Algebraic K-theory

374 Pages·1997·79.5 MB·English

by  Snaith, Victor P.

Selected Title s i n Thi s Serie s
16 Victo r P. Snaith, Editor, Algebrai c K-Theory, 1997
15 Stephe n P. Braham, Jack D. Gegenberg, and Robert J. McKellar, Editors , Sixt h
Canadian conference on general relativity and relativistic astrophysics, 1997
14 Moura d E. H. Ismail, David R. Masson, an d Mizan Rahman, Editors , Specia l
functions, g-series and related topics, 1997
13 Pete r A . Fillmore and James A. Mingo , Editors, Operato r algebras and their
applications, 1997
12 Dan-Virgi l Voiculescu, Editor , Fre e probability theory, 1997
11 Collee n D. Cutler and Daniel T. Kaplan , Editors, Nonlinea r dynamics and time
series: Building a bridge between the natural and statistical sciences, 1997
10 Jerrol d E. Marsden, Georg e W. Patrick , an d William F. Shadwick, Editors ,
Integration algorithms and classical mechanics, 1996
9 W . H . Kliemann, W. F . Langford, an d N. S . Namachchivaya, Editors , Nonlinea r
dynamics and stochastic mechanics, 1996
8 Larr y M. Bate s and David L. Rod, Editors, Conservativ e systems and quantum
chaos, 1996
7 Willia m F. Shadwick, Perinkula m Sambamurth y Krishnaprasad , an d Tudo r
Stefan Ratiu, Editors, Mechanic s day, 1996
6 Ann a T. Lawniczak an d Raymond Kapral , Editors, Patter n formation and lattice
gas automata, 199 6
5 Joh n Chadam, Marti n Golubitsky, Willia m Langford, an d Brian Wetton ,
Editors, Patter n formation: Symmetry methods and applications, 1996
4 Willia m F. Langford and Wayne Nagata , Editors , Norma l forms and homoclinic
chaos, 1995
3 Anthon y Bloch , Editor, Hamiltonia n and gradient flows, algorithms and control, 1994
2 K . A. Morris, Editor, Contro l of flexible structures, 1993
1 Michae l J. Enos, Editor, Dynamic s and control of mechanical systems: The falling cat
and related problems, 1993
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http://dx.doi.org/10.1090/fic/016
FIELDS INSTITUT E
COMMUNICATIONS
THE FIELD S INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCE S
Algebraic K-Theor y
Victor P. Snaith
Editor
American Mathematical Society
Providence, Rhode Island
The Fields Institut e
for Research in Mathematical Science s
The Fields Institute is named in honour of the Canadian mathematician Joh n Charle s
Fields (1863-1932) . Field s was a visionary who received man y honour s for his scientifi c
work, including election to the Royal Society of Canada in 1909 and to the Royal Society of
London in 1913. Among other accomplishments in the service of the international math -
ematics community, Field s was responsible fo r establishin g the world's most prestigiou s
prize for mathematics research—the Fields Medal .
The Fields Institute for Research in Mathematical Sciences is supported by grants fro m
the Ontario Ministry of Education and Training and the Natural Sciences and Engineerin g
Research Counci l o f Canada . Th e Institut e i s sponsored b y McMaste r University , th e
University of Toronto, the University of Waterloo, and York University and has affiliate d
universities in Ontario and across Canada .
1991 Mathematics Subject Classification. Primar y 19-06 .
Library of Congress Cataloging-in-Publication Dat a
Algebraic K-Theory / Victor P. Snaith, editor.
p. cm. — (Fields Institute Communications, ISSN 1069-5265 ; v. 16)
Papers from the Second Great Lakes Conference on Algebraic K-Theory, held Mar. 1996 at the
Fields Institute for Research in Mathematical Sciences in memory of Robert Wayne Thomason.
Includes bibliographical references.
ISBN 0-8218-0818-4 (alk. paper)
1. K-theory—Congresses. 2 . Geometry, Algebraic—Congresses . 3 . Topology—Congresses .
