Algebraic K-Theory

328 Pages·1991·8.263 MB·English

by  V. Srinivas (auth.)

Progress in Mathematics
Volume90
Series Editors
J. Oesterle
A. Weinstein
V. Srinivas
Algebraic K-Theory
1991 Springer Science+Business
Media, LLC
V. Srinivas
School of Mathematics
Tata Institute of Fundamental Research
Bombay, India
ISBN 978-1-4899-6737-4 ISBN 978-1-4899-6735-0 (eBook)
DOI 10.1007/978-1-4899-6735-0
Printed on acid-free paper.
©Springer Science+Business Media New York 1991
Originally published by Birkhauser Boston in 1991
Softcover reprint of the hardcover 1st edition 1991
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Dedicated to my parents.
Introduction
These notes are based on a course of lectures I gave at the Tata Institute
during 1986-87. The aim of the course was to give an introduction to
higher K-theory, and in particular, to expose in <iletail the results of Quil
len, contained in the following basic papers:
1. D. Quillen: "Higher Algebraic K-Theory I," Lect. Notes in Math.
No. 341, Springer-Verlag, New York (1973).
2. D. Grayson: "Higher Algebraic K-Theory II (after Daniel Quil
len)," Lect. Notes in Math. No. 551, Springer-Verlag, New York
(1976).
The audience consisted of colleagues and some graduate students who
were mainly algebraists and algebraic geometers, and were interested in
learning about K-theory because of its applications to these fields. Most
members of the audience had a limited background in topology. As such,
one of my aims during the course was to give proofs of the topological
results needed, assuming the minimum possible. The two applications
(beyond Quillen's results) which are discussed also reflect the tastes of
the audience (and the lecturer). In a few places, I chose not to prove
results in the maximum possible generality, when I felt that the ideas
behind the proofs might be obscured by technical details.
Algebraic K-theory is an active area of research, which has connec
tions with algebra, algebraic geometry, topology and number theory.
Some recent interesting results in algebra and related fields proved using
K-theoretic methods are the following:
(i) Mercurjev and Suslin's theorem on Brauer groups of fields, its gen
eralisations due to Merkurjev and Suslin, and Levine, and the work
of Rost and others on Milnor's conjecture relating K and Witt
2
groups; these results have interesting consequences for the Chow
groups of algebraic cycles modulo rational equivalence on a smooth
algebraic variety.
(ii) Serre's conjecture on the vanishing of intersection multiplicities for
modules over a regular local ring, proved by Gillet and Soule (this
was independently proved by Paul Roberts, using' intersection the
ory, as developed in Fulton's book).
vii
viii Introduction
(iii) Levine's computation in terms of K for the Grothendieck group of
1
modules of finite length and finite projective dimension over the
local ring of an isolated Cohen-Macaulay singular point of a variety,
which leads to a new proof of the results of Dutta, Hochster and
MacLaughlin on modules of finite projective dimension with nega
liv~ intersection multiplicity. Levine's results also have applications
to Chow groups of singular varieties. A generalisation of Levine's
results has been announced by Thomason and Trobaugh, which
should have similar applications.
Chapters 8 and 9 touch on the first and third 'algebraic' applications
mentioned above.
Lack of knowledge prevents me from giving a detailed list of results
in topology and number theory, but I will mention W aldhausen' s alge
braic K-theory of spaces, which is a key ingredient in the recently an
nounced proof by Hsiang and Madsen of the Novikov conjecture on the
surgery obstruction map; the higher dimensional generalisation of class
field theory due to Bloch, Kato, Saito, and others; work of Bloch, Colli
ot-Thelene, Sansuc, and others on the torsion and cotorsion of Chow
groups of varieties over number fields and local fields; and finally, results
of Bloch, Beilinson and others relating the ranks of K-groups and Chow
groups of varieties over number fields to the orders of vanishing of L
functions, leading to the celebrated Beilinson conjectures. For the alge
braic geometers, I should also mention Beilinson's generalisation of the
Hodge conjectures, which relate certain groups of transcendental coho
mology classes to K-theory.
The work of Quillen, cited above, provides the foundation for much
of this work, and forms the core of these notes.
The more algebraically minded reader II¥1Y prefer, at a first reading,
to read Chapters 1, 3, 4 and 5, skip the more topological Chapters 2, 6,
7, and go on to applications of interest. The somewhat long Appendix A
should, I hope, help such a reader to eventually work through the 'topo
logical' chapters.
The detailed contents of the notes are as follows (this is for the benefit
of readers who may already have some acquaintance with K-theory, say
at the level of K and to state where we have omitted topics from Quil
0,
len's papers).
Chapter 1 contains a quick review of "classical" K-theory (i.e., K0,
Ku K mainly based on Milnor's book. In chap. 2, Quillen's plus con
2),
struction is given, leading to the definition K (R) = Tr;(BGL(RV) fori
1
~ 1, for the higher K-groups of a ring. We construct products K;(R) ®
