Knebusch Specialization of Quadratic and Symmetric Bilinear Forms

2010
ISBN: 978-1-84882-242-9
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 11, 192 Seiten

Reihe: Algebra and Applications

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).

Autoren/Hrsg.

Knebusch, Manfred

Weitere Mitwirkende

Unger, Thomas

Inhaltsverzeichnis

1;Preface;6
2;Contents;11
3;1 Fundamentals of Specialization Theory;13
3.1;1.1 Introduction: on the Problem of Specialization of Quadratic and Bilinear Forms;13
3.2;1.2 An Elementary Treatise on Symmetric Bilinear Forms;15
3.3;1.3 Specialization of Symmetric Bilinear Forms;19
3.4;1.4 Generic Splitting in Characteristic 2;28
3.5;1.5 An Elementary Treatise on Quadratic Modules;34
3.6;1.6 Quadratic Modules over Valuation Rings;38
3.7;1.7 Weak Specialization;48
3.8;1.8 Good Reduction;60
4;2 Generic Splitting Theory;66
4.1;2.1 Generic Splitting of Regular Quadratic Forms;66
4.2;2.2 Separable Splitting;73
4.3;2.3 Fair Reduction and Weak Obedience;76
4.4;2.4 Unified Theory of Generic Splitting;86
4.5;2.5 Regular Generic Splitting Towers and Base Extension;90
4.6;2.6 Generic Splitting Towers of a Specialized Form;97
5;3 Some Applications;102
5.1;3.1 Subforms which have Bad Reduction;102
5.2;3.2 Some Forms of Height 1;107
5.3;3.3 The Subform Theorem;114
5.4;3.4 Milnor's Exact Sequence;119
5.5;3.5 A Norm Theorem;124
5.6;3.6 Strongly Multiplicative Forms;129
5.7;3.7 Divisibility by Pfister Forms;136
5.8;3.8 Pfister Neighbours and Excellent Forms;144
5.9;3.9 Regular Forms of Height 1;149
5.10;3.10 Some Open Problems in Characteristic 2;152
5.11;3.11 Leading Form and Degree Function;155
5.12;3.12 The Companion Form of an Odd-dimensional Regular Form;162
5.13;3.13 Definability of the Leading Form over the Base Field;169
6;4 Specialization with Respect to Quadratic Places;176
6.1;4.1 Quadratic Places; Specialization of Bilinear Forms;176
6.2;4.2 Almost Good Reduction with Respect to Extensions of Quadratic Places;181
6.3;4.3 Realization of Quadratic Places; Generic Splitting of Specialized Forms in Characteristic 2;183
6.4;4.4 Stably Conservative Reduction of Quadratic Forms;186
6.5;4.5 Generic Splitting of Stably Conservative Specialized Quadratic Forms;192
7;References;195
8;Index;198