Quadratic Algebras, Clifford Algebras, and Arithmetic Witt Groups

295 Pages·1994·10.5 MB·English

by  Alexander J. Hahn (auth.)

Un iversitext
Editorial Board
(North America):
J.H. Ewing
F.W. Gehring
P.R. Halmos
Universitext
Editors (North America): 1.H. Ewing, F.W. Gehring, and P.R. Halmos
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(continued after index)
Alexander J. Hahn
Quadratic Algebras,
Clifford Algebras, and
Arithmetic Witt Groups
Springer-Verlag
New York Berlin Heidelberg London Paris
Tokyo Hong Kong Barcelona Budapest
Alexander 1. Hahn
Department of Mathematics
University of Notre Dame
Notre Dame, IN 46556
USA
Editorial Board
(North America):
1.H. Ewing F. W. Gehring
Department of Mathematics Department of Mathematics
Indiana University University of Michigan
Bloomington, IN 47405 Ann Arbor, MI 48109
USA USA
P.R. Halmos
Department of Mathematics
Santa Clara University
Santa Clara, CA 95053
USA
Mathematics Subject Classifications (1991): 15A66, 16H05, 13A20
Library of Congress Cataloging-in-Publication Data
Hahn, Alexander, 1943-
Quadratic algebras, Clifford algebras, and arithmetic Witt groups/
Alexander J. Hahn.
p. cm. - (Universitext)
Includes bibliographical references.
ISBN-13: 978-0-387-94110-3 e-ISBN-13: 978-1-4684-6311-8
DOl: 10.1007/978-1-4684-6311-8
I. Clifford algebras. 2. Commutative rings. 3. Forms, Quadratic.
1. Title. II. Series.
QA199.H34 1993
512' .57 - dc20 93-5139
Printed on acid-free paper.
© 1994 Springer-Verlag New York, Inc.
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987654321
To my parents
with great affection and esteem
Preface
This volume had its starting point with certain questions about the structure of
the Clifford algebra over a commutative ring, in particular with the existence of
certain "special elements" in the centralizer of the even subalgebra. These - see
Chapter 7 of Hahn-O'Meara The Classical Groups and K-Theory, Springer
Verlag 1989 - play a role in the isomorphism theory of linear groups of small
rank. When I realized that the structure theory of Clifford algebras over
commutative rings could be developed around special elements and the closely
related concept of quadratic algebras, these early investigations had developed into
the theme for this volume. The flow of ideas was refined, expanded, and finalized
in the course of discussions, lectures, and seminars at the University of Notre
Dame. In this context, I would like to express a warm thank you to my
colleagues at the University of Notre Dame, especially to Timothy O'Meara,
who introduced me to the theory of quadratic forms years ago and has been a
source of inspiration since; to Stephan Stolz and Gudlaugur Thorbergsson for
contributing their considerable expertise to Chapter 15, especially for providing
Chapters 15F and 15G, respectively; to Ken Grant, who provided valuable
assistance with the number theory required for the table in Chapter 14E; and to
Warren Wong for the many discussions on the topics of this text A word of
gratitude goes also to my colleagues at the Mathematisches Institut der
Universiutt Innsbruck, particularly to Ottmar Loos and Ulrich Oberst, for their
hospitality both during the academic years 1987-88 (toward the beginning of the
present project) and again in 1992-93 (at its conclusion). Thanks, too, to my
student Zhang Qi for her diligent proofreading, and to the staff at Springer
Verlag, New York, for its courtesy and professionalism. To my wife Marianne:
many thanks for your understanding throughout this enterprisel Last, but
certainly not least, I wish to thank the National Security Agency for supporting
my efforts with a two-year research grant awarded under its Mathematical
Sciences Program.
