Table Of Content147
Graduate Texts in Mathematics
Editorial Board
S. Axler F.W. Gehring P.R. Halmos
Springer
New York
Berlin
Heidelberg
Barcelona
Budapest
Hong Kong
London
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Jonathan Rosenberg
K-
Algebraic Theory
and Its Applications
, Springer
Graduate Texts in Mathematics
ｔａｋｅｕｔｕｾｇＮ＠ Introduction to 33 HIRSCH. Differential Topology.
Axiomatic Set Theory. 2nd ed. 34 SPITlER. Principles of Random Walk.
2 OXTOBY. Measure and Category. 2nd ed. 2nd ed.
3 SCHAEFER. Topological Vector Spaces. 35 WERMER. Banach Algebras and Several
4 HILTON/STAMMBACH. A Course in Complex Variables. 2nd ed.
Homological Algebra. 36 KELLEy/NAMIOKA et al. Linear
5 MAc LANE. Categories for the Working Topological Spaces.
Mathematician. 37 MONK. Mathematical Logic.
6 HUGHES/PIPER. Projective Planes. 38 GRAUERT/FRITlSCHE. Several Complex
7 SERRE. A Course in Arithmetic. Variables.
8 ｔａｋｅｾｇＮ＠ Axiomatic Set Theory. 39 ARvnsON. An Invitation to C*-Algebras.
9 HUMPHREYS. Introduction to Lie Algebras 40 KEMENy/SNELLlKNAPP. Denumerable
and Representation Theory. Markov Chains. 2nd ed.
10 COHEN. A Course in Simple Homotopy 41 ApOSTOL. Modular Functions and
Theory. Dirichlet Series in Number Theory.
11 CONWAY. Functions of One Complex 2nd ed.
Variable I. 2nd ed. 42 SERRE. Linear Representations of Finite
12 BEALS. Advanced Mathematical Analysis. Groups.
13. ANDERSON/FuLLER. Rings and Categories 43 GILLMAN/JERISON. Rings of Continuous
of Modules. 2nd ed. Functions.
14 GOLUBITSKy/GUlLLEMIN. Stable Mappings 44 KENDIG. Elementary Algebraic Geometry.
and Their Singularities. 45 LOtVE. Probability Theory I. 4th ed.
15 BERBERIAN. Lectures in Functional 46 LotVE. Probability Theory 11. 4th ed.
Analysis and Operator Theory. 47 MOISE. Geometric Topology in
16 WINTER. The Structure of Fields. Dimensions 2 and 3.
17 ROSENBLATT. Random Processes. 2nd ed. 48 SACHs/WU. General Relativity for
18 HALMOS. Measure theory. Mathematicians.
19 HALMOS. A Hilbert Space Problem Book. 49 GRUENBERo/WEIR. Linear Geometry.
2nd ed. 2nd ed.
20 HUSEMOLLER. Fibre Bundles. 3rd ed. 50 EDWARDS. Fermat's Last Theorem.
21 HUMPHREYS. Linear Algebraic Groups. 51 KLINGENBERG. A Course in Differential
22 BARNEs/MACK. An Algebraic Introduction Geometry.
to Mathematical Logic. 52 HARTSHORNE. Algebraic Geometry.
23 GREUB. Linear Algebra. 4th ed. 53 MANIN. A Course in Mathematical Logic.
24 HOLMES. Geometric Functional Analysis 54 GRAVERlWAlKINS. Combinatorics with
and Its Applications. Emphasis on.the Theory of Graphs.
25 HEWITT/STROMBERG. Real and Abstract 55 BROWN/PEARCY. Introduction to Operator
Analysis. Theory I: Elements of Functional
26 MANEs. Algebraic Theories. Analysis.
27 KELLEY. General Topology. 56 MASSEY. Algebraic Topology: An
28 ZAIusKIlSAMUEL. Commutative Algebra. Introduction.
Vol.l. 57 CROWEulFox. Introduction to Knot
29 ZAIusKIlSAMUEL. Commutative Algebra. Theory.
Vol.I1. 58 KOBLITZ. p-adic Numbers, ｰｾ｡､ｩ｣＠
30 JACOBSON. Lectures in Abstract Algebra I. Analysis, and Zeta-Functions. 2nd ed.
Basic Concepts. 59 LANG. Cyclotomic Fields.
31 JACOBSON. Lectures in Abstract Algebra 60 ARNOLD. Mathematical Methods in
II. Linear Algebra. Classical Mechanics. 2nd ed.
32 JACOBSON. Lectures in Abstract Algebra
III. Theory of Fields and Galois Theory. continued after index
Jonathan Rosenberg
Department of Mathematics
University of Maryland
College park, MD 20742 USA
Editorial Board
S. Axler F. W. Gehring P.R. Halmos
Department of Department of Department of
Mathematics Mathematics Mathematics
Michigan State University University of Michigan Santa Clara University
East Lansing, MI 48824 Ann Arbor, MI 48109 Santa Clara, CA 95053
USA USA USA
Mathematics Subject Classification (1991): 19-01, 19-02, 19AXX, 19BXX, 19CXX,
55QXX, 19EXX, 19JXX, 19KXX, 18G60
Library of Congress Cataloging-in-Publication Data
Rosenberg, J. (Jonathan), 1951-
Algebraic K-theory and its applications / Jonathan Rosenberg.
p. cm. -- (Graduate texts in mathematics; )
Includes bibliographic references and index.
ISBN-13:978-1-4612-8735-3 e-ISBN-13:978-1-4612-4314-4
DOl: 10.1007/978-1-4612-4314-4
1. K-theory. 1. Title. II. Series.
QA612.33.R67 1994
512.55--dc20 94-8077
Printed on acid-free paper.
@ 1994 Springer-Verlag New York, Inc.
Reprint of the original edition 1994
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9 8 76 5 4 3 2 (Corrected second printing, 1996)
ISBN 0-387-94248-3 Springer-Verlag New York Berlin Heidelberg
ISBN 3-540-94248-3 Springer-Verlag Berlin Heidelberg New York
Preface
Algebraic K-theory is the branch of algebra dealing with linear algebra
(especially in the limiting case of large matrices) over a general ring R
instead of over a field. It associates to any ring R a sequence of abelian
groups Ki(R). The first two of these, Ko and K1, are easy to describe in
concrete terms; the others are rather mysterious. For instance, a finitely
generated projective R-module defines an element of Ko(R), and an invert
ible matrix over R has a "determinant" in Kl(R). The entire sequence of
groups Ki (R) behaves something like a homology theory for rings.
Algebraic K-theory plays an important role in many areas, especially
number theory, algebraic topology, and algebraic geometry. For instance,
the class group of a number field is essentially Ko(R), where R is the ring
of integers, and "Whitehead torsion" in topology is essentially an element
of K1(Z1I"), where 11" is the fundamental group of the space being stud
ied. K-theory in algebraic geometry is basic to Grothendieck's approach
to the Riemann-Roch problem. Some formulas in operator theory, involv
ing determinants and determinant pairings, are best understood in terms
of algebraic K-theory. There is also substantial evidence that the higher
K -groups of fields and of rings of integers are related to special values of
L-functions and encode deep arithmetic information.
This book is based on a one-semester course I gave at the University
of Maryland in the fall of 1990. Most of those attending were second- or
third-year graduate students interested in algebra or topology, though there
were also a number of analysis students and faculty colleagues from other
areas. I tried to make the course (and this book) fairly self-contained, and
to assume as a prerequisite only the standard one-year graduate algebra
course, based on a text such as [Hungerford], [Jacobson], or [Lang], and the
standard introductory graduate course on algebraic and geometric topol
ogy, covering the fundamental group, homology, the notions of simplicial
and OW-complex, and the definition and basic properties of manifolds. As
taught at Maryland, the graduate algebra course includes the most basic
definitions and concepts of category theory; a student who hasn't yet seen
these ideas could consult any of the above algebra texts or an introduc
tion to category theory such as [Mac Lane]. Since many graduate algebra
courses do not include much in the way of algebraic number theory, I have
vi Preface
included many topics such as the basic theory of Dedekind rings and the
Dirichlet unit theorem, which may be familiar to some readers but not
to all. I've tried in this book to presuppose as little topology as possible
beyond a typical introductory course, and to develop what is needed as
I go along, but to give the reader a flavor of some of the important ap
plications of the subject. A reader with almost no topology background
should still be able to follow most of the book except for parts of Sections
1.6, 1.7, 2.4, 4.4, and 6.3, and most of Chapter 5 (though I would hope
this book might encourage him or her to take a more systematic course
in topology). A problem one always has in writing a book such as this
is to decide what to do about spectral sequences. They are usually not
mentioned in first-year graduate ｣ｯｵｲｳｾＬ＠ and yet they are indispensable
for serious work in homological algebra and K-theory. To avoid having
to give an introduction to spectral sequences which might scare. off many
readers, I have avoided using spectral sequences directly anywhere in the
text. On the other hand, I have made indirect reference to them in many
places, so that the reader who has heard of them will often see why they
are relevant to the subject and how they could be used to simplify some of
the proofs.
For the most part, this book tends to follow the notes of the original
course, with a few additions here and there. The major exceptions are
that Chapters 3 and 5 have been greatly expanded, and Chapter 6 on
cyclic homology has been added even though there was no time for it in
the original course. Cyclic homology is a homology theory for rings which
may be viewed as the "linearized version" of algebraic K-theory, and it's
becoming increasingly clear that it is both a useful computational tool and
a subject of independent interest with its own applications.
Each chapter of this book is divided into sections, and I have used a
single numbering system for all theorems, lemmas, exercises, definitions,
and formulas, to make them easier to locate. Thus a reference such as
1.4.6 means the 6th numbered item in Section 4 of Chapter 1, whether
that item is a theorem, a corollary, an exercise, or a displayed formula.
The exercises are an integral part of the book, and I have tried to put at
least one interesting exercise at the end of every section. The reader should
not be discouraged if he finds some of the exercises too difficult, since the
exercises vary from the routine to the very challenging.
I have used a number of more-or-less standard notations without special
reference, but the reader who is puzzled by them will be able to find most
of them listed in the Notational Index in the back of the book.
Why This Book?
The reader might logically ask how this book differs from its "competi
tion." [Bass] remains an important reference, but it is too comprehensive
to use as a text for an elementary course, and also it predates the defini
tion of K2, let alone of higher K-theory or of cyclic homology. My original
course was based on the notes by Milnor [Milnor], which are highly recom
mended. However, I found that [Milnor] is hard to use as a textbook, for
Preface vii
the following three reasons:
(1) Milnor writes for a working mathematician, and sometimes leaves
out details that graduate students might not be able to provide for
themselves.
(2) There are no exercises, at least in the formal sense.
(3) The subject has changed quite a bit since Milnor's book was writ
ten.
For the working algebraist already familiar with the contents of [Milnor]
who wants to learn about Quillen K-theory and its applications in alge
braic geometry, [Srinivas] is an excellent text, but it would have been far
beyond the reach of my audience. The notes of Berrick [Berrick] give a
more elementary introduction to Quillen K-theory than [Srinivas], but are
rather sketchy and do not say much about applications, and thus again are
not too suitable for a graduate text. And [LluisP] is very good for an up
to-date survey, but is, as the title says, an overview rather than a textbook.
For cyclic homology, the recent book by Loday [LodayCH] is excellent, but
to be most useful requires the reader already to know something about
K-theory. Also, I do not believe that there is any book available that dis
cusses the applications of algebraic K-theory in functional analysis (which
are discussed here in 2.2.10-2.2.11,4.4.19-4.4.24,4.4.30,6.3.8--6.3.17, and
6.3.29-30). Thus for all these reasons it seemed to me that another book
on K-theory is needed. I hope this book helps at least in part to fulfill that
need.
Acknowledgements
I would like to thank Mike Boyle for making his notes of my lectures
(often much more readable and complete than my own) available tome, and
to thank all the others who attended my lectures for their useful feedback.
I would also like to thank the National Science Foundation for its support
under grants # DMS 90-02642 and # DMS 92-25063, which contributed
substantially to the research that went into the writing of this book, both
directly and indirectly. Finally, I would like to thank several anonymous
referees and numerous colleagues, including in particular Andrew Ranicki
and Shmuel Weinberger, for useful suggestions on how to improve my early
drafts.
Added August, 1995. I am quite grateful for the opportunity to correct
a few mistakes in the second printing. I thank several colleagues for their
comments on this book, and especially thank Bruce Blackadar, Andreas
Rosenschon, and Dan Voiculescu for their corrections.
Contents
Preface v
Chapter 1. Ko of Rings 1
1. Defining Ko 1
2. Ko from idempotents 7
3. Ko of PIDs and local rings 11
4. Ko of Dedekind domains 16
5. Relative Ko and excision 27
6. An application: Swan's Theorem and topological
K-theory 32
7. Another application: Euler characteristics and the Wall
finiteness obstruction 41
Chapter 2. Kl of Rings 59
1. Defining K 1 59
2. Kl of division rings and local rings 62
3. Kl of PIDs and Dedekind domains 74
4. Whitehead groups and Whitehead torsion 83
5. Relative Kl and the exact sequence 92
Chapter 3. Ko and Kl of Categories, Negative
K -Theory 108
1. Ko and Kl of categories, Go and G1 of rings 108
2. The Grothendieck and Bass-HeIler-Swan Theorems 132
3. Negative K-theory 153
Chapter 4. Milnor's K2 162
1. Universal central extensions and H2 162
Universal central extensions 163
Homology of groups 168
2. The Steinberg group 187
3. Milnor's K2 199
4. Applications of K2 218
Computing certain relative Kl groups 218
K2 of fields and number theory 221
Almost commuting operators 237
Pseudo-isotopy 240
x Contents
Chapter 5. The +-Construction and Quillen
K -Theory 245
1. An introduction to classifying spaces 245
2. Quillen's +-construction and its basic properties 265
3. A survey of higher K-theory 279
Products 279
K-theory of fields and of rings of integers 281
The Q-construction and results proved with it 289
Applications 295
Chapter 6. Cyclic homology and its relation to
K -Theory 302
1. Basics of cyclic homology 302
Hochschild homology 302
Cyclic homology 306
Connections with "non-commutative de Rham theory" 325
2. The Chern character 331
The classical Chern character 332
The Chern character on Ko 335
The Chern character on higher K-theory 340
3. Some applications 350
Non-vanishing of class groups and Whitehead groups 350
Idempotents in C* -algebras 355
Group rings and assembly maps 362
References 369
Books and Monographs on Related Areas of Algebra,
Analysis, Number Theory, and Topology 369
Books and Monographs on Algebraic K -Theory 371
Specialized References 372
Notational Index 377
Subject Index 383
