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Algebraic K-Theory Second Edition
V. Srinivas
R e p r i n t o f the 1995 Second E d i t i o n
Birkh~iuser Boston 9 Basel 9 Berlin
V. Srinivas Tara Institute of Fundamental Research School o f Mathematics H o m i B h a b h a Road, Colaba M u m b a i 4 0 0 005 India
Originally published as Volume 90 in the series P r o g r e s s in M a t h e m a t i c s
Cover design by Alex Gerasev. Mathematics Subject Classification: 18F25, 19-XX, 19Dxx, 19D55, 19E08, 19E15
Library of Congress Control Number: 2007937324 ISBN-13:978-0-8176-4736-0
e-ISBN-13:978-0-8176-4739-1
Printed on acid-free paper. 9 Birkh~iuser Boston All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (BirkMuser Boston, c/o Springer Science+Business Media LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 987654321 www. birkhause r. corn
(IBT)
V. Srinivas
Algebraic K-Theory Second Edition
Birkh~iuser Boston 9 Basel 9 Berlin
V. Srinivas School o f Mathematics Tata Institute of Fundamental Research B o m b a y , India
Library of Congress Cataloging-in-Publication Data Srinivas, V. Algebraic K-theory / V. Srinivas. -- 2nd ed. p. cm. -- (Progress in mathematics ; v. 90) Includes bibliographical references. ISBN 0-8176-3702-8 1. K-theory. I. Title. II. Series: Progress in mathematics QA612.33.$67 1993 93-9417 512'.55--dc20 CIP
Printed on acid-free paper 9 Birkhltuser Boston 1996
Birkhiiuser
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Dedicated to my parents.
Contents
Preface to t h e F i r s t E d i t i o n
...............................
Preface to t h e Second Edition 1. "Classical" K - T h e o r y
.............................
xi xvii
.....................................
Review of parts of Milnor's book: definitions of K0, K1, Ks of rings; computation of K1 of a noncommutative local ring; definition of symbols; statement of Matsumoto's theorem; examples of symbols (norm residue symbol, Galois symbol, differential symbol); presentation for K2 of a commutative local ring. 2. T h e Plus C o n s t r u c t i o n
....................................
18
The plus construction; computation that ~r2(BGL(R)+) ~- K2(R); H-space structure of BGL(R) + and products in K-theory (following Loday); statement of Quillen's theorem on Ki of a finite field. 3. T h e Classifying Space of a Small Category . . . . . . . . . . . . . . . . . . .
31
Simplicial sets; geometric realization; classifying space of a small category; elementary theorems about classifying spaces (compatibility with products, natural transformations give homotopies, adjoint functors give homotopy inverses, filtering categories are contractible); example of the classifying space of a discrete group as the classifying space of the category with one object, whose endomorphisms equal the group. 4. E x a c t C a t e g o r i e s a n d Quillen's Q - C o n s t r u c t i o n
...............
Exact categories; admissible mono- and epimorphisms; definition of QC for a small exact category C; definition of K~(C) for a small exact category C; statements of theorems about Ki (Ko agrees with that defined "classically", theorem on exact sequences of functors, resolution theorem, d~vissage theorem, localization theorem); "bare hands" construction of a homomorphism Ko(C) ~ ~rl BQC).
38
viii
Algebraic K-Theory
5. T h e K - T h e o r y of Rings and Schemes
.......................
46
Statement of the theorem comparing the definitions of Ki of a ring using the plus and Q constructions; definition of G~(A) as K~ of finitely generated A-modules, for Noetherian rings A; computations of G~(A[t]), G,(A[t,t-1]) for Noetherian A, and hence K~(A[t]), Ki(A[t,t-1]) for Noetherian regular A; definition of Ki(X), Gi(X) for schemes, using vector bundles and coherent sheaves, respectively; construction of direct image and inverse image maps for Ki and Gi of Noetherian schemes for morphisms satisfying appropriate conditions; action of K0 on Ki, Gi and projection formulas; Ki, Gi commute with filtered direct limits; localization for Gi of a closed subscheme and the open complement; Mayer-Vietoris for Gi; Gi of affine and projective space bundles; filtration by codimension of support and the BGQ spectral sequence; Gersten's conjecture for power series rings, and semilocal rings of finite sets of points on a smooth variety over an infinite field; Bloch's formula; Ki of projective bundles, of p l over a noncommutative ring, and of Severi-Brauer schemes. 6. P r o o f s of the T h e o r e m s of C h a p t e r 4 . . . . . . . . . . . . . . . . . . . . . . . .
89
Proofs of the following theorems: 7r~BQC) ~- Ko(C); Theorems A and B of Quillen; the theorem on exact sequences of functors; the resolution theorem; the d6vissage theorem; the localization theorem. 7. C o m p a r i s o n of t h e Plus and Q - C o n s t r u c t i o n s
................
126
Monoidal categories; localization of the action of a monoidal category on a small category; computation of the homology of the classifying space of a localized category; the S - 1 S construction, viewed as a "functoriar' version of the plus construction; construction of the homotopy equivalence 8 - 1 3 ---, 12BQC for any exact category C in which all exact sequences are split, where 3 is the category of isomorphisms in C; corollary that the plus and Q-constructions yield the same K-groups for projective modules over a ring. 8. T h e M e r k u r j e v - S u s l i n T h e o r e m
............................
145
The Galois symbol; statement of the Merkurjev-Suslin theorem; Hilbert's Theorem 90 for K2; proof of the Merkurjev-Suslin theorem; torsion in K2; torsion in CH 2. 9. Localization for Singular Varieties
..........................
Quillen's localization theorem for the complement of an effective Cartier divisor in a quasi-projective scheme with affine complement; discussion of naturality of this sequence (after Swan); proof of the "Fundamental Theorem" on Ki of polynomial and Laurent polynomial rings; Levine's localization theorem; computation of K0 of the category of modules of finite length and finite projective dimension over the local ring of a normal surface singularity, in terms of Hi(K:2) of the resolution; computation of this Ko for quotient singularities; Chow groups of surfaces with quotient singularities.
194
Contents
ix
230 Appendix A. Results from Topology . . . . . . . . . . . . . . . . . . . . . . . . . . (A.1) Compactly generated spaces; (A.2)-(A.6) Homotopy groups, Hurewicz theorems; (A.7) Products; (A.8)-(A.12) GW-complexes, Whitehead theorem, Milnor's theorem on the homotopy type of mapping spaces, comparison of singular and cellular homology and cohomology; (A.13)-(A.15) Local coefficients, homology and cohomology with local coefficients for CW-complexes via cellular chains; (A.16) Obstruction theory for maps and homotopies between CW-complexes (which may not be simply connected); (A.17)-(A.22) Fibrations, the homotopy lifting property, long exact homotopy sequence, fiber homotopy equivalence, fibrations over a contractible base are fiber homotopy equivalent to a product, local coefficient systems of the homology and cohomology groups of the fibers of a fibration; (A.23)-(A.26) Leray-Serre spectral sequence for homology and cohomology of a fibration over a CW-complex; (A.27) Homotopy fibers; (A.28) Spectral sequences for the homology and cohomology of a covering space; (A.29)-(A.35) Quasi fibrations (some results of Dold and Thorn); (A.36)-(A.42) NDR-pairs and cofibrations (following Steenrod); (A.43)-(A.47) H-spaces; (A.48)(A.50) Covering spaces of simplicial sets; (A.51)-(A.54) nurewicz and Whitehead theorems for non-simply connected H-spaces; (A.55) Milnor's theorem on the geometric realization of a product of simplicial sets. Appendix B. Results from Category Theory . . . . . . . . . . . . . . . . . . . .
276
Small categories; equivalences; Abelian categories; construction of the quotient of a small Abelian category by a Serre subcategory; examples of quotients; adjoint functors; filtering categories and direct limits. Appendix C. Exact Couples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 The spectral sequence of an exact couple; bigraded couples; elementary discussion of convergence; the BGQ spectral sequence; the spectral sequence of a filtered complex. Appendix D. Results from Algebraic Geometry . . . . . . . . . . . . . . . . . 295 (D.1)-(D.14) Sheaves; (D.15)-(D.20) Schemes; (D.21)-(D.41) Some properties of schemes; (D.42)-(D.59) Coherent and quasi-coherent sheaves; (D.60)-(D.66) Cohomology and direct images of quasi-coherent and coherent sheaves; (D.67)-(D.70) Some miscellaneous topics. Bibliography
...............................................
339
Preface to the First Edition 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 detail the results of Quillen, contained in the following basic papers: 1. D. Quillen: Higher Algebraic K-Theory I, Lecture Notes in Math. No. 341, Springer-Verlag, New York (1973). 2. D. Grayson: Higher Algebraic K-Theory II (after Daniel Quillen), Lecture 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 connections with algebra, algebraic geometry, topology, and number theory. Some recent interesting results in algebra and related fields proved using Ktheoretic methods are the following: (i) Merkurjev and Suslin's theorem on Brauer groups of fields, its generalizations due to Merkurjev and Suslin, Levine, and the work of Rost and others on Milnor's conjecture relating K2 and Witt 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 theory, as developed in Fulton's book). (iii) Levine's computation in terms of KI, for the Grothendieck group of modules of finite length and finite projective dimension over the local
xii
Algebraic K-Theory 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 negative intersection multiplicity. Levine's results also have applications to Chow groups of singular varieties. A generalization 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 Waldhausen's algebraic K-theory of spaces, which is a key ingredient in the recently announced proof by Hsiang and Madsen of the Novikov conjecture on the surgery obstruction map; the higher dimensional generalization of class field theory due to Bloch, Kato, Saito, and others; work of Bloch, Colliot-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 algebraic geometers, I should also mention Beilinson's generalization of the Hodge conjectures, which relate certain groups of transcendental cohomology 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 may prefer, at a first reading, to read Chapters 1, 3, 4, and 5, skip the more topological Chapters 2, 6, and 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 "topological" 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 K0, and to state where we have omitted topics from Quillen's papers). Chapter 1 contains a quick review of "classical" K-theory (i.e., K0, K1, K2), mainly based on Milnor's book. In Chapter 2, Quillen's plus construction is given, leading to the definition K~(R) - 7h(BGL(R) +) for i >_ 1, for the higher K-groups of a ring. We construct products Ki(R) | Kj(R) --, K~+j(R), following Loday. We also show that K1, K2 agree with the "classical" K1, K2 of Chapter 1. Chapters 3-7 contain our exposition of "Higher Algebraic K-Theory I", and the comparison of the plus and Q constructions for Ki. Chapter 3
Preface to the First Edition
xiii
introduces the language of simplicial sets, and leads to the basic notion of the classifying space BC of a small category C. This leads to a "dictionary" between category theory and topology, under which we have the following correspondences: small category functor natural transformation adjoint functors category with an initial or final object
; topological space (CW complex) ; cellular map , homotopy , homotopy inverse pair of maps ~ contractible space
In Chapter 4, we give Quillen's Q construction, using the K-groups of a small exact category C in terms of the homotopy groups of the classifying space of its associated Quillen category QC, Ki(C) = ~r,+l (BQC). We then state a number of purely "K-theoretic" results, contained in the first part of "Algebraic K-Theory r' (the computation of K0, and the theorems on exact sequences of functors, resolution, ddvissage, and localization). Given that K0 is the same as that defined "classically", these theorems express standard facts about K0. However, the extensions to the 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 (or a Noetherian ring), let Ki(R) (or G~(R), respectively) be the K-groups of the category of finitely generated projective R-modules (or of all finitely generated R-modules, respectively). We first prove the formulas G,(R[t]) ~- G,(R), G,(R[t,t-1]) ~- G,(R) $ G,_I(R). (Quillen deduces these formulas from more general results about filtered rings, which we omit; we also omit the applications to computing Ki for certain division rings.) For a (Soetherian separated) scheme X, define K~(X) (or G,(X), respectively) using the category of vector bundles (or coherent sheaves, respectively). We first study the groups Gi(X), and in particular construct the BGQ-spectral sequence
E[ 'q = (~codimx=p K_p_q(k(x)) ~
C_p_q(X)
using the filtration of the category of coherent sheaves given by "codimension 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 set of
xiv
Algebraic K-Theory
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
CHv(X) = Hv(X, lCv,x) where GHv(X) is the Chow group of Mgebraic cycles of codimension p on X modulo rational equivalence, and/Cp,x is a sheaf for the Zariski topology with stalks Kv(O~,x). This generalizes the familiar formula Pic(X) HI(x, 0~) for the Picard group of invertib]e sheaves on X. Chapter 5 ends with the computation of Ki for a projective bundle, and for a SeveriBrauer scheme. Chapter 6 contains the proofs of the results stated in Chapter 4. We begin by computing K0 using simplicia] coverings. We then prove Quillen's Theorem A, which gives a criterion for a functor to induce a homotopy equivalence on classifying spaces, and Theorem B, which identifies the homotopy fiber of such a map on classifying spaces, in certain cases. These proofs make use of various topological results proved (or discussed) in Appendix A. Theorems A and B are then used to give "algebraic" proofs of the remaining results of Chapter 4 (exactness, resolution, d6vissage, and localization). Chapter 7 is devoted to the proof that "+ -- Q", using the notions of monoida] categories, and actions of these on other categories. A monoidal category is, roughly speaking, a (small) category S, together with a functor + : S x S ~ S which is "associative" and has an "identity". Thus the classifying space B S becomes an H-space ("homotopy monoid"). The basic example for us is ,~ = IsoP(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 ~0(BS) of path components is just the monoid of isomorphism classes of projective R-modules. The construction of S - 1 S yields an H-group ("group up to homotopy") with an H - m a p B,~ ~ B S - 1 S , which is "universal up to homotopy" among such maps; on ~0, this just yields the Grothendieck group Ko(R). One first shows that there is a homotopy equivalence Ko(R) x BGL(R) + ~- B S - I S by computing the homology of B S - t S . Then, using Quillen's extension construction, one shows that B 8 - 1 , ~ is homotopy-equivalent to the loop space of BQT'(R). This yields the isomorphisms ~i(BGL(R) +) ~- ~i+1 (BQP(R)), relating the two definitions of K~(R), and in particular identifying K1 (R), K2(R) with the groups of Chapter 1. Chapter 8 gives the proof of the theorem of Merkurjev and Suslin, relating/(2 and the Brauer group of a field. Let F be a field containing a primitive n-th root of unity. Then the theorem states that the natural
Preface to the First Edition
xv
map (the "Galois Symbol" or "Norm Residue homomorphism") gives an isomorphism: K2(F) | Z / n Z ~- , S r ( F ) . We give the proof of this theorem in detail, based on an expository article by Suslin. We omit only one step in the proof r a t h e argument (see Prop. (8.7)(c)) using Gillet's Riemann-Roch theorem for the vanishing of certain differentials in the BGQ spectral sequence, up to torsion; the proof of the Riemann-Roch theorem involves tools from K-theory and topology beyond the scope of these notes (e.g., the homotopy theory of simplicial sheaves). We then prove the relevant easy case of the results of Bloch and Ogus, and deduce the result that if X is a smooth variety over an algebraically closed field, then the n-torsion subgroup nCH2(X) of the Chow group of codimension-2 cycles on X is finite. We also prove Roitman's theorems on torsion-zero cycles. Chapter 9 begins with Quillen's localization theorem for singular varieties, contained in the latter half of "Higher Algebraic K-Theory II", leading to the Fundamental Theorem (computation of K~(R[t]), K~(R[t,t-~])). Next, we give a generalization of Quillen's localization theorem due to Levine, and use it to obtain a presentation for Ko(C.n), where Cn is the category of R-modules of finite length and finite projective dimension over the local ring R of a normal surface singularity. This is used to show that quotient singularities do not contribute to the Chow group of zero cycles on a normal surface. Appendix A discusses the topological results needed in the main text. We sketch proofs for standard results when these are not too long, and give references in other cases. We also give more detailed proofs for the results of Dold and Thorn on quasi fibrations, and of some results on H-spaces, and on simplicial sets, which are perhaps less standard. Appendix B discusses category theory, and in particular contains the construction of the quotient of a small Abelian category by a Serre subcategory. Appendix C deals with spectral sequences from the point of view of exact couples. Though this is standard in topology, it seems to be less familiar to algebraists. We give an ad hoc treatment of convergence that suffices for our purposes. A word about sources w Chapters 4-7, and the first half of Chapter 9, are based on the work of Quillen cited above. For the other chapters, we list a few main sources at the beginning of the chapter, and give other references in the course of the text. The absence of a specific reference, however, does not imply any claim to originality on my part; in fact, all the material covered in these notes (with the exception of parts of Chapter 9 on singular surfaces) is based on other sources.