Innsbruck,Austria
vii
Contents
Preface ............................................................................................. vii
Introduction ................ ........................................................................ 1
Notation and Terminology ..................................................................... 3
Chapter 1. Fundamental Concepts in the Theory of Algebras ..... ...... ............ 5
A. Free Quadratic Algebras .... ........ ........... ............. ....... ......... ...... 5
B. Involutions on Algebras ........... ............ ........... ....... ..... .... ...... 9
C. Gradings on Algebras ............................................................ 10
D. Tensor Products and Graded Tensor Products..... ... ..... ....... ..... ...... 11
E. Exercises ............................................................................. 13
Chapter 2. Separable Algebras ............................................................. 18
A. Separability of Algebras ..... ...... ............ ...... ......................... .. 18
B. Separability Idempotents ..... ....... ............. ... ........ ....... ......... .... 20
C. Separable Free Quadratic Algebras ........................................... 21
D. Properties of Conjugation .. ............. ....... .................... ....... .... 24
E. Exercises ....... ..... .............. ............ ........... .... .... ....... ............ 27
Chapter 3. Groups of Free Quadratic Algebras ........................................ 29
A. The Group QUt\R) .... ...... ................... ................................... 29
a ..................... ... ...... .................................
B. The Discriminant 32
C. The Group QUt\R) ............................................................... 33
D. Another Look at (a, b)E • (b, c)" ............................................. 36
E. Exercises ......... ........................... ... ........ ............................. 40
Chapter 4. Bilinear and Quadratic Forms ............................................... 45
A. Localization ... .......... ........ .......... ......... ............. ................... 45
B. Bilinear Forms .... ...... ........ ......... ........... ............. ...... ... ........ 48
C. The Group Dis(R) ................................................................. 52
D. Quadratic Forms ................................................................... 56
E. Exercises ............................................................................. 61
ix
x Contents
Chapter 5. Clifford Algebras: The Basics ............................................... 65
A. I>efinition and Existence ..... .............. ............... ................ ...... 65
B. Generation, Grading, and Involutions . ........... ................. .......... 66
C. Graded Tensor Product ........................................................... 69
D. Exterior Algebras ................................................................. 73
E. Exercises . . . .... . . .. . . . .. . ... . . . . . ... . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Chapter 6. Algebras with Standard Involution .... ............. ....................... 77
A. Standard Involutions ............... ... .......... ............. ... ........ ......... 77
B. Free Quaternion Algebras ...................................................... 81
C. Separability of Free Quatemion Algebras ................................. 84
D. Nonsingular Algebras .......................................................... 86
E. Exercises ............................................................................ 88
Chapter 7. Arf Algebras and Special Elements ........................................ 92
A. TIle Arf Algebra ................................................................. 92
B. The Arf Algebra of an Orthogonal Sum ............... ..................... 96
C. Special Elements ................................................................. 99
D. Exercises .............. .......... .............. ......... .............. .... ..... ..... 104
Chapter 8. Consequences of the Existence of Special Elements ................ 106
A. Connections between C(M) and Co(M) ................................... 106
B. Gradings I>efmed by Roots of X2 - aX - b....... ............. ......... 108
C. Linear Maps with Polynomial X2 - aX - b..... .............. .... ...... 109
D. Graded Properties of Representations ..................................... 114
E. Comparing the Tensor and Graded Tensor Products ................... 118
F. Exercises ..... . .... . . ... ... .. .. . . ..... ..... . . . . . . . ... . . .. . . . . . . .... . . . . . . . . . . . .. . . 121
Chapter 9. Structure of Clifford and Arf Algebras .................................. 123
A. More on Separable Algebras ................................................. 123
B. The Separability of C(M) and Co(M) .................................... 126
C. The Even-Odd Splitting of C(M) ............................................ 127
D. The Structures ofCen C(M), Cen Co(M), and A(M) ................. 131
E. Exercises ................................... , .... .. . .. ... .. ..... . . .. ..... .. . .. . ... . 133
Chapter 10. The Existence of Special Elements .................................... 137
A. Separable Quadratic Algebras ................................................ 137
B. The Discriminant Module of S .............................................. 142
C. Criteria for the Existence of Special Elements .......................... 144
D. Special Elements and the Discriminant .................................... 146
E. Exercises ........................................................................... 150
Contents xi
Chapter 11. Matrix Theory of Oifford Algebras over Fields .................... 153
