Lecture Notes in Mathematics Edited by A Dold and B. Eckmann 342 Algebraic K.:rheory 11- "Ciassical" Algebraic K-Theory, and Connections with Arithmetic Proceedings of the Conference held at the Seattle Research Center of the Battelle Memorial Institute, Aug. 28-Sept. 8, 1972 Edited by H. Bass Springer-Verlag Berlin Heidelberg New York Tokyo Editor Hyman Bass Department of Mathematics, Columbia University NewYork, N.Y. 10027, USA 1st Edition 1973 2nd Printing 1986 Mathematics Subject Classification (1970): 13D 15, 14F 15, 16A54, 18F25 ISBN 3-540-06435-4 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-06435-4 Springer-Verlag New York Heidelberg Berlin Tokyo This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. ©by Springer-Verlag Berlin Heidelberg 1973 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210 Introduction A conference on algebraic K-theory was held at the Battelle Seattle Research Center from August 28 to September 8, 1972, with the joint support of the National Science Foundation and the Battelle Memorial Institute, The present volume consists mainly of papers presented at, or stimulated by, that conference, plus some closely related papers by mathematicians who did not attend the conference but who have kindly consented to publish their work here. In addition there are several papers devoted to surveys of subjects treated at the conference, and to the formulation of open research problems. It was our intention thus to present a reasonably comprehensive documentation of the current research in algebraic K-theory, and, if possible, to give this research a greater coherence than it has heretofore enjoyed. It was particularly grati~fying to see the latter aim largely achieved already in the course of preparing these Proceedings. Algebraic K-theory has two quite different historical roots both in geometry, The first is concerned with certain topological obstruction groups, like the Whitehead groups, and the L-groups of surgery theory. Their computation, which is in principle an algebraic problem about group rings, is one of the original missions of algebraic K-theory. It remains a rich source of new problems and ideas, and an excellent proving ground for new techniques. The second historical source of algebraic K-theory, from which the subject draws its name, is Grothendieck's proof of the Riemann-Roch theorem, and the topological K-theory of Atiyah-Hirzebruch, which has the same point of departure, Starting from the analogy between projective modules and vector bundles one is led to seek a K-theory for rings analogous to that of Atiyah-Hirzebruch for spaces. This enterprise made, at first, only very limited progress. In the few years preceding this conference, however, several interesting definitions of higher K-groups were proposed; the relations between them were far from clear. Meanwhile the detailed study of K and K had revealed some beautiful 1 2 arithmetic phenomena within the classical groups. This contact with algebraic number theory had become a major impulse in the subject as well as a theme for IV conjectures about the significance of the higher K-groups. More recently there have appeared definitions and potential applications of higher K-theory in the framework of algebraic geometry. As this brief account suggests, a large number of mathematicians, with quite different motivations and technical backgrounds, had become interested in aspects of algebraic K-theory. It was not altogether apparent whether the assembling of these efforts under one rubric was litte more than an accident of nomenclature. In any case it seemed desireable to gather these mathematicians, some of whom had no other occasion for serious technical contact, in a congenial and relaxed setting, and to leave much of what would ensu.e to mathematical and human chemistry. A consensus of those who were present is that the experiment was enormously successful. Testimony to this is the fact that many of the important new results in these volumes were proved in the few months following the conference, growing out of collaborative efforts and discussions begun there. One major conclusion of this research is that all of the higher K-theories which give the "classical" Kn 1 s for n s; 2 coincide. Thus, in some sense, the subject of higher algebraic K-theory "exists'~ an assertion some had begun to depair of making. Moreover one now has, thanks largely to the extraordinary work of Quillen, some very effective tools for calculating higher K-groups in interesting cases. The papers that follow are somewhat loosely organized under the headings: I. Higher K-theories; I I. "Classical" algebraic K-theory, and connections with arithmetic; and Ill. Hermitian K-theories and geometric applications. Certain papers, as their titles indicate, contain collections of research problems. The reader should be warned, however, that because of the vigorous activity ensu.