Specialization of Quadratic and Symmetric Bilinear Forms

Specialization of Quadratic and Symmetric Bilinear Forms

von: Manfred Knebusch

Springer-Verlag, 2011

ISBN: 9781848822429 , 192 Seiten

2. Auflage

Format: PDF

Kopierschutz: Wasserzeichen

Windows PC,Mac OSX Apple iPad, Android Tablet PC's

Preis: 80,24 EUR

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Mehr zum Inhalt

Specialization of Quadratic and Symmetric Bilinear Forms


 

Preface

6

Contents

11

1 Fundamentals of Specialization Theory

13

1.1 Introduction: on the Problem of Specialization of Quadratic and Bilinear Forms

13

1.2 An Elementary Treatise on Symmetric Bilinear Forms

15

1.3 Specialization of Symmetric Bilinear Forms

19

1.4 Generic Splitting in Characteristic 2

28

1.5 An Elementary Treatise on Quadratic Modules

34

1.6 Quadratic Modules over Valuation Rings

38

1.7 Weak Specialization

48

1.8 Good Reduction

60

2 Generic Splitting Theory

66

2.1 Generic Splitting of Regular Quadratic Forms

66

2.2 Separable Splitting

73

2.3 Fair Reduction and Weak Obedience

76

2.4 Unified Theory of Generic Splitting

86

2.5 Regular Generic Splitting Towers and Base Extension

90

2.6 Generic Splitting Towers of a Specialized Form

97

3 Some Applications

102

3.1 Subforms which have Bad Reduction

102

3.2 Some Forms of Height 1

107

3.3 The Subform Theorem

114

3.4 Milnor's Exact Sequence

119

3.5 A Norm Theorem

124

3.6 Strongly Multiplicative Forms

129

3.7 Divisibility by Pfister Forms

136

3.8 Pfister Neighbours and Excellent Forms

144

3.9 Regular Forms of Height 1

149

3.10 Some Open Problems in Characteristic 2

152

3.11 Leading Form and Degree Function

155

3.12 The Companion Form of an Odd-dimensional Regular Form

162

3.13 Definability of the Leading Form over the Base Field

169

4 Specialization with Respect to Quadratic Places

176

4.1 Quadratic Places; Specialization of Bilinear Forms

176

4.2 Almost Good Reduction with Respect to Extensions of Quadratic Places

181

4.3 Realization of Quadratic Places; Generic Splitting of Specialized Forms in Characteristic 2

183

4.4 Stably Conservative Reduction of Quadratic Forms

186

4.5 Generic Splitting of Stably Conservative Specialized Quadratic Forms

192

References

195

Index

198