xvi
Algebraic K-Theory
Acknowledgements I must thank the people who attended the original course of lectures, and made many comments that clarified my ideas, primarily the members of the "algebra school" at the Tata Institute. I began learning algebraic geometry and K-theory when I was a graduate student at Chicago, from Bloch, Murthy, and Swan; at the Tata Institute, I have also learned a lot from Mohan Kumar and Madhav Nori; to all these people, I owe a considerable debt. I received generous help from several people on topics connected with these notes: Balwant Singh, Bhatwadekar, Coombes, Dalawat, Esnault, Lemaire, Levine, Parimala, Paranjape, Pati, Raghunathan, Ramanan, Roy, Simha, Soule, Sridharan, Stienstra, Suslin, and Vaserstein. Thanks are also due to M.K. Priyan for the major effort of typing the first version of the notes, and to K.P. Shivaraman for help with the final version.
Preface to the Second Edition Apart from the improved typescript, now rendered in TEX, there are two main changes in the second edition. Several people with more of a background in topology, but who were unfamiliar with algebraic geometry, commented to me that the first edition of the book was difficult for them to follow. In response to these comments, I have added Appendix D, entitled "Results from Algebraic Geometry," which contains all the definitions and results (with sketches of proofs, and/or suitable references) needed to understand Chapter 5. I have also added suitable cross-references to this appendix throughout Chapter 5. The reader who masters this material ought to be able to also read Chapters 8 and 9, consulting the references given there. Secondly, I have rewritten major parts of Chapter 8, based on a later proof of the Merkurjev-Suslin theorem given by Merkurjev in Uontemp. Math. 55, Part II, A.M.S. (1986). This proof is more elementary than the one given in the first edition of this book, in that higher K-theory and K-cohomology are needed only to prove Hilbert's Theorem 90 for Kz. The treatment of the Merkurjev-Suslin theorem now given is thus self-contained (at least with respect to K-theory), as we do not need to invoke the theory of Chern classes or the Riemann-Roch theorem for higher K-groups. A reader interested in the connections with Cyclic Homology (or "additive K-theory") can consult the book J.-L. Lodny, Cyclic Homology,Grundlehren Math. 301, SpringerVerlag (1992). The reader would also do well to consult the book J. Rosenberg, Algebraic K-Theory and Its Applications, Graduate Texts No. 147, Springer-Verlag (1994). His treatment of K-theory assumes less background of the reader than mine, but covers different ground. The book also gives some historical background, discusses the connections with number theory, C*-algebras, etc. and contains a more complete list of references. Thanks are due to Birkh~iuser-Boston, and in particular Ann Kostant, for the job of rendering the manuscript into TF_/X, to obtain the muchimproved result now before the reader. V. Srinivas Bombay, 1995
1. "Classical"
K-Theory
The main reference used here is Milnor's book: Introduction to Algebraic K-Theory, Annals of Mathematical Studies, No. 72, Princeton University Press (1971). Let R be an associative ring (with 1), and let 7~(R) denote the category of finitely generated projective R-modules. We define the Grothendieck group Ko(R) to be the quotient
go(R) = y / n , ~" = free Abelian group on the isomorphism classes of projective modules in P(R), = subgroup generated by elements [P (9 Q ] - [ P ] - [Q], for all P, Q E P(R). Thus, for any P, Q E P(R), [P] = [Q] in Ko(R) ~ P (9 P' ~- Q (9 P' for s o m e P ' E 79(R) ~ P ( g R = ~ Q ( g R n for s o m e n >_ 0. Indeed, if [p] = [Q] in K 0 ( R ) , then we have a relation in jr of the form T
8
i=1
j=l
[P]- [Q] = ~([P/(9 Q/]- [P/]- [Q/I)- Z([P~ (9 Q~]- [P~I- [Q~]). Hence T
[Q] + ~ [ e ,
8
(9 Q,] + ~ ( [ P ; ] + [Q;]) =
i=1
[P]
j-----1
T
8
+ Z ( [ P , ] + [Q,]) + ~ [ P ; i=1
(9 Q;]
j----1
in ~ , the free Abelian group on isomorphism classes in 7)(R). Hence the terms on the right must be a permutation of the set of terms on the left. In particular, if we let
P'=
P, (gQ,)
(9
(gQ~)
then P (9 P ' ~ Q (9 P'. Thus, we have shown that [P] = [Q] in Ko(R) --> P $ P' ~ Q (9 P ' for some P ' E P(R). The converse is obvious. Further, we can find Q' E 7~(R) such that P' (9 Q' ~ R" for some n, since P ' is a
2
Algebraic K-Theory
quotient of some R n ( P ' is finitely generated) and p t is projective. Hence p $ p ' ~- Q ~ P' ~ P @ Rn ~ Q (g R n. If f : R -4 S is a homomorphism of rings, f induces a functor P ( R ) --. P ( S ) given by P : ; S | P. This preserves direct sums, and hence induces a homomorphism f . : K0 (R) --, K0 (S). E x a m p l e (1.1). Let (R,j~4) be a local ring, i.e., R is a possibly noncommutative ring (with 1), .h4 C R is a 2-sided maximal ideal, and R - . h 4 = R*, the group of units. Then Ko(R) = Z, with a generator given by the class of the free R-module of rank 1. Indeed, there is a natural h o m o m o r p h i s m p" g o ( R )
, go(R/M)
= Z,
since R / f l 4 is a division ring, and a finitely generated projective (left) R / f l 4 - m o d u l e is a left vector space, i.e., a free R / M - m o d u l e of rank equal to t h e dimension of the vector space. T h u s p is surjective, since p([Rn]) is the cla~s of a vector space of dimension n; we prove t h a t in fact every projective R-module is free, so t h a t p is an isomorphism. Let P E P ( R ) , and let d i m R / ~ P / . M P = n. Choose x l , . . . ,xn E P whose images :~I,...,:E n E P / f l 4 P give a basis for the ( R / A d R ) - v e c t o r space. We claim t h a t X l , . . . ,xn give a basis for P as a free R-module. Let Q E :P(R) such that P (9 Q ~ R ~+m is a free module, where we m u s t have m = dimR/y~Q/A/IQ. Let X n + l , . . . , x m + n E Q m a p to a basis of Q / A 4 Q over R / M R , so t h a t x l , . . . , x m + n E R m+n give a basis of ( R / f l 4 R ) m+n. We claim then t h a t x l , . . . , x m + ~ give a basis for the free R-module R re+n, which immediately implies t h a t X l , . . . ,xn E P , X ~ + l , . . . , xm+n E Q give bases, so t h a t P , Q are free R-modules of ranks n, m, respectively. Let xi = (ail,...,ai,m+n) E R m+n. It suffices to prove t h a t A = [aij] e GLm+n(R), i.e., A has a 2-sided inverse. If a q : ; ~q e R / . M , and ft. = [5ij], then A E GLm+n(R/.h4), so there exists B E U m + n ( R ) , with B : ; B E GLm+n(R/A4), and A B = B A = Ira+n, where Im+n is the identity matrix. Then A B - B A - Im+,,(modA4), and so A B = [c~j] with c~i E R*, c~i E f14 for i ~ j. Hence, there exists an elementary m a t r i x E with A B E = d i a g ( c l , . . . ,cm+n) for suitable c l , . . . , c ~ + n E R*, i.e., A B can be diagonalized by right column operations, involving adding a right multiple of a column to another c o l u m n - - w e m a y first add a suitable multiple of the first column to each of the other columns to make all offdiagonal entries on the first row vanish; this does not alter the condition t h a t other off diagonal entries lie in r and diagonal entries are units, since R is local; now perform a similar reduction of the second row, etc.
1. "Classical" K-Theory
3
Since d i a g ( c l , . . . , am+n) is invertible, A has a right inverse. By a similar argument using row operations on B A, we see that A has a left inverse. Since matrix multiplication is associative, A is invertible. E x a m p l e (1.2). Let R be a Dedekind domain, i.e., R is a commutative Noetherian integrally closed domain such that every non-zero prime ideal of R is maximal. Then Ko(R) ~ Z (9 Ct(R) where Ct(R) is the ideal class group of R, defined to be the group of isomorphism classes of invertible ideals (with tensor product as the group operation--see Milnor's book for details). D e f i n i t i o n of K1. Let R be an associative ring (with 1), GLn(R) the group of invertible matrices of size n over R; let En(R) be the subgroup of elementary matrices, defined to be the group generated by the matrices e~. ) (A) , 1 < _(n) (A) is the unipotent matrix whose - i # j < - n, A e R, where ~ij only non-trivial off-diagonal entry is A in the (i,j)th position. Thus, if i < j , then ~ij -(") (A) has the form j t h column
!
"1
" " ' ~
0
0 0
0
, ith row.
...1...
~ . . .
0
0
......
0
1
Let GLn(R) "--. GLn+I(R) by A:
,[Ao
0]1
and let GL(R) = lim GLn(R). Similarly, let E(R) = lim En(R). Since ; _(n+l) e~)(A) : vq (A) under En(R) "-, En+I(R), we obtain matrices eq(A)
E E(R) as the common image of all the e~)(A) for n >_ i, j, and E ( R ) is the subgroup of GL(R) generated by the eq(A). The eq(A) satisfy the following identities:
(1.3) b) legaCY), ktCt,)l = 1 for j # k, i # t
4
Algebraic K-Theory
These identities are deduced from similar identities for the e~)(A). We deduce immediately that En (R) is perfect for n > 3 (i.e., [En (R), En (R)] En(R)) and E(R) is perfect. L e m m a (1.4) (Whitehead). For any A e GLn(R),
[a 0] 0 A-'
Proof.
e E2n(R).
For any B E Mn(R), it is easy to see that
[~ ,] [,. 0] o
I,,'
B
I,~
lie in E2n(R). Now use the identity
[0~ 0] [ ,. A -~
=
0][,.,~][ ,. 0][,. All
A-I-I,~
In
0
In
A-In
In
0
I,~
"
P r o p o s i t i o n (1.5). E(R) = [E(R),E(R)] = [GL(R), GL(R)]. P r o o L We have already noted that E ( R ) is perfect, so it suffices to note that for any A , B e GLn(R),
[A,AI~I 0] [A~ 0 ][Ai 0][~1 0] 0
In
=
0
(AB) -1
0
A
0
B
and the right side lies in E~n(R) by Lemma (1.4). Definition.
K1 (R) -- GL(R)/[GL(R), GL(R)] -- G L ( R ) / E ( R ) - H I ( G L ( R ) , Z ) (the definitions are equivalent by Proposition (1.5), and the isomorphism
gl (G, Z) ~ G a b - G/[G, G] for any G). E x a m p l e (1.6). Let (R,.h4) be a (possibly non-commutative) local ring. Then the natural map GLI(R) -~ K1 (R) induces an isomorphism
R*/[R*, R*] ~ K, (R). Indeed, let
~. = (R*) -b If A = [ao] E GL,,(R), then by adding another column to the first one if necessary, we may assume all E R* (any row or column must have an entry in R*, since A = [hij] e G L n ( R / . M ) has a non-zero entry in every row and column). Adding multiples of the first column to the others, we
1. "Classical" K-Theory
5
can make a li = 0 for all i > 1. Similarly, we work on the second row to make a22 E R*, a2i = 0 for i # 2, etc. Thus, after column operations, given by right multiplication by an element of E n ( R ) , we can make A diagonal without changing its image in K1 (R). Now by the Whitehead lemma (1.4), we deduce t h a t the image of A in K1 (R) lies in the image of R* = GL1 (R). Hence R* --~ g l (R). Since g l (R) is Abelian, we have an induced surjection n* = n ' / [ n * , n * ]
.
(n).
Next, if x, y E R such t h a t 1 - x y E R*, we claim t h a t 1 - y x E R* also, and (1 - xy)(1 - y x ) -1 E [R*, R*]. If 1 - xy, y e R* then
with yl E [R*,R*] proving our claim in this case. A similar a r g u m e n t works if x E R*. So we may assume x, y E j~4, in which case clearly 1 - y x , 1 - x y E R*. If xl = l + x - xy, then x l , 1 - y x l , 1 - x l y E R*, and we compute t h a t (1-xy)(1-y)=l-y-xy+xy (1 -
y
)(1 -
y) = 1 - y-
and
2=l-xly, +
= 1 -
Hence (1 - x y ) ( 1 - y x ) - 1 - (1 - x l y ) ( 1
-
y X l ) - 1 E [R*, R*].
as claimed. (I learned this argument from L. Vaserstein, who has used analogous arguments to compute K1 for most semi-local rings.) We now show t h a t there is a well-defined determinant homomorphism det " G L ( R ) ~ R*, satisfying (i) d e t ( A B ) = det A . det B (ii) det A = 1 for all A e E ( R ) (iii) the composite R* = G L l ( R ) tient map.
, GL(R)
det
~. is the natural quo-
Our construction of 'det' follows the treatment given in Artin's book Geometric Algebra of the determinant over division rings. It suffices to construct a compatible family of maps detn 9 G L , ( R ) --. R* such t h a t m
1),~ if A E G L n ( R ) , and A r is obtained from A by multiplying a column on the right by ~ E R*, then detn A t = ~ - d e t n A, where # ; ; ~ E R* m
2),~ if A E G L n ( R ) and A ~ is obtained from A by adding a right multiple of a column to another column, then detn A ~ = detn A 3),, if In E G L n ( R ) is the identity matrix, detn In = 1, the image of 1 E R* in R*. m
6
Algebraic K-Theory
We prove by induction on n that detn exists, and observe at once that detn, if it exists, is characterized by the above properties, since any A E GLn(R) can be transformed by operations as in 2)n to a matrix a
0 1 1
,
~
0
a E R*,
1
so that detn A -- 5 E R*, by 1)n and 3)n. Further, if detn exists, it must satisfy: a) detnIGL._l(R) = detn-1 b) detn(AB) = detn(A)" detn(B) c) if A' is obtained from A by interchanging 2 columns, detn A ~ = (---1) 9 detn A. We establish b), and leave a), c) to the reader (see Artin's book for details)" write B = diag(b, 1 , . . . , 1). B' with b E R*, B' E En(R). Then detn B = b by 2)n. Similarly
detn(AB) = detn(A, diag(b, 1 , . . . , 1)) (by 2)n) =b.det, A (byl)n) = detnA- detnB, since R* is commutative 9 P r o o f T h a t detn Exists. Clearly detl exists and is the natural quotient R* -+ R*. Assume n > 1, and by induction, detn_ 1 exists, and hence is well determined and satisfies a), b), c) above. Let A E GLn(R) have columns A t , . . . ,An, so that A = [A1,... ,An]. Then the columns Ai give a basis for the free right R-module R n of columns of length n, so that there is a unique 1 0 linear combination ~ AiAi = . , the first standard basis vector. Clearly 0
atleastoneA,
E R* 9 W r i t e A , = [ aBi li],B,
E R n-1 , ati E R, so that
aliA~ = 1 and ~ BiA~ = 0. For any i, let C~ = [B1,... ,B~-I, BI++,...] be the matrix with columns B1,..., Bi_l, B i + l , . . . , Bn. If A~ E R*, then 1 0 clearly A 1 , . . . , Ai_l, A i + t , . . . , An, 9 form another basis for the free 0
right module of columns R ~, so that B 1 , . . . , Bi-1, B i + l , . . . , Bn form a
1. "Classical" K-Theory
7
basis for the free right module of columns R n-1. Hence A~ E R* ~
Ci E
GLn_,(R). If Ai, Aj E R* with i < j, then by 2)n-i we get det~_, [B,,..., B~_I, B~A~,..., Bj_,, Bj+,,..., B~]
= detn_I[B,,...,B~_,,-BjAj,B~+I,...,Bj_,,Bj+I,...,B~], where both matrices lie in GL,~_,(R) since Ai, Aj e R*, and Ci, Cj e GL,~_I(R). Thus we have (the o verbar denotes the image in R*) d e t n - l C j = (----T)J-i-lA~-l(-Aj)detn-1 (Ci)
i.e., (_-_---T)i + 1L - 1detn_ 1(Ci) = (---T)~+ 1~j - 1detn_ 1(C~), if AiA/ E R*. Define detn A = (-'ZT)i+IA~-I det,,_l(Ci) for any i such that Ai E R*; this is well defined by the above remarks. We have to verify 1)n, 2)n, 3),, of which 3)n is obvious. 1)n" Suppose Ai is replaced by Aip, for some/~ E R*. Then Ai is replaced by/~-1Ai, and the remaining Aj are unchanged. If Ai E R*, Ci is unchanged, and #-1)ki E .R*, SO that detnA' = ( - ' ~ ) i + I ( # - I A i ) - I
detn-1 (Ci)
= p detnA. If A1 E R* with j r i, a column of Cj is multiplied by/~ to get C~, so t h a t d e t , A' = (----1)i+1 ~ 1 . detn-1 (C~) ---- ( - - T ) i + l , ~ , j l ( / ~ ,
9 det,_l(Ci))
= ~ . detnA.
2)n: Suppose Ai is replaced by A~ + Ai# for some # in R. Then Aj is replaced by Aj - pAi, while Ak is unchanged for k ~ j, since (A~ + Aj/~)A~ + Aj(A~ -/~A~) = A~A~ + AjAj. If some Ak E R*, k ~ i,j then C~ is obtained from Ck by a column operation, and so detn A' = (---T)k+IA~ 1 detn C~ = detn A. If Ai E R*, (7' = Ci and A' = A~, so that detn A' = det~ A. Finally, if Aj e R* and Ak E A4 for k ~- j, then we compute that if i < j (the computations when i > j are similar), C~ = [B1,... ,B~_I,B~ + Bjp, B~+~,...,Bj_~,B.~+I,...] = [B~, ..., Bi_l, B,(I
- ~
-
AiA;1]~)
BkAkA-~'p,B,+,,...,Bj_,,Bj+,,...]
[B~,...,B~_~,B,(1 - A~j-~#), B~+~,... ,B.i_I,Bj+~,...],
8
Algebraic K-Theory
where + ; denotes the result of a column operation. This last matrix is obtained from Cj by multiplying the ith column of the right by 1 - AiAf 1# E R* (since Ai E r Hence detnA' = (---T)J+I(Aj - # ) q ) - l ( 1 - AiA~x~u) 9detnCj = (Aj - #Ai) - 1 . (1 - AiA~l/z) 9 A j - d e t n A . So we must show t h a t
Aj - #Ai = A = Aj (1 - AiA~-1) in/~* r
(I - #uAiAj -I) = (I - AiA~-I#)
since
r
(1 - # . (AiAj-1)) = (1 - (A,Aj-X)/z)
--is
a homomorphism
which has the form (1 x y ) = (1 - yx), with x = / z , y = A+A~-x ~ A4. But we have seen that this relation holds in R*. This completes the proof t h a t detn has all the required properties. C o r o l l a r y (1.7) (Dieudonnd). If D is a division ring, then K I ( D ) D*/[D*,D*], induced by the Dieudonnd determinant G L ( D ) ~ (D*) ab. E x a m p l e (1.8). Let R be a commutative ring. The determinant gives a surjection G L ( R ) --+ R* split by GL1 (R) ~ G L ( R ) . Let S L ( R ) C G L ( R ) be the group of matrices with determinant 1. Then [ S L ( R ) , S L ( R ) ] = E ( R ) . Let S K i ( R ) = S L ( R ) / [ S L ( R ) , SL(R)] = S L ( R ) / E ( R ) . Then we have a natural split exact sequence
O:
; SKI(R) ~
KI (R)
, R*
~0.
E x a m p l e (1.9) (Mennicke symbol). Let R be a commutative ring, a, b ~ R such that a R + bR = R. Choose c, d E R so that a d - be = 1, and define the Mennicke symbol [a, b] e SK1 ( R)
tobethectassof[a c
db] i n S K l ( R ) .
Then
(i) [a, b] is well defined. (ii) [a, b] = [b, a]; if a e R*, [a, b] = 1 for all b e R. (iii) [ala2, b] = lax, b][a2, b] if a,a2R + bR = R. (iv) [a, b] = [a + Ab, b] for all A e n. (v) If R is Noetherian of Krull dimension < 1, then the Mennicke symbols generate S K i ( R ) (see Bass: Algebraic K-Theory).
1. "Classical" K-Theory
9
(vi) If R is Noetherian of Krull dimension < 1 with finite residue fields at all maximal ideals, then SK1 (R) is torsion. (vii) If R is a Euctidean domain
(e.g., Z, Z[i], k[t] where k is a field) then
SKI(R) -- O. (viii) If R is the ring of algebraic integers in an algebraic number field (a finite algebraic extension of Q) then SK~(R) = 0 (see Milnor's book, Ch. 16). D e f i n i t i o n o f K 2 . Let R be a ring. The nth Steinberg group Stn(R) is defined to be the quotient of the free group on symbols ~ _(n) (~) for 1 < i,j < n, i ~ j, and for all A E R, modulo the normal subgroup generated by the words:
(i)
- ( " ) ( # ) . x~; ) (A + #) - ' for all i, j, for all A, # e R x,j
(ii) [x~. ) (A), x(~ ) (#)1 for i ~/~, k ~ j, for all A,/z e R (iii) [x~. ) (A), ~jk "(n) (#)]" ~ik ,,.(n) (,X#)- 1 for i r k, for all X, # E R. From (1.3) we have a natural surjection Cn : Stn(R) --, En(R), given by Cn(x~-J(X)) = e~)(A). We also have natural homomorphisms Stn(R) --' Stn+t(R) (which need not be injective), and so we obtain the infinite Steinberg group St(R) = aim Stn(R), and the surjection r
St(R)
} E(R).
D e f i n i t i o n . K2(R) - ker r Let xij(X) E St(R) be the common image of x_(n) ij (X), for any n > i, j, where i r j, and X E R. L e m m a (1.9). K2(R) /s the center of St(R). P r o o f . It is easy to see that the center of E(R) is trivial, so that the center of St(R) lies in K2(R). Hence it suffices to prove t h a t K2(R) is central in
St(R).
Suppose a e K2(R)f3 image(Stn_,(R) --, St(R)), i.e., a can be expressed as a word in the xo(A ) with i , j < n. Let Pn be the subgroup of St(R) generated by all the xin(#), 1 <_ i < n - 1, # E R. Then from the relation (ii) in Stm(R), m > n, we deduce that 1), is commutative. From (i), x~i(0 ) = 1, so t h a t each element of Pn has a unique expression in the form Zln(]-tl) ...Xn-l,n(l~n-1),~l,...,l~n-I
E R,
10
Algebraic K-Theory
SLn(R) C GL(R)
and r is a n i s o m o r p h i s m o n t o t h e g r o u p of m a t r i c e s in of t h e f o r m 1 0 --- 0 /zl 1
--- 0 #2 9 -. 1 ~tn-1 9 -- 0 1 N o w we c o m p u t e t h a t if i, j < n t h e n 9
"
ifj#k
=
ifj =k
so t h a t t h e given class a e K2(R) n o r m a l i z e s Pn. Since CIR. is injective, while r = 1, c~ centralizes Pn, i.e., [cr, xm(A)] = 0 for all 1 < i < n - 1, A E R. B y a similar a r g u m e n t we see t h a t [c~, xnj(A)] = 0 for all 1 < j < n - 1, A e R. H e n c e a c o m m u t e s w i t h [xjn(A),xn,(~)] = xji(A~t) for all A, # E R, 1 < j # i < n - 1. Since n c a n b e t a k e n t o b e a r b i t r a r i l y large, we are d o n e . T h u s
,, st(n)----, E(R)
o - - , K (n)
o
is a c e n t r a l e x t e n s i o n of E ( R ) . Definition.
If G is a g r o u p , a c e n t r a l e x t e n s i o n
(E)... is called a
0
;g
-H
universal central extension o f
(E')...
0
,K'J
;G
;0
G if for a n y o t h e r c e n t r a l e x t e n s i o n
~H'
t h e r e is a u n i q u e h o m o m o r p h i s m f : H
,G---,0, ; H ' o v e r G.