Ki (R)-+ K;+j(R), following Loday. We also show that K1, K2 agree with
the "classical(' Ku K of chap. 1.
2
Chapters 3-7 contain our exposition of ''Higher Algebraic K-Theory
Introduction ix
I'', and the comparison of the plus and Q constructions for K;. Chapter
3 introduces the language of simplicial sets, and leads to the basic notion
of the classifying space ~ of a small category '€. This leads to a '' dic
tionary" between category theory and topology, under which we have
the following correspondences:
small category ~ topological space (CW complex)
functor ~ cellular map
natural transformation ~ homotopy
adjoint functors ~ homotopy inverse pair
of maps
category with an initial or ~ contractible space.
final object
In Chapter 4, we give Quillen's Q construction, using the K-groups of
a small exact category '€ in terms of the homotopy groups of the classify
ing space of its associated Quillen category QC(6,
K;('€) = '1Ti+t (BQ'€).
We then state a number of purely "K-theoretic" results, contained in the
first part of "Algebraic K-Theory I" (the computation of K and the
0,
theorems on exact sequences off unctors, resolution, devissage and local
isation). Given that K is the same as that defined "classically," these
0
theorems express standard facts about K However, the extensions to the
0•
higher K; involve a lot of new machinery. As such, we postpone the
proofs of these results to Chapter 6.
In Chapter 5, we apply the results of Chapter 4 to study the K-theory
of rings and schemes (the second part of ''Higher Algebraic K-Theory
I"). If R is a ring (respectively, a Noetherian ring), let K;(R) (respec
tively, G;(R)) be the K-groups of the category of finitely generated projec
tiveR-modules (respectively, o=f a ll finitely generatedR=-m odules).We first
prove the formulas G;(R[t]) G;(R), G;(R[t,t-1]) G;(R) EB G;_1(R)
(Quillen deduces these formulas from more general results about filtered
rings, which we omit; we also omit the applications to computing K; for
certain division rings).
For a (Noetherian separated) scheme X, define K;(X) (respectively,
G;(X)) using the category of vector bundles (respectively, coherent
sheaves). We first study the groups G;(X), and in particular construct the
BGQ-spectral sequence
EBcodim
FJ:•q = x=p K -p-q(k(x)) ~ G -p-q (X)
using the filtration of the category of coherent sheaves given by ''codi
mension of support." We give Quillen's proof of Gersten's Conjecture
for a power series ring, and for the semi-local ring obtained from a finite
X Introduction
set of smooth points on a variety over an infinite field (Quillen proves it,
more generally, assuming only that the variety is regular over a field).
This is used to obtain Bloch's formula
Clf'(X) = lf'(X,'Xp.x)
where Clf'(X) is the Chow group of algebraic cycles of codimension p
on X modulo rational equivalence, and 'Xp.x is a sheaf for the Zariski
topology with stalks Kp(Ox,x>· This generalizes the familiar formula
Pic(X) = H1(X, O*x) for the Picard group of invertible sheaves on X.
Chapter 5 ends with the computation of K for a projective bundle, and
1
for a Severi-Brauer scheme.
Chapter 6 contains the proofs of the results stated in chap. 4. We begin
by computing K using simplicial coverings. We then prove Quillen's
0
Theorem A, which gives a criterion for a functor to induce a homotopy
equivalence on classifying spaces, and Theorem B, which identifies the
homotopy fibre of such a map on classifying spaces, in certain cases.
These proofs make use of various topological results proved (or dis
cussed) in Appendix A. Theorems A and Bare then used to give "alge
braic" proofs of the remaining results of Chapter 4 (exactness, resolu
tion, devissage, and localisation).
Chapter 7 is devoted to the proof that "+ = Q", using the notions
of monoidal categories, and actions of these on other categories. A mon
oidal category is, roughly speaking, a (small) category [:/, together with
a functor + : ~ x ~- g which is "associative" and has an "identity".
Thus the classifying space B[:f becomes an H-space ("homotopy mon
=
oid"). The basic example for us is [:! Iso(/}(R), the category whose
objects are finitely generated projective modules, and whose arrows are
all isomorphisms of projective modules; the operation + is the direct
sum. The set 7r0(B[:f) of path components is just the monoid of isomor
phism classes of projective R-modules. The construction of [:/-1[:/ yields
an H-group ("group upto homotopy") with an H-map B[:f- B[:f-1[:/,
which is "universal upto homotopy" among such maps; on 7r0, this just
yields the Grothendieck group K (R). One first shows that there is a ho
0
motopy equivalence K (R) x BGL(R)+ = BS-1S, by computing the ho
0
mology of Bg>-1g>. Then, using Quillen's extension construction, one
shows that B[J-1g> is homotopy equivalent to the loop space of BQCJ}(R).
This yields the isomorphisms Tr/BGL(R)+) =:: 7r1+ 1(BQCJ}(R)), relating the
two definitions of K(R), and in particular identifying KlR), KlR) with
1
the groups of Chapter 1.
Chapter 8 gives the proof of the theorem of Mercurjev and Suslin,
relating K and the Brauer group of a field. Let F be a field containing a
2
primitive nth root of unity. Then the theorem states that the natural map
(the "Galois Symbol" or "Norm Residue homomorphism") gives an
isomorphism: