23.3. Rigidity 213

§23.4. Demuˇskin κ-structures 214

Chapter 24. Milnor K-Rings of Fields 217

§24.1. Definition and basic properties 217

§24.2. Comparison theorems 219

§24.3. Connections with Galois cohomology 221

Chapter 25. Milnor K-Rings and Orderings 225

viii CONTENTS

§25.1. A K-theoretic characterization of orderings 225

§25.2. Cyclic quotients 228

Chapter 26. K-Rings and Valuations 231

§26.1. Valuations and extensions 231

§26.2. The Baer–Krull correspondence 234

§26.3. Totally rigid subgroups 235

§26.4. Sizes of multiplicative subgroups 236

§26.5. HS and the K-ring 238

§26.6. Bounds in the totally rigid case 240

§26.7. Fans 242

§26.8. Examples of totally rigid subgroups 244

Chapter 27. K-Rings of Wild Valued Fields 247

§27.1. The discrete case 247

§27.2. A vanishing theorem 248

§27.3. The general case 250

Chapter 28. Decompositions of K-Rings 253

§28.1. The basic criterion 253

§28.2. Topological decompositions 256

§28.3. Local pairs 257

§28.4. Arithmetical decompositions 259

Chapter 29. Realization of κ-Structures 263

§29.1. Basic constructions 263

§29.2. K-rings modulo preorderings of finite index 265

§29.3. κ-structures of elementary type 267

Bibliography 269

Glossary of Notation 275

Index 281