Remark. In the above situation, f l K " K ~ K' in fact d e t e r m i n e s ( E ' ) u p t o i s o m o r p h i s m , in t h e following sense. If K ' is a n y A b e l i a n g r o u p , g: K .~ K ' a n y h o m o m o r p h i s m , t h e n t h e p u s h o u t o f K
~
H
Kl yields a c e n t r a l e x t e n s i o n (E") ...
0
, K'
.- H x K K '
: G ~
0.
If g = f[K w h e r e f - H ~ H ' is as a b o v e , t h e n b y t h e universal p r o p e r t y of t h e p u s h o u t , t h e r e is a m a p H x K K ' ." H ' g i v i n g a d i a g r a m 0
;
K'
;
0
}
K'
~
II
HXKK'
H'
~
G
~
0
=~- G
~
0
II
1. "Classical" K-Theory
11
Hence ( E " ) and ( E ' ) are isomorphic central extensions of G. F r o m homological algebra, it is s t a n d a r d t h a t central extensions of G by an Abelian group K ~, up to isomorphism, are classified by elements of H Z ( G , K ' ) , where K ' is regarded as a G-module with trivial action. From t h e above remarks, if (E) is a universal central extension with kernel K , t h e n we have an isomorphism H2(G, K ' ) ~- H o m ( K , K ' ) of functors on the c a t e g o r y of Abelian groups. From the proposition below, we also have H1 (G, Z) = 0, since G has a universal central extension; hence by the universal coefficient theorem, H 2 ( G , K ') ~- Hom(H:2(G,Z),K') is an isomorphism of functors. Thus, g ~ H2(G, Z). Proposition
( 1 . 1 0 ) . (a) A central extension
(E)...
0
,K
,H
,a
,0
is universal r H is perfect (i.e., H I ( H , Z ) = H/[H,H] = O) and every central extension of H is split (i.e., H2(H, Z) = O, from the above remarks). (b) G has a universal central extension ~ G is perfect. Outline of Proof. G: (E')...
(a) (r 0
let (E') be an arbitrary central extension of ,K'
,H'
,G
,0.
T h e pull-back of ( E t) along H ~. G is a cemral extension of H , which must be trivial. Hence the projection
p2 : H ' x c H
~.H
has a section s : H ~ H ' x G H, and f = pl o s : H ~. H ' is a m a p over G. To prove t h a t f is unique, we see that if f~ is another m a p H ~H ~ over G, and x, y E H are arbitrary, then f ' ( x ) = a f ( x ) , f ' ( y ) = b f ( y ) with a, b E K ' n c e n t e r ( n ' ) . T h u s f'([x, V]) = [f'(x), f'(v)] = I f ( x ) , f(v)] = f([x, y]) since f, f ' are homomorphisms. Since H = [H, HI, f ' = f . ( ~ ) If
(E) . . .
0
,K
,n
0 ~G
-~0
is a central extension such t h a t H is not perfect, then there is a non-zero map r : H ~ A for some Abelian group A. If H ' = A x G is t h e split central extension of G by A, then f = (0, 0) and f ' = (r 9) are 2 distinct homomorphisms H ---4 H ' over G. Hence (E) cannot be universal, unless H is perfect. If (E) is universal, and if 0
~K'
; H'
~ H
~0
12
Algebraic K-Theory
is a central extension, then one shows t h a t the induced surjection H ' ~G has a central kernel; hence there is a (unique) map H ; H ' over G, which must be a section of H ' ; H (one first shows that [H', H'] is perfect, and r H; if x0 e ~ - x ( g ) , then x ~-. x o x x o 1 gives a map [H', g ' ] ----* [g', H'] over H). (b) If G has a universal central extension
(E)...
0
~.g
,H---,G---.O
then H = [ g , g ] by (a) above, so G = [G, G]. Conversely, if G = [G, G], and F , G is a surjection from a free group to G, giving a presentation 0
~.R---*F
DO,
~.G
then o
, R / t F , R] - - 4 F / [ f , R]
, G --- 0
is a central extension, and o
'
R n IF, F]
IF, R]
, [F, FI/[F, R]
, G ---, 0
is a central extension with [F, F]/[F, R] as a perfect group. One directly verifies t h a t this is a universal central extension: given any central extension
(E)...
0
~.K---~H
~G
;0,
there is a map F ; g over G (as F is free), which kills IF, R] since (E) is central; as in (a), the restriction of this map to [F, F] is independent of the choice of F , H over G. P r o p o s i t i o n (1.11). (a) S t ( R ) and S t n ( R ) , n > 3 are perfect. (b) S t ( R ) and Stn(R), n > 5 have no non-split central extensions. Corollary
(1.12). The extension 0
, K2(R)
; St(R)
, E(R)----, 0
is a universal central extension of E ( R ) . In particular, K2(R) = H 2 ( E ( R ) , Z ) . P r o o f o f (1.11). (a) is clear from the definitions of Stn(R), S t ( R ) . (b) (Outline) Let
(E)...
0
:~. g ~
g
r
~0
be a central extension. For each i ~ j and each xij(X), choose h ~ i, j, and let yl e ~b-l(xih(1)), Y2 e r Then y,j(X) -- [Yl,Y2] e
1. "Classical" K-Theory
13
r is independent of the choices of yl, y2, since (E) is a central extension. One checks that (i) yij(A) does not depend on the choice of h ~ i,j (ii) {y,j(A)} satisfy the Steinberg identities (1.3)(a), (b), (c). Thus, xij(A) 9 ; yij (A) gives a section of r St,,(R), n > 5.
A similar argument works for
P r o d u c t s . Suppose R is a commutative ring. Then the tensor product (P, Q) ---' P | Q on projective modules induces a pairing
K0(R) |
K0(R)
, Ko(R)
making Ko(R) into a commutative, associative ring, with identity element given by the class [R] of the free module of rank 1. We claim that there are natural pairings
Ko(R) |
K,(R)
, Ki(R), i = 1, 2
making K~(R), i = 1,2 into K0(R)-modules. Given a projective module P and a matrix A E GLn(R), choose a projective module Q such t h a t P ~ Q '~ R m is free, and fix such an isomorphism. Then we have an automorphism (A | 1p) $ (1R~ | 1Q) e Aut(R n |
(P 9 Q)) ~- GLmn(R).
If hp : GLn(R) ~. GL(R) is the resulting map, then it is well defined up to an inner conjugation on GL(R). If h p ' G L n ( R ) } KI(R) is induced by composing with the natural quotient map, then h p l e p 2 = hpl + hp2, so hp depends only on the class [P] e Ko(R). Thus one gets a well defined product Ko(R) | K1 (R) ~ K1 (R). Next, we note that hp(En (R)) lies in the commutator subgroup E ( R ) C GL(R), and one checks~ that the induced map En(R) ----, E(R) is well defined up to an inner conjugation of E(R). Hence hp induces a m a p (hp). : H2(En(R),Z) } H2(E(R),Z). Further, one can check (hp| -(h-pp). + (hQ).. Hence the induced map
(hp). " H2(E(R), Z) = lim H2(En(R), Z) --4
; H2(E(R), Z)
depends only on the class [P] E Ko(R). This gives a product Ko(R) |
K2(R)
, K2(R).
I To verify that hp 9 En(R) , E(R) is well defined up to an inner conjugation, we note that it factors through E,~(R); now conjugation by B E GLmn(R) agrees with conjugation by [B0 B-1 0 ] E E2m~(R) C E(R) on the subgroup
14
Algebraic K-Theory
E x a m p l e (1.13). Let R be a commutative ring, R[t, t -1] the Laurent polynomials over R. Change of rings yields a homomorphism ~ : Ko(R) go(R[t,t-1]). Since t e R[t,t-1] * C GL(R), t gives a class in g l (R[t, t - l ] ) . Composing c~ with multiplication by It] E g l (R[t, t - l ] ) yields a map r g o ( R ) -~. g l ( R [ t , t-l]). It turns out that r is always injective, and its image is (functorially) a direct summand in KI(R[t, t - ' ] ) . Next, for commutative rings R, we construct a pairing
KI (R) ~Ko(R) K1 (R)
, K2(R).
Suppose a, fl E E(R) commute; we define a 'product' a , fl e K2(R) by a , fl = [&, fl], where 5, fl e St(R) are inverse images in St(R) of a, respectively. Then a , fl does not depend on the specific lifts 5, f], and has the following properties: (a) if a l and a2 both commute with fl, then ( ~ 1 ~ 2 ) * t~ : ( ~ l * f ~ ) - ( ~ 2 * f~)
(b) for any v e E(R), r a y -1 . vflv -1 = a .
(c) ~ * f~ = - ( ~ .
~).
Now let A e GLn(R), B e GLm(R), and let In E GLn(R), I m e GLm(R) be the respective identity matrices. Then
a = (A | I m , A -1 | Im, I , | 13= (In | B, In | Im, In | B -~) give commuting elements of E3mn(R). We define
{A, B} = a , fl. One checks that this determines a well-defined skew-symmetric pairing g l (R) @ g l (R)
.~K2(R)
which is K0(R)-bilinear (see Milnor's book, Ch. 8 for details). In particular, if m = n = 1, we have a product R* |
R*
." K2(R)
[O 0 0] [b0 0]
called the Steinberg symbol, given by (a ~ b):
, {a, b} =
0
a -1
0
0
0
1
*
0
1
0
0
0
b -1
.
One knows (Lemma (9.8) of Milnor's book) that the Steinberg symbol has the property that if a E R* such that 1 - a E R*, then the Steinberg relation {a, 1 - a } = 1 is valid in K2(R); also, for any a e R*, { a , - a } = 1.
1. "Classical" K-Theory
15
Theorem
(1.14) (Matsumoto). IfF is a (commutative) field, then K2(F) has a presentation as the free AbeHan group on the symbols {a,b} with a, b E F* subject to the relations: (i)
b} =
b}
b}/o
b e F*
(ii) {a, b} = {b, a } - I for all a, b E F* (iii) {a, 1 - a} = 1 for all a ~. F* - { 1}. S y m b o l s o n a F i e l d . Let F be a field. A symbol on F with values in an Abelian group G is a map (,):F*xF*
~G
satisfying (i) (ala~, b) = (al, b) (a~, b) for all al, a~, b ~ F* (ii) (a, b) = (b, a ) - 1 for all a, b E F* (iii) (a, 1 - a) -- 1 for all a E F* - { 1}. From Matsumoto's theorem, there is a bijection between symbols on F with values in G, and homomorphisms K2(F) ~ G. There are m a n y interesting examples of symbols arising "in nature". We list some below. E x a m p l e (1.15) (Tame Symbol). Let F be a field with a discrete valuation v, whose residue field is k(v). Let U c F* be the kernel of v : F* ~ Z, and let 0 : U ". k(v)* be the quotient map. Define the tame symbol :Iv : F* x F* --~ k(v)* by the formula Tv(a, b) - r (We also denote the resulting map K2(F) , k(v)* by Tv and call it the tame symbol; we follow this practice in the other examples too.) E x a m p l e (1.16) (Galois symbol). Let F be a field, n a positive integer 1 such that -~ E F. Assume that F contains the nth roots of unity, and let E F* be a primitive nth root of unity. The nth order Galois symbol is a map F* x F* ~ nBr(F), where ~ B r ( F ) is the n-torsion subgroup of the Brauer group of similarity classes of central simple algebras over F. The map is given by (a, b)
, [D((a, b)! e B r ( F )
16
Algebraic K-Theory
where De(a, b) is the cyclic algebra of dimension n 2 over F with generators X, Y satisfying X n = a, y n = b, X Y = ~ Y X . E x a m p l e (1.17)(Differential symbol). Let F be a field, 121F = VllF/Z the F-vector space of absolute Khhler differentials of F; thus l)lF is spanned by symbols da, for a E F, subject to the relations d(ala2) = al 9da2 + a2 . dal for all al,a2 e F. Let ~2F "-- / k2F Q 1F . The differential symbol is the map F*
x
F*
.~~2~-
given by (a, b) --~ d. A ~ for all a, b e F* ~
9
E x a m p l e (1.18) (Norm-Residue symbols). Let k be a local field with __t E k, and containing the mth roots of unity. Let k' be the extension m field obtained by adjoining the ruth roots of all elements of k. By K u m m e r theory, we have an isomorphism k* |
Z / m Z ~= H o m ( G a l ( k ' / k ) , pro),
and by local class-field theory, we have an isomorphism Gal(k'/k) ~= k*/Nk,/k(k')* where Nk,/k is the norm (note that k l / k is a finite extension, since k* | Z / m Z is finite). Thus we have a pairing k* |
k* ('); #m
called the local ruth-power norm-residue symbol; one can check that it does indeed satisfy the conditions (i)-(iii) in the definition of a symbol. In fact, if k ~ C, then the m-torsion subgroup of B r ( k ) is known to be isomorphic to Z / m Z ~ pro; for a suitable choice of this isomorphism, one can identify the local norm residue symbol with the Galois Symbol (Ex. (1.16)). Now let K be a number field. For any place v of K which is either real or non-Archimedean, if re(v) is the number of roots of unity in the local field K~, let ( , )v denote the composition of K 2 ( K ) ----, K 2 ( K , ) with the local m(v)th power norm-residue symbol, taking values in the group /z. C K~, of roots of unity. If m is the order of the group #K of roots of unity in K, we have surjections #v ;/zN given by raising to the (m--~)-th k m / power. T h e o r e m (Moore Reciprocity Law). The sequence K2(K)
a ; (~ tp
#v
;#K-
;0
1. "Classical" K-Theory
17
is exact, where c~ is the s u m of the maps ( , )v (and in particular, almost all ( , )v are trivial on a given class in K2(K)), and Z is the s u m of the surjections I~v ~. I~K above. Example
( 1 . 1 9 ) . Let F be a finite field. Then K 2 ( F ) = 0.