-'ing the conference, some of the research problems posed below are in fact resolved elsewhere in these volumes. The editional effort necessary to eliminate such instances would have cost an excessive delay in publication. I am extremely grateful to the following participants who contributed V to the preparation of the survey and research problem articles: S, Bloch, J. Coates, Keith Dennis, S. Gersten, M. Karoubi, M.P. Murthy, Ted Petrie, L. Roberts, J. Shaneson, M. Stein, and R. Swan. On behalf of the participants I express our thanks to the National Science Foundation and the Battelle Memorial Institute for their generous financial support. For the splendid facilities and setting of the Battelle Seattle Research Center, and for the efficient and considerate services of its staff, the conference participants were uniformly enthusiastic in their praise and gratitude. Finally, I wish to thank Kate March of Columbia University for her invaluable secretarial and administrative assistance in organizing the conference, and Robert Martin of Columbia University for his aid in editing these Proceedings. H. Bass Paris, April, 1973 LIST OF PARTICIPANTS AND AUTHORS Dr. Neil Paul Aboff Dr. Spencer J. Bloch Department of Mathematics Department of Mathematics Harvard University Fine Hall 2 Divinity Avenue Princeton University Cambridge, MA 02138 Princeton, NJ 08540 Dr. Yilmaz Akyildiz Dr. Armand Borel Department of Mathematics Institute for Advanced Study University of California Princeton University at Berkeley Princeton, NJ 08540 Berkeley, CA 94704 Dr. Kenneth S. Brown Dr. RO<:Jer Alperin Department of Mathematics Department of Mathematics Cornell University Rice University Ithaca, NY 04850 6100 Main Street Houston, TX ·nool Dr. Sylvain Cappell Department of Theoretical Dr. Donald w. Anderson Mathematics Department of Mathematics The Weizman Institute of Science University of California Rehovoth, Israel at Sctn Diego San Diego, CA 92037 Mr. Joe Carroll Department of Mathematics Dr. David M. Arnold Harvard University Department of Mathematics 2 Divinity Avenue New Mexico State University Cambridge, MA 02138 Las Cruces, NM 88001 Dr. A. J. Casson Dr. Anthony Bak Department of Mathematics Departement des Math~matiques Trinity College 2-4 rue du Lievre Cambridge, London, England Geneve, Switzerland Dr. Stephen U. Chase Dr. Hyman Bass Department of Mathematics Department of Mathematics Cornell University Columbia University White Hall Broadway and West 116th Street Ithaca, NY 14850 New York, NY 10027 Dr. K. G. Choo Dr. Israel Berstein Department of Mathematics Department of Mathematics University of British Columbia White Hall Vancouver, British Columbia Cornell University Canada Ithaca, NY 14850 VIII Dr. John Henry Coates Dr. Charles H. Giffen Department of Mathematics Department of Mathematics Stanford University University of Virginia Stanford, California 94305 Charlottesville, VA 22904 Dr. Edwin H. Connell Mr. Jimmie N. Graham Department of Mathematics Department of Mathematics University of Miami McGill University Coral Gables, FL 33124 P. o. Box 6070 Montreal 101, Quebec Canada Dr. Francis X. Connolly Department of Mathematics Dr. Bruno Harris University of Notre Dame Department of Mathematics Notre Dame, IN 46556 Brown University Providence, RI 02912 Dr. R. Keith Dennis Department of Mathematics Dr. Allen E. Hatcher cornell University Department of Mathematics White Hall Fine Hall Ithaca, NY 14850 Princeton University Princeton, NJ 08540 Dr. Andreas w. M. Dress Fakultat fur Mathematik Dr. Alex Heller Universitat Bielefeld, FRG Department of Mathematics 48 Bielefeld Institute for Advanced Study Postfach 8640, Germany Princeton University Princeton, NJ 08540 Dr. Richard Elman Department of Mathematics Dr. Wu-chung Hsiang Rice University Department of Mathematics 6100 Main Street Fine Hall Houston, TX 77001 Princeton University Princeton, NJ 08540 Dr. E. Graham Evans, Jr. Department of Mathematics Dr. James E. Humphreys University of Illinois Courant Institute Urbana, Illinois 61801 New York University 251 Mercer Street Dr. Howard Garland New York, NY 10012 Department of Mathematics State University of New York Dr. Dale Husemoller at Stony Brook Department of Mathematics Stony Brook, NY