K-Theory of Forms PDF
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Annals of Mathematics Studies Number98 K-THEORY OF FORMS BY ANTHONY BAK PRINCETON UNIVERSITY PRESS AND UNIVERSITY OF TOKYO PRESS PRINCETON, NEW JERSEY 1981 Copyright© 1981 by Princeton University Press ALL RIGHTS RESERVED The Annals of Mathematics Studies are edited by Wu-chung Hsiang, Robert P. Langlands, John Milnor, and Elias M. Stein Corresponding editors: M. F. Atiyah, Hans Grauert, Phillip A. Griffiths, and Louis Nirenberg Published in Japan exclusively by University of Tokyo Press; In other parts of the world by Princeton University Press 41 William Street Princeton, New Jersey Printed in the United States of America by Princeton University Press, Princeton, New Jersey Library of Congress Cataloging in Publication data will be found on the last printed page of this book In memory of my father TABLE OF CONTENTS §1. INTRODUCTION 3 A. General remarks 3 B. Quadratic modules 5 C. Hermitian modules 11 D. The necessity for refined definitions 15 §2. HYPERBOLIC AND METABOLIC MODULES 17 §3. AUTOMORPHISM GROUPS OF NONSINGULAR MODULES 24 §4. K-THEORY GROUPS OF NONSINGULAR MODULES 60 A. The Ki -functors 60 B. The K2-functors 62 C. Relativization 66 §5. HOMOLOGY EXACT SEQUENCES 70 A. Homology groups and central extensions 70 B. The relative sequence of a homomorphism 74 C. The Mayer-Vietoris sequence of a fibre square 78 D. Excision 85 §6. K-THEORY IN CATEGORIES WITH PRODUCT 93 A. Fibre product categories 93 B. The relative sequence of a product preserving functor 97 C. The Mayer-Vietoris sequence of a fibre square 107 D. Excision 110 §7. K-THEORY OF NONSINGULAR AND PROJECTIVE MODULES 113 A. Approximation squares 114 B. Arithmetic and localization, completion squares 126 C. Conductor and related squares 133 D. Fibre product categories 139 E. Restricted direct products 145 F. Orders 151 §8. COMPARISON EXACT SEQUENCES 155 A. Change of K 1-torsion 155 B. Change of K2-torsion 161 C. Kernels and cokernels of hyperbolic and metabolic maps 178 vii viii TABLE OF CONTENTS §9. SCALING AND MORITA: THEORY 180 §10. REDUCTION MODULO A COMPLETE IDEAL 184 §11. CHANGE OF FORM PARAMETER 190 A. The group S(f'I A) 190 B. The group T(f'I A) 206 §12. INDUCTION THEORY 232 A. Frobenius modules 232 B. Induction machine 242 §13. ALTERNATE DEFINITIONS OF QUADRATIC MODULES 251 §14. REMARKS ON NOTATION 256 §15. WALL'S SURGERY GROUPS 258 BIBLIOGRAPHY 259 SUBJECT INDEX 262 NOTATION INDEX 266
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