This is easily proved using Matsumoto's theorem (1.14), and the fact t h a t the multiplicative group F* of non-zero elements of F is a cyclic group. Indeed, let u E F* be a generator. It suffices to prove t h a t {u, u} is trivial in K 2 ( F ) . If F has characteristic 2, then {u, u} = { u , - u } is trivial. So we may assume F has odd characteristic. If F has cardinality q, then u (q-1)/2 = - 1 , so {u, u (q+1)/2} = { u , - u } is trivial. Hence {u, u} has order dividing (q + 1)/2, as well as q - 1 (as u has order q - 1 in F*), i.e., {u, u} has order at most 2. Now F* - {1 } is invariant under the bijection v H 1 - v. Since F* - { 1 } has ( q - l ) / 2 non-squares, and ( q - 3 ) / 2 squares, there m u s t be a non-square v such t h a t 1 - v is also a non-square. Then v = u i, 1 - v = u j with i , j odd, so t h a t the triviality of (v, 1 - v} implies that {u, u} has odd order. Hence {u, u} is trivial. R e m a r k . M. Stein has shown t h a t if R is a commutative semi-local ring, which is generated by R* as an additive group, then K 2 ( R ) is generated by Steinberg symbols (see "Surjective stability in dimension 0 for K2 and related functors", Trans. A . M . S . 178 (1973), 165-191). W. van der Kallen has further shown t h a t the relations (i), (ii), (iii), in T h e o r e m (1.14) determine a presentation for K2(R), provided all residue fields have at least 5 elements (see "The K2 of rings with many units", Ann. Sci. Ecole Norm. Sup. 10 (1977), 473-515).
2. T h e P l u s C o n s t r u c t i o n In this section, all spaces, pairs, etc. have the homotopy type of a CW-complex. The main reference for this chapter is: J.-L. Loday, Kth~orie alg~brique et representations de groupes, Ann. Sci. Ecole Norm. Sup. 9 (1976). A general reference for topology is: G.W. Whitehead, Elements of Homotopy Theory, Grad. Texts in Math., No. 61, SpringerVerlag. These are cited below as [L] and [W], respectively. For any associative ring R, we regard GL(R) as a topological group with the discrete topology, and let BGL(R) denote the 'classifying space' of GL(R). For our purposes, it is only i m p o r t a n t to know that B G L ( R ) is an Eilenberg-MacLane space K(GL(R), 1), i.e., BGL(R) is a connected space with ~I(BGL(R)) ~- GL(R), lr,(BGL(R)) = 0 for i > 2, and t h a t these properties characterize BGL(R) up to homotopy equivalence (since we are assuming that all spaces considered here have the homotopy type of a CW-complex). We give a construction of the classifying space of a discrete group in the next chapter (Example (3.10)). Since ~t (BGL(R)) ~ GL(R), ~rl(BGL(R)) has a perfect normal subgroup E(R). We will construct below a space BGL(R) + by applying the plus construction of Quillen to the pair (BGL(R), E(R)). There is an inclusion i" BGL(R) ; BGL(R) + such t h a t
(i)
9 rl(BGL(R)) ~ ~I(BGL(R) +) is the natural quotient m a p GL(R) ~GL(R)/E(R). i.
(ii) for any local coefficient system L on BGL(R) +,
i. " H,,,(BGL(R),i*L)
~ Hn(BGL(R)+,L)
is an isomorphism for all n > 0. These properties will characterize B G L ( R ) + up to homotopy equivalence. Quillen's first definition of higher K-groups is: Ki(R) = ~,(BGL(R)+), for i>1. T h e plus construction is described in the following result. (2.1) (Quillen). Let ( X , x ) be a path connected space, N ~rl (X, x) a perfect normal subgroup. Then there exists a continuous map of pairs f " ( X , x ) ". (X+,x +) such that Theorem
2. The Plus Construction
19
(a) there is an exact sequence
0
}N
~ 1rl(X, x) I. 7rl(X+, x +)
,0
(b) for any local coe]ficient system L on X +,
I. " Hn(X, f'L)
, H,(X+,L)
is an isomorphism for any n > 0 (c) if g" (X, x)
~ (Y, y) is a continuous map such that N C ker(g. "1rl ( X , x )
; 7rl(Y, y)),
then there exists a continuous map h" ( X + , x +) to homotopy, making the diagram
} (Y, y), unique up
(Y,Y) commute. P r o o L We construct X + by attaching 2-cells and 3-cells to X, so t h a t ( X + , X ) is a CW-pair of (relative) dimension 3, and take x + = x. We construct X + in a number of steps: S t e p (i). First choose classes ea E lrt(X,x), for a running over a suitable index set ,4, such that ea generate N as a normal subgroup. For each a E A, choose a loop 7a representing e~, and attach a 2-cell a~ to X using 7a on the boundary. Let X1 be the resulting space. Then by van Kampen's theorem, Irl (X1, x) is the quotient of lrl(X, x) by the normal subgroup generated by the e~, i.e., l r l ( X l , x ) = l r l ( X , x ) / N . S t e p (ii). Let )(1 .~X1 be the universal covering, and let )~ ;X be the induced covering, so that we have a pullback diagram
-,~
1
X
'
~
.Xl
l"
X1
Then Xl is obtained from X by attaching 2-cells; since )(1 is connected, so is )(. Thus )~ " X is the covering space corresponding to the subgroup N C ~rl(X,x), and is Galois with group ~ r l ( X , x ) / Y . For each 2-cell aa of ( X 1 , X ) , ~r~(X,x) acts transitively on the 2-cells in ~r-l(aa), with the isotropy of any of these 2-cells being N. Hence H 2 ( ) f l , ) ~ ; Z ) is a free Z[lrl(X,x)/N]-module on generators [ha], where 5a is a 2-cell in l r - l ( a ~ ) .
20
Algebraic K-Theory
S t e p (iii). We have the following diagram, whose vertical maps are Hurewicz maps,
-2(2)
7r2(Xl)
--,
1
H2(X, Z)
}
1
l
H2(X,, Z)
,
~, H2(X,,X; Z)
J = ~r,(R) ~b = N "b = 7rl (X1) = 0, lr2(X1) ~ H2(X1,Z) by onto, we can thus find maps f~ 9S 2 ILl e ~(2~) m~p~ t o [a~l e H 2 ( 2 ~ , 2 ; Now HI(R,Z)
~2(21)
71"2(21,)~)
; H2(X1,7/~)
}
7rl(j3~)
'
H1(2,Z).
1
0 since N is perfect.
Next, since the Hurewicz theorem. Since j is , 2 1 such t h a t the homotopy class Z ) u n d e r t h e composite
J} U2(Xl, )~; 7/~).
Let f a 9 S 2 , X1 be the composite ~ o ]a, and let X + be the space obtained by attaching 3-cells ba to X1 using fa along the boundary, for each c~ E ,4. By construction, (X +, XI) is a relative CW-complex with only 3-cells, so t h a t r l ( X l , x ) ~ l r l ( X + , x ) ~ lr,(X,x)/N. This proves (a) of the theorem for (X,x) } (X+,x +) where x + = x. Let )( + be the universal cover of X + ; then we can extend our earlier diagram as follows"
2
, 2,
I
i
, 2+
i
X ~ XI ~ X+ The 2 squares are Cartesian (i.e., are pullbacks). T h e relative cellular chain complex (see (A.14)) for ()C+, X ) looks like:
, c3(~: + , 2 )
9"" 0
a,
c2(2+,2)
.~0 - . -
with non-zero terms only in dimensions 2 and 3. We also have isomorphisms (from the definition of the relative cellular chain complex)
C3(X§ , X) ~-C3(2§ Xl) -~-/-/3(2§ 21; ~), c~(2+,~:) ~- c2(2,,)~) u H~(2,,)~; Z), such t h a t d is identified with the boundary map in the homology sequence of the triple ()~+,)fl, )()" 9""
' H3(X1,X; Z)
}H3(X+,X;Z)
d H2()~l, X; ~)
} H 3 ( X + , ) ~ I ; Z)
}
9 ~
From the construction of the homology sequence of a triple, d = j o 0 where 0 is the boundary map 9 ..
~3(2 + z)~ ,
~H3(s + 2,;Z) ,
o H2(21,Z) }
o~
~
2. The Plus Construction
21
in the homology sequence of the pair ()~+,)~1), and j is the natural m a p H2()~1, Z) , H2(X1, )~; Z) (used in (ii) above to construct X +). Let ba be the 3-cell of X + with attaching map ],,. We have a diagram with vertical Hurewicz maps .~
;
~'3(2 + )
'
1
H3(X + , Z)
;
II'3(2"~, Xl)
I
H3(X + , X I , Z )
.~
~-2(21)
1
0 ~. H2(X,, - Z).
Then ba also determine elements of 7r3()(+, X1), which we also denote by [b~], whose images under the Hurewicz map are the class Ibm] e Ha()~ +, X1;Z) corresponding to the 3-cells b~,. If []a] e 71"2(X1) is the class determined by ]a, the attaching map of b~, then [ba] maps to []a] under the boundary m a p 7 r 3 ( 2 + , X l ) ~ 712(X1) , from the definition of this boundary map. By construction, [fa] e 71"2(X1) maps to [g~] E H2()~1, )~; Z) under the composite
~2(x1)
* H2(X1, Z) J;
H2(XI,X; Z).
We conclude that d" H3()(+,X1; Z) ~ H2(X,,)(; Z) maps [ba] to [ha]. As in (ii) above, H3()~+,)(1;Z) is a free [Th(X,x)/N]-module on the generators [ba]. Hence, C3()~+,X) and C2()~+,)~) are both free (~rl (X, x) /N)-modules on generators indexed by the same set ,4, and d puts the two sets of generators in bijection. Hence d is an isomorphism of (7rl (X, x)/N)-modules. Now let L be a local coefficient system on X +, i.e., L is an Abelian group on which Irl(X+,x +) acts. Equivalently, L is a (~l(X,x)/g)-module. The relative homology groups Hr,(X+, X; L) are computed as the homology groups of the tensor product complex
L|
C.()~+,)~) 9
But this complex is acyclic, so Hn(X+,X;L) = 0 Vn. From the exact homology sequence for (X+,X) with local coefficients in L, we see t h a t H,.,(X,i*L) ~ Hn(X+,L) Vn, where i : X -4 X +. This proves (b). The proof of (c) is by obstruction theory; we omit the details (however,
see (A. 16)). (2.2). Let X = BS~, where S~o = U , > I So is the infinite permutation group. Let Aoo c So~ be the infinite alternating group [Jn>_l An. Then Aoo ,~ Soo = lh(BS~) is a perfect normal subgroup, so we can form the space B S +, with lrl(BS +) = Z/2Z. In fact, by
Example
the theorem of Barrat, Priddy and Quillen, one has a homotopy equivalence Z x B S + "~ fl~176176176176 where S o is the 0-sphere (discrete 2 point
22
Algebraic K-Theory
space) (see S. Priddy, On ~~176176 and the infinite symmetric group, Proc. Syrup. Math. 22, A. M. S. (1971), pp. 217-220). We note that for any ( X , x ) , if ]EX is the reduced suspension and f~X the loop space of loops based at x, with natural base points, then there is a natural continuous map X ; f ~ X ; by applying this to ~ n X and applying the functor f~n, we obtain a direct system X
} ~"~"],X
} ~'~2~"]2X
}''"
} ~'~n~,,nX
}...
whose direct limit is called f~~176176176 Then we have
so that we have a direct system of groups /ri(X)
' 7ri+l()-]-X)
, 7ri+2()-]2X)
,-.-
w h o s e direct limit is the ith stable homotopy group of X.
Let lri be the ith stable homotopy group of S ~ 7r, = lim r,+k(S k) ~ 7r,(BS+), --t
i > 1.
Now Soo c GL(Z) as the group of permutation matrices, and A ~ C E(Z). Hence, by the functoriality of the plus construction, there is a map B S + ---, B G L ( Z ) +. This induces maps on homotopy groups
lr~ ~- lr~(BS+)
} 7ri(BGL(Z) +) = g i ( z ) .
For i = 1, the image of the generator of ~rl = Z / 2 Z is ( - 1 ) e K1 (Z).
Let (f(,~) } ( X , x ) be the covering of X corresponding to the subgroup N ,~ 7rl (X, x), and let ( X +, ~.+) be the universal covering of (X +, x +). Then (f(+, ~+) is the result of applying the plus construction to ()(, ~).
P r o p o s i t i o n (2.3).
P r o o f . This was built into our construction of X +. (2.4). Let fi 9 (Xi,xi) } (X + , x i+), i = 1,2 be obtained from the plus construction, for given perfect normal subgroups Ni <~ 7rl(X~,x,). Then ( f l , f 2 ) " (X, x X2, (Xl,X2)) ; ( X + x X +, (x+,x+)) is homotopy equivalent to the result of applying the plus construction to
Proposition
N1 x N2 <] 7rl(X1 • X 2 , ( X l , X 2 ) ) . P r o o f . The properties (2.1) (a), (b) are easily verified. By (c), this characterizes the homotopy type of the resulting space. We now return to the special case when X -- B G L ( R ) , N = E(R).
2. The Plus Construction
23
Let F ( R ) be the homotopy fiber of B G L ( R ) ~ BGL(R)+; since B G L ( R ) + is connected, the homotopy type of F(R) is independent of the choices of the base points (see (A.27)). P r o p o s i t i o n (2.5). (a) F(R) is acyclic, i.e., Hn(F(R),Z) = 0 for all n > O. (b) ~I(F(R)) ~ St(R), the Steinberg group.
