Stable Domination and Independence in

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 one of the book is a study of stably dominated types and it begins with an introduction to the key ideas of stability theory for stably dominated types. Part two 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.

Dugald Macpherson is a Professor in the School of Mathematics at the University of Leeds.

Preface	p. ix
Introduction	p. 1
Stable Domination
Some background on stability theory	p. 13
Saturation, the universal domain, imaginaries	p. 15
Invariant types	p. 17
Conditions equivalent to stability	p. 18
Independence and forking	p. 20
Totally transcendental theories and Morley rank	p. 23
Prime models	p. 23
Indiscernibles, Morley sequences	p. 24
Stably embedded sets	p. 25
Definition and basic properties of St[subscript C]	p. 27
Invariant types and change of base	p. 41
A combinatorial lemma	p. 53
Strong codes for germs	p. 59
Independence in ACVF
Some background on algebraically closed valued fields	p. 69
Background on valued fields	p. 69
Some model theory of valued fields	p. 71
Basics of ACVF	p. 72
Imaginaries, and the ACVF sorts	p. 73
The sorts internal to the residue field	p. 77
Unary sets, 1-torsors, and generic 1-types	p. 78
One-types orthogonal to [Gamma]	p. 83
Generic bases of lattices	p. 85
Sequential independence	p. 87
Growth of the stable part	p. 99
Types orthogonal to [Gamma]	p. 103
Opacity and prime resolutions	p. 115
Maximally complete fields and domination	p. 123
Invariant types	p. 137
Examples of sequential independence	p. 137
Invariant types, dividing and sequential independence	p. 143
A maximum modulus principle	p. 151
Canonical bases and independence given by modules	p. 161
Other Henselian fields	p. 171
References	p. 177
Index	p. 181
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Stable Domination and Independence in Algebraically Closed Valued Fields

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