Stable Domination and Independence in Algebraically Closed Valued Fields

Haskell D., Hrushovski E., MacPherson D.0511371047

“This book addresses a gap in the model-theoretic understanding of valued fields that has, until now, limited the interactions of model theory with geometry. It contains significant developments in both pure and applied model theory.”.”Part I of the book is a study of stably dominated types. These form a subset of the type space of a theory that behaves in many ways like the space of types in a stable theory. This part begins with an introduction to the key ideas of stability theory for stably dominated types. Part II continues with an outline of some classical results in the model theory of valued fields and explores the application of stable domination to algebraically closed valued fields. The research presented here is made accessible to the general model theorist by the inclusion of the introductory sections of each part.”–BOOK JACKET. Read more… Ch. 1. Introduction — Pt. 1. Stable Domination — Ch. 2. Some background on stability theory — Ch. 3. Definition and basic properties of St[subscript C] — Ch. 4. Invariant types and change of base — Ch. 5. A combinatorial lemma — Ch. 6. Strong codes for germs — Pt. 2. Independence in ACVF — Ch. 7. Some background on algebraically closed valued fields — Ch. 8. Sequential independence — Ch. 9. Growth of the stable part — Ch. 10. Types orthogonal to [Gamma] — Ch. 11. Opacity and prime resolutions — Ch. 12. Maximally complete fields and domination — Ch. 13. Invariant types — Ch. 14. A maximum modulus principle — Ch. 15. Canonical bases and independence given by modules — Ch. 16. Other Henselian fields

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Table of contents :
HALF-TITLE……Page 3
SERIES-TITLE……Page 5
TITLE……Page 7
COPYRIGHT……Page 8
CONTENTS……Page 9
PREFACE……Page 11
Acknowledgments……Page 13
CHAPTER 1 INTRODUCTION……Page 15
PART 1 STABLE DOMINATION……Page 25
CHAPTER 2 SOME BACKGROUND ON STABILITY THEORY……Page 27
2.1. Saturation, the universal domain, imaginaries……Page 29
2.2. Invariant types……Page 31
2.3. Conditions equivalent to stability……Page 32
2.4. Independence and forking……Page 34
2.6. Prime models……Page 37
2.7. Indiscernibles, Morley sequences……Page 38
2.8. Stably embedded sets……Page 39
CHAPTER 3 DEFINITION AND BASIC PROPERTIES OF StC……Page 41
CHAPTER 4 INVARIANT TYPES AND CHANGE OF BASE……Page 55
CHAPTER 5 A COMBINATORIAL LEMMA……Page 67
CHAPTER 6 STRONG CODES FOR GERMS……Page 73
PART 2 INDEPENDENCE IN ACVF……Page 81
7.1 Background on valued fields
……Page 83
7.2 Some model theory of valued fields
……Page 85
7.3. Basics of ACVF……Page 86
7.4. Imaginaries, and the ACVF sorts……Page 87
7.5 The sorts internal to the residue field
……Page 91
7.6. Unary sets, 1-torsors, and generic 1-types……Page 92
7.7. One-types orthogonal to Γ……Page 97
7.8. Generic bases of lattices……Page 99
CHAPTER 8 SEQUENTIAL INDEPENDENCE……Page 101
CHAPTER 9 GROWTH OF THE STABLE PART……Page 113
CHAPTER 10 TYPES ORTHOGONAL TO Γ……Page 117
CHAPTER 11 OPACITY AND PRIME RESOLUTIONS……Page 129
CHAPTER 12 MAXIMALLY COMPLETE FIELDS AND DOMINATION……Page 137
13.1. Examples of sequential independence……Page 151
13.2. Invariant types, dividing and sequential independence……Page 157
CHAPTER 14 A MAXIMUMMODULUS PRINCIPLE……Page 165
CHAPTER 15 CANONICAL BASES AND INDEPENDENCE GIVEN BY MODULES……Page 175
CHAPTER 16 OTHER HENSELIAN FIELDS……Page 185
REFERENCES……Page 191
INDEX……Page 195