(c)
t ,i lly dimensions > 2.
i > 2
F(R)
P r o o f . (a) If we replace BGL(R) + by its universal cover, and B G L ( R ) by the induced covering, this does not change the homotopy type of F ( f ) (see (A.27)); we again use F(R) to denote the homotopy fiber of
BGL(R)
~ B G L ( R ) +,
where B G L ( R ) + is the universal cover the induced covering of BGL(R), which ciated to the subgroup E(R) c GL(R) has the homotopy type of BE(R), the
of BGL(R) +, and B G L ( R ) is is just the covering space asso= ~I(BGL(R)) (thus B G L ( R ) Eilenberg-MacLane space with lrl(BE(R)) ~- E(R), lr~(BE(R)) = O, i ~ 1). We have a spectral sequence (see (A.27)) E2,q = Hp(BGL(R) ~ + , Hq(F(R), Z)) ~
Hp+q(BGL(R), Z)
(where the E2-term is the usual homology group with coefficients in
Hq(F(R),Z), since the local coefficient system associated to H q ( F ( R ) , Z ) is trivial on the simply connected space BGL(R)+). Further, from Proposition (2.3) and Theorem (2.1)(b), the edge homomorphisms
Hn(BG~"~L(R),Z)
:: En,~176 " E~,o = Hn(BG~L(R) +, Z)
are isomorphisms, i.e., E~, 0 = EC~ and E~,q = 0 for q ~ 0. Now suppose F(R) is not acyclic; since ~ ( R ) + is simply connected and B G L ( R ) is connected, F(R) is path connected. Thus if q is the smallest integer such t h a t I-Iq(F(R), Z) r 0, then q > 0. Then Ep,q, 2 = Ep,q, r = 0 for all p, and all q' with 0 < q' < q. Since dr : E~,q * E~-r,q+r-l, we see t h a t E2o,q ~- Eq,+q1 E 0,q q+2 ~- E 0,q' ~176and E 0,q q+2 is the cokernel of dq+l 9E q+l,0 q+l E 0,q q+l 9 B u t E ~ , 0 = E n,0, ~176so that dq+l = 0 also. Hence E~,q = E 2O,q ~ =0as seen above. But on the other hand, E,O,q 2 =
Ho(BGL(R) + Hq(F(R),Z) ~ 0
as we assumed Hq(F(R), Z) - Hq(F(R), Z) ~ 0. This contradiction proves t h a t F(R) is acyclic.
Algebraic
24
K-Theory
(b), (c)" Let G - vr,(F(R)), and consider the spectral sequence (see (A.28)) for the universal covering F ( R ) "~F(R). This has the form
E~,q = Hp(G, Hq(F(R), Z)) ~
gp+q(F(R), Z)
where by (a) we have H , ( F ( R ) , Z) = 0 Yn. Also, F ( R ) is simply connected, so HI (F(R), Z) = 0. This forces E 2 , 0 = E ~176 1,0 = 0 ,
2 - E 2,0 ~176 =0; E2,o
further E32,0 ~ E3,0 da * E03,2 t,j = E 02,2 must be an isomorphism, since the kernel and cokernel are respectively E~176= 0 and E ~176- 0 . Thus H1 (G, Z) = Ha(G, Z) -- 0, and H3(G, Z) ~ Ho(G, H2(F(R)). The homotopy exact sequence (see (A.lS)) for
F(R) ~
BG"'~L(R)
~ BG~L(R) +
together with lr,(~(R)
i >_ 2,
+) ~ 1ri(BGL(R)+),
~r~(BGL(R)) = O, i >_ 2,
and 7r, (BGL(R)) = E(R) yields isomorphisms Ir~+I(BGL(R)+) ~ 7q(F(R)), i >_ 2, and an exact sequence
0
~. 7r2(BGL(R) +)
,
7q(F(R))
(,)
II
--4
E(R)
----,
O.
G By Lemma (A.26), the conjugation action of G on itself is trivial on ker(G ~. E(R)), and G acts trivially on ~r,(F(R)), i > 2 (i.e., F ( R ) i s simple in dimensions i > 2). Thus the above exact sequence (,) is acentral extension of E(R), such that H1 (G, Z) = H2(G, Z) - 0, i.e., by Proposition (1.10) is isomorphic to the universal central extension
0
, K2(R)
~ St(R)
~.E ( R )
~ O.
In particular G ~- St(R). C o r o l l a r y (2.6). vr~(BGL(R)+) ~- Ki(R), {= 1,2, and
Ir3(BGL(R)+) ~- H3(St(R), Z).
2. The Plus Construction
25
P r o o L The only thing left to prove is the formula for ~r3(BGL(R)+). But lr3(BGL(R) +) ~- lr2(F(R)) ~ ~2(F(R)) ~- H2(F(R),Z) (by Hurewicz); we also had an isomorphism H3(G, Z) ~ Ho(G, H2(F(R))) "~ Ho(G, lr2(F(R))) "~ ~2(F(R)), since F(R) is simple in dimension 2. This corollary motivates Quillen's first definition of higher K-theory (for projective modules over a ring). Definition. Ki(R) = ~(BGL(R)+), i >_ 1. As a consequence, we obtain C o r o l l a r y (2.7). K 3 ( R ) - - H3(St(R),Z). R e m a r k . We note that there are Hurewicz homomorphisms
Ki(R)
= ~r,(BGL(R)
+)
~ H~(BGL(R)+,Z)
~ H,(BGL(R),Z).
Following Loday, we describe below a natural H-space structure on
BGL(R) +, which we then use to construct products gi(R) | Kj(R) - , gi+j(R). One other consequence of the H-space structure is that by a theorem of Milnor and Moore (see J. Milnor, J.C. Moore, On the Structure of Hopf Algebras, Ann. Math. 81 (1965), 211-264), the Hurewicz maps above are injective up to torsion, and the Q-subspace
K~(R) |
Q c H~(BGL(R),
Q) = H~(GL(R),
Q)
is identified with the subspace of primitive elements for the comultiplication of the natural Hopf algebra structure on H.(GL(R), Q) (for any connected space X, there is a comultiplication
A. : H.(X, Q)
; H.(X, Q) •
H.(X, Q)
induced by the diagonal A : X : X x X, and the Kunneth isomorphism H.(X x X, Q) ~ H.(X, Q) | H.(X, Q); the primitive elements H,(X, Q) are those elements x satisfying A(x) = x | 1 + 1 | x). This result has been used by A. Borel (Stable Real Cohomology of Arithmetic Groups, Ann. Sci. E.N.S. 7 (1974), 235-272) to compute the ranks of the higher K-groups of the ring of algebraic integers in a number field. The following discussion is based closely on ILl. Let R be a ring; for a,/3 6 GL(R) let c~ (t)~ be defined by (a~fl)ij=
{
c~ke i f i = 2 k - l , j = 2 s ~k~ i f i = 2 k , j = 2 t 0 otherwise.
26
Algebraic K-Theory
Schematically, if
9
Lemma
X
X
X
-'-
9 ""
X
X
X
'''
o
,
9
o
9
9
o
9
o
9
o
o
then
a(gf~
9 ""
=
,
0
,
0
0 , 0
x 0 x
0 * 0
x 0 x
9
o
.
.
( 2 . 8 ) . @ ' G L ( R ) • GL(R)
o o
o
o
6 GL(R).
GL(R) is a homomorphism.
P r o o f . Left as an easy exercise. Remark.
Let c~ 6 GL,.,(R), ~3 6 GLm(R),
7=
[oo] 0
/3 6 G L m + n ( R ) ,
and let a s, /3s, 78 denote their images in GL(R). T h e n a" @/3 s ~ 7 s, but the two homomorphisms G L , ( R ) x G L , , ( R ) ; GL(R), given by (~,/3) ~-* c~s @/38, and (~,/3) ~-. 7 s, are conjugate by an element of GL(R) (even by one of E(R), since the two maps factor through GLd(R), d 2 m a x ( m , n), on which conjugation by A e GLd(R) equals conjugation by
0
A -1
6 E2d(R)). In particular, for any c~ 6 GL(R), ot@o~-1 e E ( R ) .
We define a product on BGL(R), as follows. By Proposition (2.4), the natural m a p
k : B(GL(R) x GL(R)) +
; BGL(R) + x BGL(R) +
is a h o m o t o p y equivalence. Choose a h o m o t o p y inverse k -1, and let + be the composite
+" BGL(R) + x BGL(R) + k-__~ B(GL(R)
x GL(R)) + ~+~ BGL(R) +,
where we note t h a t (by L e m m a (2.8)) there is a m a p B ( G L ( R ) x GL(R)) B G L ( R ) , such that the induced map on fundamental groups carries E ( R ) x E(R) into the commutator subgroup E ( R ) C GL(R), and hence induces a map ~ + between the plus constructions.
2. The Plus Construction
27
P r o p o s i t i o n (2.9). ( B G L ( R ) +, +) is a homotopy commutative and associative, connected H-space, hence a commutative H-group. The proof will depend on a few simple lemmas, which we prove first. Let u 9N - , N be an injective self-map of the set of positive integers. Define u ~ GL(R) ~ GL(R) by
u~
=
akt 8i1
if (i, j) = (u(k),u(s (Kronecker delta) otherwise.
We call u~ a 'pseudo-conjugation' of GL(R). L e m m a (2.10). For each pseudo-conjugation u~ there is an induced map u+ 9 BGL(R) + ; B G L ( R ) + which is a homotopy equivalence. P r o o f . Since u. is a homomorphism, by an easy computation, it induces a map B G L ( R ) + ; BGL(R)+; it also induces a GL(R)/E(R)-equivariant self-map on the universal covering space B G L ( R ) + ~- B E ( R ) + (by Prop. (2.3), and the fact t h a t B E ( R ) ; B G L ( R ) is precisely the covering associated to E ( R ) C G L ( R ) = ~rl(BGL(R))). Let Ct : B E ( R ) + B E ( R ) + be this induced map; since B E ( R ) + is simply connected, if we show that it induces an isomorphism on integral homology groups, then it is a homotopy equivalence (see (A.10)). Let x E H n ( B E ( R ) , Z) be the class of the cycle r~n,(g~i)),..., g(ni)) (in the standard complex for E(R), say). Then ft.(x) is the class of the cycle En,(u.(g~ ') ...,u~ Now the g!')j range over a finite set in E ( R ) , contained in Era(R), say. The map u~ Era(R) ~. E ( R ) is equal to conjugation by some C E E ( R ) (we can take C to be an even permutation matrix), so that ft.(x) is also represented by C(F~n,(g~O,...,g(O))C-1. But inner conjugation induces the identity map on group homology, so that ft.(x) = x. L e m m a (2.11). Let M be the monoid (under composition) of injective self-maps of the set N of natural numbers. Then the Grothendieck group Ko(M) = 0 (i.e., any monoid homomorphism from M to a group is trivial). P r o o f . Suppose u E M has infinitely many fixed points; let i E M be given by i(n) = n t h fixed point of u. Then u o i = i, so t h a t the class [u] e Ko(M) is trivial. In general, for any v e M, we claim there exists u E M such that u, vu both have infinitely many fixed points. The proof of the claim is left as an exercise to the reader. C o r o l l a r y (2.12). For any u e M, u+ " B G L ( R ) + ~ B G L ( R ) + is homotopic to the identity (through maps preserving the base point).
28
Algebraic K-Theory
P r o o f . T h e assignment u ---~ [u +] gives a monoid homomorphism from M to the group of homotopy classes of base point preserving self-homotopy equivalences of B G L ( R ) +. R e m a r k . The fact that fi: B E ( R ) + , B E ( R ) + induces the identity on homology groups does not suffice to conclude t h a t it is homotopic to the identity. In fact, there exist spaces with self-maps not homotopic to the identity, but which induce the identity on homology and homotopy (where we assume the map fixes a base point, to make sense of the statement about homotopy groups). M. Lemaire has constructed such an example. D e f i n i t i o n . Let G be a group. Two homomorphisms f, g" G ". G L ( R ) are pseudo-conjugate if there exists u E M such t h a t either u. o f and g, or u. o g and f, are conjugate by an element of GL(R). C o r o l l a r y (2.13). /] f, g 9 G the induced maps f +, g+ 9 B G preserving the respective base points.
G L ( R ) are pseudo-conjugate, then B G L ( R ) + are homotopic as maps
P r o o f . For any map f : G ; GL(R), let f ~ 1 denote the map given by x ~-. f ( x ) ~ l , for any x E G. Then f + is homotopic to ( f ~ l ) +, preserving the base point, because f ~ 1 = (u0)~ o f , where u0 E M is defined by 0(i) = 2 i -
1.
Now suppose the given maps f, g satisfy g = (u~ o f ) a for ~ e G L ( R ) (where ( u ~ ~(u~ -1 for a l l x e G). If/~ = a~c~ -1 E E ( R ) , then g @ 1 = ((u~ o f ) ~ 1) ~. Thus, the induced maps B G , BGL(R) are freely homotopic, such that under a homotopy, the image of the base point of B G is a loop homotopic to [/~] E Irl(BGL(R)) ~ GL(R) (Lemma (A.50)). T h e induced maps B G ~. B G L ( R ) + are homotopic preserving the base points, since [/~] ~-. 0 in l r l ( B G L ( R ) +) (see (A.41)). P r o o f o f P r o p o s i t i o n (2.9). Let u0, v0 E M be given by Uo(i) = 2i - 1, vo(i) = 2i, for all i E N. Then (U0)o(a) = a 9 1, and (V0)o(a) = 1 ~ a for any c~ E GL(R). By construction, the map + : B G L ( R ) + x B G L ( R ) + B G L ( R ) +, when restricted to B G L ( R ) + x {,}, is homotopic to (u0) +, and when restricted to {,} x B G L ( R ) +, is homotopic to (v0) +. Hence the base point * E B G L ( R ) + is a 2-sided identity, up to homotopy, for the operation +. By definition, this makes ( B G L ( R ) +, + ) a H-space. Next, there exist w,w t E M such t h a t for any x , y , z E GL(R), x ~ y = W o O ( y S x ) , and ( x @ y ) @ z = w~ o ( x ~ ( y S z ) ) . Hence B G L ( R ) + is homotopy commutative and associative. Since it is connected, it is an
2. The Plus Construction Loday's Product products
in K - T h e o r y .