A. Matrix Connections between C(M) and Co(M) ......................... 153
B. Basics about Quadratic Spaces .............................................. 157
C. Quaternion Algebras ............................................................ 159
D. Periodicity Phenomena. ....... ......................... ... .... ................. 162
E. Local and Global Fields .. ...... ............................................ ... 164
F. Excr-cises.......... .... ..... ..... ........................... ..... ..... .... ..... ..... 169
Chapter 12. Dis(R) and Qu(R) ........................................................... 172
A. 1be Quadratic Group Qu(R) ..... .......................................... ... 172
B. More about Dis(R) .............................................................. 175
C. Connecting Qu(R) with Dis(R) ............................................. 177
D. The Case of an Integrally Closed Domain ............................... 183
E. The Classical Discriminant ..... ................ ..... ... ..... ................ 187
F. Ex€7Cises ........ .... ..... ..... .... ................... ..... ..... .... ..... ..... ..... 190
Chapter 13. Brauer Groups and Witt Groups ...................................... ... 194
A. Brauer and Brauer-Wall Groups .............................................. 194
B. 1be Graded Quadratic Group QU(R) ....................................... 199
C. The Witt Group of Quadratic Forms ....................................... 204
D. The Witt Group of Symmetric Bilinear Forms ......................... 209
E. The Classical Situations ....................................................... 214
F. Excr-cises..... ....................... ..... ..... ..................................... 218
Chapter 14. 1be Arithmetic ofWq(R) ................................................. 223
A. Arithmetic Dedekind Domains ............................................ ... 223
B. The Arithmetic of Br(R)2 ......... ................ ......... ........ ........... 225
C. AnalyzingWq(R) ................................................................ 229
D. Computing Qu(R....) and Wq(R....) ............................................ 233
E. Connections between W(R) and Wq(R) .................................... 237
F. Excr-cises ..... ................ ... .... ....................... ..... ................... 242
Chapter 15. Applications of Clifford Modules ...................................... 249
A. Clifford Modules ................................................................ 249
B. Vector Fields on Spheres ...................................................... 251
C. Connections with Topological K-Theory ................................. 253
D. Lie Groups and Lie Algebras ................................................. 257
E. DinIc Operators ............................. .................. ................ ... 258
F. Spin Manifolds ............. ............................. .................. ...... 260
G. Isoparametric Hypersurfaces ... ....... ....................... .... ..... ......... 263
Bibliography................... ..... .... .... .... ..... ............. ... ..... .... .... .... ........ 267
Index 281
Introduction
The goal of this volume is to introduce the reader in an elemental and accessible
way to the large and dynamic area of algebras and forms over commutative rings.
Quadratic algebras and their analysis give the volume its direction. Indeed, its
defming moment is the fact that quadratic algebras lie at the heart of the theory
of quadratic forms and Clifford algebras over commutative rings.
The first two chapters set out the main themes: algebras over a commutative
ring R, involutions on algebras, gradings and tensor products of algebras, and
separable algebras. These concepts are illustrated in the concrete case of free
quadratic algebras, i.e., algebras of the form R[X]/(X2 - aX - b). The third
chapter gathers the isomorphism classes of separable free quadratic algebras into
a group, the "free quadratic group," and studies its main properties. This chapter
ends the introductory phase of the book. It is very elementary, but gives a flavor
of the material that is investigated later.
Chapter 4 quickly recalls the localization of rings and modules, introduces
bilinear and quadratic modules, and develops the group of "discriminant
modules," i.e., the group of isomorphism classes of projective bilinear modules
of rank 1. The elementary properties of the Clifford algebra of a quadratic module
are collected in Chapter 5 mostly without proof. The chapter that follows
investigates algebras with "standard" involution. The most important examples
are the quaternion algebras, i.e., Oifford algebras of quadratic modules of rank 2.
They provide concrete examples of the basic concepts and constructions in the
theory of Oifford algebras.
The third part of the book interrelates the first two. It begins in Chapter 7,
with the study of the Arf algebra, i.e., the centralizer of the "even" part of the
Clifford algebra in the full Clifford algebra. This algebra is quadratic and
establishes the connection between quadratic algebras, Clifford algebras, and
quadratic forms. Properties of the Arf algebra, and in particular certain "special
elements" in it, have impact on the representation theory of Clifford algebras and
clarify the connection between the tensor and graded tensor products. This is
taken up in Chapter 8. In Chapter 9, the structure of the Clifford algebra C(M)
and its even part Co(M) for a general fmitely generated projective nonsingular
quadratic module M is analyzed. Both are shown to be separable. If the rank of
M is even, then C(M) is central, and if the rank of M is odd, then the even
part is central. Chapter 10 proves that special elements exist whenever M is
1