Specialization of Quadratic and Symmetric Bilinear Forms (eBook)

Manfred Knebusch (Autor)

eBook Download: PDF

2011 | 2010
XIV, 192 Seiten
Springer London (Verlag)
978-1-84882-242-9 (ISBN)

Infos

A Mathematician Said Who Can Quote Me a Theorem that's True? For the ones that I Know Are Simply not So, When the Characteristic is Two! This pretty limerick ?rst came to my ears in May 1998 during a talk by T.Y. Lam 1 on ?eld invariants from the theory of quadratic forms. It is-poetic exaggeration allowed-a suitable motto for this monograph. What is it about? At the beginning of the seventies I drew up a specialization theoryofquadraticandsymmetricbilinear formsover ?elds[32].Let? : K? L?? be a place. Then one can assign a form? (?)toaform? over K in a meaningful way ? if? has 'good reduction' with respect to? (see§1.1). The basic idea is to simply apply the place? to the coe?cients of?, which must therefore be in the valuation ring of?. The specialization theory of that time was satisfactory as long as the ?eld L, and therefore also K, had characteristic 2. It served me in the ?rst place as the foundation for a theory of generic splitting of quadratic forms [33], [34]. After a very modest beginning, this theory is now in full bloom. It became important for the understanding of quadratic forms over ?elds, as can be seen from the book [26]of Izhboldin-Kahn-Karpenko-Vishik for instance. One should note that there exists a theoryof(partial)genericsplittingofcentralsimplealgebrasandreductivealgebraic groups, parallel to the theory of generic splitting of quadratic forms (see [29] and the literature cited there).

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A Mathematician Said Who Can Quote Me a Theorem that's True? For the ones that I Know Are Simply not So, When the Characteristic is Two! This pretty limerick ?rst came to my ears in May 1998 during a talk by T.Y. Lam 1 on ?eld invariants from the theory of quadratic forms. It is-poetic exaggeration allowed-a suitable motto for this monograph. What is it about? At the beginning of the seventies I drew up a specialization theoryofquadraticandsymmetricbilinear formsover ?elds[32].Let? : K? L?? be a place. Then one can assign a form? (?)toaform? over K in a meaningful way ? if? has "e;good reduction"e; with respect to? (see1.1). The basic idea is to simply apply the place? to the coe?cients of?, which must therefore be in the valuation ring of?. The specialization theory of that time was satisfactory as long as the ?eld L, and therefore also K, had characteristic 2. It served me in the ?rst place as the foundation for a theory of generic splitting of quadratic forms [33], [34]. After a very modest beginning, this theory is now in full bloom. It became important for the understanding of quadratic forms over ?elds, as can be seen from the book [26]of Izhboldin-Kahn-Karpenko-Vishik for instance. One should note that there exists a theoryof(partial)genericsplittingofcentralsimplealgebrasandreductivealgebraic groups, parallel to the theory of generic splitting of quadratic forms (see [29] and the literature cited there).

Inhalt

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
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
4.4 Stably Conservative Reduction of Quadratic Forms 186
4.5 Generic Splitting of Stably Conservative Specialized Quadratic Forms 192
References 195
Index 198

Produkt-Details

• Erscheint lt. Verlag: 22.1.2011
• Reihe/Serie: Algebra and Applications
• Reihe/Serie: Algebra and Applications
• Übersetzer: Thomas Unger
• Zusatzinfo: XIV, 192 p.
• Verlagsort: London
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Technik
• Schlagworte: Addition • Character • DEX • Form • Generic splitting theory • Integral • quadratic form • quadratic forms • quadratic places • Specialization theory • Symmetric bilinear forms
• ISBN-10: 1-84882-242-1 / 1848822421
• ISBN-13: 978-1-84882-242-9 / 9781848822429