K,(R) | K i( R )
29
Following [L], we construct natural
, K,+i(R),
i,j > 1
which generalize the product for i = j = 1 considered in w First, fix a choice of a bijection r 9N x 1~ , N. Given rings R, R ~ and elements a e GL(R), a' e GL(R'), r enables us to convert a | a ' into an element of G L ( R | R~) 9 Thus we obtain a homomorphism GL(R) x GL(R') , G L ( R | R'), which clearly maps E ( R ) x E(R') into the commutator subgroup E ( R | R~), and hence induces a continuous map g: B G L ( R ) + x B G L ( R ' ) + ) B G L ( R | R') +. Let xo, x~o be the respective base points of BGL(R) +, B G L ( R ' ) + and define a new map
~" B G L ( R ) + x B G L ( R ' ) +
; BGL(R |
R') +
by ~(x, x') = g(x, x') - g(xo, x') - g(x, x~o)+ g(xo, xto) where the operations on the right side are performed using the H-group structure on the space B G L ( R | R') + given by Proposition (2.9). Then by construction ~ is null homotopic when restricted to B G L ( R ) + VBGL(R') + - B G L ( R ) + • {X~o}U {x0} x BGL(R~) +. Hence there exists a mapping on the smash product
h" B G L ( R ) + A B G L ( R ' ) +
~ BGL(R |
R') +,
unique up to homotopy, making the diagram below commute up to homotopy:
BGL(R) + x BGL(R') +
BGL(R |
~ BGL(R) + A B G L ( R ' ) +
R') +
This induces bilinear pairings on homotopy groups (see (A.7))
h.
Ki(R) |
Kj(R)
~ Ki+j(R |
R'),
from the homeomorphisms S ~ A SJ -~ S ~+j. Further, the "switch map" S ~AS j - - ~ S j AS ~ has degree ( - 1 ) ~j (compute it on homology) so that if R is commutative, the homomorphism R | R ; R given by multiplication induces bilinear products
, " Ki(R) |
Kj(R)
, Ki+j(R),
i,j > 1
satisfying x , y = ( - 1 ) i J y , x for x e Ki(R), y E Kj(R). Loday computes t h a t for i = j = 1, this product is the negative of the product constructed in w following Milnor (see [L], Proposition (2.2.3)).
30
Algebraic K-Theory
Finally, we remark that Quillen has computed the K-groups of a finite field in the following paper: D. Quillen, On the Cohomology and K - T h e o r y of the General Linear Group over a Finite Field, Ann. Math. 96 (1972), 552586. In fact, the computations in this paper motivated the definition of the plus construction. The result is as follows: If ]Fq denotes the field with q elements, then K0(F ) = Z;
K2~(Fq) = 0,
for i > 0;
K2,-ICFq) ~ ZICq'- 1)Z,
for i > 0.
3. The Classifying Space of a Small C a t e g o r y The references for simplicial sets are: J. Milnor, The geometric realization of a semi-simplicial complex, Ann. Math. 65 (1957), 357-362, and J.P. May, Simplicial Objects in Algebraic Topology, Midway reprints (1982). Apart from Quillen's paper, Higher Algebraic K-Theory - I, a reference for the classifying space of a category is: G. Segal, Classifying spaces and spectral sequences, Publ. Math. IHES 34 (1968), 105-112.
( S e m i - ) S i m p l i c i a l S e t s . Let A be the following category: for each nonnegative integer n, let n_n_= {0 < 1 < ..- < n} be the ordered set consisting of 0 , 1 , . . . , n; t h e objects of A are the ordered sets _n, a n d morphisrns are monotonic m a p s (i.e., m a p s f " m ~ _n such t h a t f(i) <_ f ( j ) for i < j ; in particular equality f(i) = y ( j ) is p e r m i t t e d ) . For each positive n, we have n + 1 maps in A On " n - 1 ~ n which are injective, given by a~(j)=
j j+l
ifj < i
ifj>i.
These are called the face maps. Next, we have n maps s r~--I i "n_n_ which are surjective, given by n-~ (J s~ ( J ) = , j - 1
~ n-
1
ifj < i ifj>i.
T h e s e are called t h e degeneracy maps. T h e face a n d degeneracy m a p s satisfy certain obvious identities (see May's book, pg. 1) a n d any m o r p h i s m in A can be w r i t t e n as a composition of face and degeneracy m a p s (if f 9 rn : n_n_,we can uniquely write f = g o h where h 9 m_ ;; p a n d g 9p ~-~ n are respectively surjective and injective arrows in A; now h is a composition of degeneracies and g a composition of faces). D e f i n i t i o n . A simplicial object in a category C is a c o n t r a v a r i a n t functor A , C i.e., a functor A ~ , C, where A ~ is the opposite category. A m o r p h i s m of simplicial objects in C is a natural transformation. Thus, a simplicial set is a functor Aop ; Se__~t,where Set denotes t h e category of sets; similarly a simplicial space is a functor A ~ - - - 4 Top w h e r e
32
Algebraic K-Theory
Top denotes the category of topological spaces. In the older terminology, a simplicial set was called a semi-simplicial set. Suppose F 9 A ~ .~ Set is a simplicial set. T h e n for each nonnegative integer n, F ( n ) is a set, called the set of n-simplices of F . T h e maps 0~ give rise to n + 1 maps of sets F(_n) } F ( n - 1), called the face maps, which associate to each n-simplex in F ( n ) a collection of n + 1 ( n - 1)-simplices in F ( n - 1), called its faces. Similarly the n maps n-1 give maps F ( n - 1) } F(_n) associating to each ( n - 1)-simplex a collection of n "degenerate" n-simplices; these maps F ( n - 1) ; F(n_) For e the of and F(s~)(6) E F(n.+ 1) the ith degenerate simplex of 6. (3.1). Let X be a topological space. Let S(X) denote the total singular complex of X, so t h a t Sn (X), the set of n-simplices of S(X), is just the set of singular n-simplices in X , i.e., S , ( X ) is the set of all continuous
Example
maps An
} X, where An is the s t a n d a r d n-simplex
An "- { ( t O , . . . , tn) E ~n-{-1 I ti >-- O, ~ t i :
1 }.
To make S(X) = {Sn(X)}n>o into a simplicial set, we must show t h a t to any morphism m .~ n in A, it is possible to assign a m a p of sets Sn(X) } Sin(X), compatible with compositions. If f " m ~ n is a m o r p h i s m in A, i.e., a monotonic m a p of ordered sets, we first describe a continuous map ] " Am } An, such that__re" .~ Am, f : ; f is a functor A } Top. Since Sn(X) = H o m w o p ( A , , X ) , and Y } Homwop(Y,X) is a functor (Top) ~ ~ Set, we will have shown t h a t S(X) is a simplicial set. If Am = {(so,...,sm) E Rm+l I s ~ >_ O, Esj = 1}, let ] ' A m
, An be the map ] ( ( s o , . . . , S m ) ) : (to,...,tn) where ti --
~
Sj ,
l(j)=~ where we define tj = 0 if {j I f(J) = i} is empty. One easily verifies t h a t this gives a functor A .~Top. E x a m p l e (3.2) ("The n-sphere"). set of m-simplices is given by
(S")m =
Let (S n) be the simplicial set whose
{era} {e., 1.} {era} U {all degenerate m-simplices associated to f , } ,
if 0 _< m < n ifm=n if m > n.
T h e faces of em all equal era-1 (for m >_ 1), and the degenerate simplices of em all equal em+l (for m >_ 0). T h e faces of fn all equal en-1, if n > 0 (if
3. The Classifying Space of a Small Category
33
n - 0, there are no face maps to be defined). We leave it to the reader to check t h a t these conditions do uniquely define a simplicial set; the reason for the name ("The n-sphere") is explained below (Ex. (3.4)). (3.3) G e o m e t r i c R e a l i z a t i o n . To each simplicial set F 9A ~ , Set, we can associate a topological space IF[ called the geometric realization of F. This is defined to be the quotient space
where for each n >_ 0, F(_n) is regarded as a discrete space. The equivalence relation ~ is defined as follows: given f 9m__ , n in A, let f 9 Am , An be the continuous map defined in Ex. (3.1) above. Then for any $ E F(n_), we set (~, ](y)) ..~ (F(f)~, y) for all y E Am, where (~, ](y)) E F ( n ) x An, and (F(f)~, y) E F(m__)x Am. Let ~ be the equivalence relation so generated, and let IFI be the quotient space, with the quotient topology (an open subset is a set whose inverse image in I_In F(n_) x An is open). Clearly the construction of the geometric realization is functorial, i.e., if F ---. G is a morphism of simplicial sets (i.e., a natural transformation of functors Aop , Set) we get a continuous map IFI , IGI . For any simplicial set F , a simplex ~ E F(_n) is called non-degenerate if it is not the degenerate simplex assigned to any ( n - 1)-simplex by one of the degeneracies. T h e n IF I is homeomorphic to a CW-complex, which has one n-cell corresponding to each non-degenerate n-simplex of F . If F, G are simplicial sets, let F x G denote the simplicial set whose n-simplices are F(_n) x G(_n), with obvious maps. Then by a theorem of Milnor (see the references cited earlier) the natural continuous map IF x G I , IFI x IGI (induced by the maps of simplicial sets F x G , F, F x G , G) is a continuous bijection, and is a homeomorphism if IFI x IGI is given the compactly generated topology associated to the product topology (i.e., the product is formed in the category of compactly generated spaces). In particular, if IFI or IG[ is locally compact (e.g., if F has only a finite number of non-degenerate simplices, so that IF I is compact) then IF x G I is homeomorphic to IFI x IGI. Let A(n) = n o m ~ (-,_n_n) be the simplicial set naturally associated to n_ E Ob A; then one sees easily that IA(n)l = An. Milnor's theorem follows from the special case f = A(n), G = A(m), which we prove in the Appendix (see (A.55)) (see also Lemma (6.8)). Finally, we note t h a t the homology of IFI can be computed as follows. Let Cn(F) denote the free Abelian group on F(n_), and let 0~' 9
34
Algebraic K-Theory
Cn(F) ~ Cn-I(F) be the map induced by F(O['). Ifdn = ]E(-1)'(9~, then C(F) -- (C,,,(F), d~)n>l is a chain complex. Then, for any Abelian group A, H . ( I F [ , A ) ~- H , ( C ( F ) | A), and H*([FI, A) ~- H * ( H o m z ( C ( F ) , A ) ) (see May's book, Prop. (16.2) and Corollary (22.3)). Let X be a topological space, and let S ( X ) denote the total singular complex (Ex. (3.1)). Then there is a continuous surjective map f : IS(X)I } X. If x0 E X is a base point, then S(xo) C S ( X ) i s a subcomplex (i.e., S(x0)(n) C S ( X ) ( n ) for all n) such that IS(x0)[ is a point whose image under f is x0. We have:
f " (JS(X)l, IS(x0)l) , (X, xo) induces isomorphisms on homotopy groups. Hence if X is a CW-complex, it is a homotopy equivalence
Theorem.
(see May's book, Theorem (16.6)). E x a m p l e (3.4). Let (S n) be the simplicial set described in Ex. (3.2). Then the geometric realization I(S r')l is naturally homeomorphic to the n-sphere S ~. Indeed, the only 2 non-degenerate simplices of (S n) are eo e (S n)(O) and fn e (S n)(n), so I(S")I is a C W complex with exactly 2 cells. If n > 0, all faces of f . are degenerate, so that I(S")I is obtained by attaching an n-cell to the 0-cell le0] with a constant attaching map. This is the standard description of S n as a CW-complex. (3.5) T h e C l a s s i f y i n g S p a c e of a C a t e g o r y . Let C be a small category, i.e., a category whose objects form a set. The nerve of C, denoted NC (or N(C)) is defined to be the following simplicial set: an n-simplex of NC is a diagram A0 /~ ,A1 f2 ' A2 I~ ' " " ~
A~
with Ai E ObC, fi E MorC. Given a m a p f 9m .~_n in A, the corresponding map NC(n_) ~ N C ( m ) m a p s the above n-simplex to the msimplex Bo gx }B1 g~~,92
~ - . . g'%Bm
where By = AI(j), and Bs ~- Bj+I is the composite map AI(j) } AI(j+I), where if f ( j ) = f ( j + 1), let AI(i) } Al(s+l) be the identity map. In particular the ith face of the above n-simplex is the ( n - 1)-simplex Ao I , AI
} ....
, A~-I f~+to.f~ ----* Ai+l
"---
} An,
while the ith degenerate simplex is the (n + 1)-simplex
Ao It, A1
}
. . . .Ai "
1-" Ai I,+,~ Ai+l
-~-'-
} An.