I. Snaith, V. P. (Victor Percy), 1944 - . II . Thomason, Robert Wayne, 1952-1995. III . Great
Lakes Conference on Algebraic K-Theory (2n d : 1996 : Fields Institute for Research in Mathe-
matical Sciences) IV . Series.
QA612.33.A384 199 7
512/.55—dc21 97-2024 3
CIP
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10 9 8 7 6 5 4 3 2 1 0 2 01 00 99 98 97
Contents
Preface vi i
Robert W. Thomason (1952-1995) i x
Quaternionic Exercises in K-Theory Galois Module Structure 1
TED CHINBURG, MANFRED KOLSTER, GEORG E PAPPA S
and VICTO R P. SNAIT H
The Bloch-Ogus-Gabber Theorem 3 1
JEAN-LOUIS COLLIOT-THELENE, RAYMON D T. HOOBLE R
and BRUN O KAH N
Milnor's Conjecture and Galois Theory I 9 5
WENFENG GAO and JAN MlNAC
Ultraproducts and the Discrete Cohomology of Algebraic Groups 11 1
J. F. JARDINE
Lambda-Operations, K-Theory and Motivic Cohomology 13 1
MARC LEVIN E
Quasi-Motives of Curves 18 5
STEPHEN LICHTENBAU M
A Chain Complex for the Spectrum Homology of the Algebraic 19 9
K-Theory of an Exact Category
RANDY MCCARTH Y
Hypercohomology Spectra and Thomason's Descent Theorem 22 1
STEPHEN A. MITCHEL L
Kahler Differentials of Certain Cusps 27 9
LESLIE G. ROBERT S
vi
Local Fundamental Classes Derived from Higher-Dimensiona l
K-Groups
VICTOR P. SNAIT H
Local Fundamental Classes Derived from Higher-Dimensiona l
K-Groups: II
VICTOR P. SNAIT H
Weak Approximation, R-Equivalence and Whitehead Group s
NGUYEN QUOC THAN G
Appendix I
Appendix II
Preface
The Second Great Lakes Conference on Algebraic K-Theory was hosted by The
Fields Institute for Research in Mathematical Sciences in March 1996. I t proved
to be a mathematically very exciting occasion with many interesting new results
being unveiled. The success of the meeting together with a desire to commemorate
Bob Thomason, one of K-theory's most influential figures, led to this Proceedings
volume.
As editor, I would like to express my thanks and those of the organizers of the
conference to the staff of The Fields Institute who made this meeting so rewarding
and enjoyable. In particular, I wish to extend special thanks to the President and
Scientific Director, John Chadam, for inviting the Great Lakes K-Theory Confer-
ence to The Fields Institute, and to the Publications Assistant, Erna Unrau, for
making the editing of this volume so easy for me.
Victor Snaith
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Robert Wayn e THOMASO N
1952-1995
In the mid 1970's, when Bob Thomason was a star graduate student at Prince-
ton University, algebrai c K-theory wa s in a very tentative state . I t ha d begu n
as Whitehead-ia n algebra-to-capture-geometry , showin g lot s o f low-dimensiona l
promise. Then , wit h the work of Quillen, algebrai c K-theory ha d suddenly ex -
panded into a grand cosmological entity with its own, extremely fascinating, ontol-
ogy and epistomology. It was clear that the subject constituted a major mathemat-
ical nexus but the evidence was in the form of hard-won hints about how Quillen's
K-theory might unify such diverse phenomena as homotopy types of diffeomorphism
groups of manifolds, special values of L-functions, polylogarithms, intersection the-
ory of algebraic varieties, Weil cohomology theories and so on. As a result, no one
was sure whether K-theory was the domain of topologists, algebraists or some other
mathematical species. In many places this hardly mattered, the subject could still
easily be shrugged off as a new fad with fine ambitions, lots of conjectured promise,
but with no "big theorem" to mark some post-Quillen progress.
Bob Thomason was the one who was to provide such a "big theorem".
ix