3. The Classifying Space of a Small Category
35
The classifying space of C is defined to be the geometric realization of NC and is denoted by BC (or B(C)); thus BC = INCl. Clearly, if F : C ; D is a functor between small categories C, :D then there is an induced map of simplicial sets NC ; N D , and hence an induced continuous map B F : BC ~ BD. A simple and useful example is given by the category {0 < 1}, consisting of two objects 0, 1 and a unique non-identity morphism 0 ; 1. Then one checks easily that B {0 < 1 } = I, the unit interval, since the only non-degenerate simplices in N{0 < 1} are {0}, {1}, {0 ~ 1} (the first two are 0-simplices, and the third is a 1-simplex). Let C, C~ be small categories, C x C~ their product, so that Ob(C x C')= (ObC) x (ObC'), and Homcxc,((A, B), (C, D)) = Homc(A, C) x Homc,(B, D). Then clearly N ( C x C') is the product simplicial set N C x NC ~. Hence B(C xC ~) ~. BC x BC ~ is a continuous bijection, which is a homeomorphism if the product on the right is given the compactly generated topology; in particular it is a homeomorphism if either BC or BC ~ is locally compact. Thus B(C x {0 < 1}) ~ BC x I . h functor H : C x {0 < 1} ; :/9 is just a pair of functors F, G : C ; :D (given by F ( A ) = H ( ( A , O ) ) , G(A) = H ( ( A , 1))) together with a natural transformation 77: F , G (given by r/(A) = H((1A, 0 , 1))). Hence we deduce the following lemma. L e m m a (3.6). Let F, G : C ~ l) be functors between small categories, such that there is a natural transformation F ~ G. Then B F , B G : BC
~B D
are homotopic maps.
C o r o l l a r y (3.7). Let F : C ~ l) be a functor between small categories. Suppose F has either a left or a right adjoint. Then B F is a homotopy equivalence. In particular, if C is a small category with either an initial or a final object, then BC is contractible. (See Appendix B for the definitions of adjoint functors.) L e m m a (3.8). Let I be a small filtering category, and let {C~}~ez be a family of small categories indexed by I. Let t7 = li__mCi be the direct limit I
category. Let X i ~ Ci be a family of objects such that X i 9 ; X j under
36
Algebraic K-Theory
the transition functors C~ , Ci of the given family (corresponding to morphisms i --. j in 1), and let X ~ C be the common image of the X~. Then 7rn(BC, {X}) = li__m7rn(BCi, {Xi}) for all n > O. I
(See Appendix B for the definitions of filtering category, etc.) P r o o f . Any finite diagram in C is the image of a similar diagram in some Ci. Thus NC = lirn NCi (where (lirn NC,)(_n) = lim NC,(n)), and any finite subcomplex of NC (a simplicial subset of NC with a finite number of non-degenerate simplices) is the isomorphic image of a subcomplex of some NCi. Since 7rn(BC, {X}) is the direct limit of r n of all finite subcomplexes of NC containing X (regarded as an element of NC(O)), and a similar claim holds for each BCi, the result follows easily. By abuse of terminology, a category is called contractible if its classifying space is; similarly a functor F is called a homotopy equivalence if B F is one, etc. C o r o l l a r y (3.9). Any small filtering category is contractible. P r o o f . Let I be a filtering category. For each i E I, let I / i denote the category of objects over i, consisting of pairs ( j , j ~i) with morphisms
(jl,jl
151
) il) ---* (j2,j2
U
152
) i2) being commutative diagrams:
j2
jl i
In fact, since I is filtering, one verifies easily that for this direct system i ) I / i , the naturally defined functor lim I / i ) I is an equivalence. But (i, i 1 ~. i) is a final object of I/i, so that I / i is contractible. Hence B I is weakly contractible (has vanishing homotopy groups). Since it is a CW-complex, it is contractible, by Whitehead's theorem (see (A.9)). R e m a r k . The classifying space functor is not full. An interesting example is given by the natural homeomorphism BC ~- BC ~ where C~ is the opposite category. E x a m p l e (3.10). Let G be a discrete group. Let _G be the category with one object * such that the monoid HomE(*,*) is the group G. We claim t h a t B_G is the classifying space of G. In fact, let G denote the category whose objects are in bijection with the elements of G, and the
3. The Classifying Space of a Small Category
37
following arrowswif [g] is the object of G corresponding to g E G, then Hom~([g], [h]) consists of a unique arrow 6(g,h). Then the 6(g,h) are forced to satisfy the composition rules ~(92, 93) o 6(gt, 92) = 6(gt, 93); one checks that these rules do define a category. There is a functor G , G__ given by [g] ~ ,, If(g, h) ~ h9 -1 E G = HomG__(*,,). The group G acts on (~ by g([h]) = [hg-1], g(i~(gl,g2)) = 15(gig -1, g2g-1). Since the isotropy group (for the G-action) of any object of (~ is trivial, the isotropy group of any n-simplex in N(G) is trivial. Hence G acts freely on the classifying space B(~. The functor (~ ~G is G-equivariant for the trivial G-action on G, so t h a t B(~ ~ B G is also G-equivariant for the trivial G-action on BG. From the criterion of (A.49), we see t h a t B(~ ~ BG is a locally trivial covering space with (discrete) fiber isomorphic to G, considered as the 0-skeleton of BG, and the group G acts transitively on this fiber. Thus BG__~ B G / G , and B(~ , B__Gis a Galois covering space with group G. Since G has an initial object (any object is an initial object) B G is contractible. Thus G ~ 7rl(BG), and 7ri(BG) = 0 for i # 0. One also sees that B G ~. BG..G_is a principal G-bundle with contractible total space, which is the usual defining property of the classifying space, which characterizes it up to homotopy equivalence.
4. Exact Categories and Quillen's Q-Construction
For our purposes, an exact category C is an additive category C e m b e d d e d as a full (additive) subcategory of an Abelian category ,4, such t h a t if 0 -, M ~ --, M ~M" ~ 0 is an exact sequence in j [ with M ' , M " E C, t h e n M is isomorphic to an object of C. An exact sequence in C is t h e n defined to be an exact sequence in ,4 whose t e r m s lie in C. Let ~: be t h e class of exact sequences in C. One can give an intrinsic definition of an exact category C in terms of a class s of d i a g r a m s in the additive category C, satisfying suitable axioms (see Quillen's p a p e r for details). In all cases relevant to us, the category embeds n a t u r a l l y in some Abelian category jr, such t h a t C is closed under extensions in .A. An exact functor F : C ~ l) between exact categories C, :D is an additive functor such t h a t if 0 ;Mt ~M ~M" , 0 is exact in C, then
0
~F ( M ' )
~. F ( M ) - - ~ F ( M " )
~0
is exact in D. If C is a small exact category, then we define the Grothendieck group K0(C) in t h e usual way: K0(C) - ~ / 7 ~ , where ~ is the free Abelian group on t h e objects of C, and T~ is the subgroup generated by classes [M] [ U ' ] - [ U " ] for each exact sequence 0
~M I
~M
.~ M "
t0
in C (the more standard definition involves t a k i n g the free Abelian group on isomorphism classes of objects of C, b u t is easily seen to be isomorphic to the above group, using sequences with M " - 0). If C is an exact category, we form a new category QC defined as follows: QC has the same objects as C, b u t a morphism X ~ Y is given by an isomorphism class of d i a g r a m s
(.)
x
i
q~-z~-, Y
where i is an admissible monomorphism and q an admissible epimorphism
4. Exact Categories and Quillen's Q-Construction
39
in C; by definition, this means t h a t there are exact sequences 0
~Z--~Y
~yi
~0
0
,X'----*Z
q*x
~0
in C. Another diagram X ~q' Z t ~' ; Y
(,)'
gives the same morphism (in which case we say the two diagrams (,), ( , ) ' , are isomorphic) if there is an isomorphism Z .~Z' making the d i a g r a m
x
commute. Composition of arrows in QC is defined as follows: given X --Z ~-~ Y and Y ,-- V ~ T, form the diagram (in the ambient category .At) X
T T Z
ZxyV
,,"
;
~
Y
T
~V~
~T
Since C is closed under extensions, and ker(Z x v V --* Z) -~ ker(V --~ Y) E C, (Z x y V) E C and Z x v V --~ X, Z x y V ~-* Y are respectively admissible epi and mono. Hence the diagram X , - Z x v V ~-, T defines an arrow in QC from X to T. One checks that the isomorphism class of this diagram depends only on the isomorphism classes of X ~- Z ~-, Y and Y ~- V ~ T, so t h a t we have a well-defined composition rule for morphisms; next, one verifies t h a t composition is associative. Hence if C is a small category, then QC is well defined, and is a small category. Let 0 E O b C = O b Q C be a zero object, so that {0} is a point of B Q C . Quillen's first result in this context is: T h e o r e m (4.0). There is a natural isomorphism ~ I ( B Q C , {0}) ~- Ko(C) (where K0(C) is as defined above). This motivates Quillen's second definition of higher K-theory.
40
Algebraic K-Theory
D e f i n i t i o n . K~(C) = 7r,+I(BQC, {0}), i > 0. We note t h a t this does not depend on the choice of a zero object, since given another zero object 0', there is a canonical isomorphism 0 -~; 0' in C, and hence in QC, giving a canonical choice of a path joining {0} and {0'} in B Q C . The isomorphism of Theorem (4.0) can be described explicitly as follows: given an object M E C, there are 2 canonically defined maps 1
0 --~ M in QC, given by the diagrams 0 ~- 0 ~-, M and 0 ~- M ~ M (where 1 denotes the identity map on M) 9 These give rise to a loop {0} ~~'~ {M} in .~ B Q C . The map Ko(C) ~ lrl ( B Q C , {0}) is given by mapping [M] E Ko(C) to the class of the above loop in ~rl. We defer the proof of this theorem to chapter 6 (however, see Ex. (4.10) below). At this point, we merely list certain properties of the K-groups defined above, whose proofs will be given in chapter 6. To illustrate the theory, and to provide motivation to work through the details of the proofs (if it is required!), we will use these properties in chapter 5 to get interesting results on the K-theory of rings and schemes; a high point will be the proof of the formula H P ( X , 1Cp,x) ~ C H P ( X ) for the Chow groups of a smooth variety. These properties of Ki are generalizations of familiar properties of the Grothendieck group functor K0. Let C be a small exact category, $ the category of (short) exact sequences in C. If C C ,4 is an embedding as a full exact subcategory of an Abelian category, then $ can be regarded as a full additive subcategory of the Abelian category of complexes in .4 of the form 0
; M'
,M
, M"
,0,
where a morphism of complexes is a (commutative) diagram in ,4 M'
~
M
~
N
1
~
M"
~
N"
l
N r
~
0
~
O.
1
Hence ~ is an exact category in a natural way, such that if E ' , E, E " E $, then 0 - ~ E ' , E ~E" , 0 is exact precisely when the corresponding diagram in C has exact rows and columns. There are exact functors s, t, q : t; , C such that if E E $, then 0
, sE
, tE ~
qE
".0
is the corresponding exact sequence in C. We can now state the first property of the functors Ki. Theorem
(4.1). (s,q)" QC
QC x Q c is a homotopy equivalence 9
4. Exact Categories and Quillen's Q-Construction Corollary
(4.2).
t. = s.
+ q. " Ki(C)
41
, Ki(C) for all i.
C o r o l l a r y ( 4 . 3 ) . Let 0 } F' ". F ; F" } 0 be an exact sequence of exact functors C ; l ) between small exact categories (i.e., there are natural transformations F ' } F and F ~ F " such t h a t 0 ----, F ' ( M ) , F(M) ~. F " ( M ) , 0 is exact in 79 for any M E C). Then F. = F~. + Ft.': K~(C) , K,(:D) for all i. ---. :/9 is an additive functor between small exact categories, an admissible filtration 0 = F0 C F1 C ..- C F,,-1 C F,~ = F by subfunctors is a sequence of additive subfunctors Fi 9C ; ~9 such t h a t F n_ 1( M ) , F p ( M ) is an admissible monomorphism in :D for each M E C. Hence the quotients F v / F n_ 1, p > 1, make sense as additive functors C } :/9. I f F 9C
C o r o l l a r y (4.4). I f {Fi}0<~
F. = ~ ( F n / F n _ I ) .
" Ki(C)
". K~(19) f o r all i.
p=l
C o r o l l a r y ( 4 . 5 ) . If 0
,Fo
}F~
,...
;Fn
,0
is an admissible long exact sequence of exact functors C obvious sense), then for all i > 0,
} 1) (in the
n
~ - ~ ( - 1 ) J ( F j ) . = 0" g i ( c ) j=0
, gi(D).
Next, we give the resolution and d6vissage theorems which allow us to replace one exact category by another, in certain situations, w i t h o u t changing the K-groups. Let ~4 be an exact category, 79 C ~4 a full additive subcategory, closed under extensions in ~4; then P is an exact category in a natural way such t h a t the inclusion :P ; ~4 is an exact functor. T h e o r e m (4.6) (Resolution theorem). A s s u m e that 7:' C A4 are as above, and assume f u r t h e r that i/o , M' then M ' E 79
, M
, M"
; 0 is exact in .&4 and M , M " ~_ 79,
(b) f o r any object M E .M, there is a finite resolution 0 with P~ E 79.
-'Pn
,Pn-1
'""
}Pc
,M
,0,
42
Algebraic K-Theory
Then BQ'P ~ BQ./t4 is a homotopy equivalence, and hence K i ( P ) K i ( . M ) for all i. C o r o l l a r y (4.7). Let {Zi}i>_l be a 5-functor from the exact category A4 to the Abelian category .A (i.e., Ti : A4 , ~4 are additive functors such t h a t if 0 .~ M ~ , M , M" .~ 0 is an exact sequence in A4, then there are functorial b o u n d a r y maps T~+~(M") , T~(M'), i > 1, giving a long exact sequence ; Ti+I(M") , T~-I(M')
, T~(M') ~ ,...
Ti(M)
, T~(M")
, T~(M")).
Let 7~ C M be the full exact subcategory of T-acyclic objects P with Ti(P) = 0 V i). Assume that for every object M E an admissible epimorphism P ". M with P E 79, and that for all su.O~ciently large n (depending on M ) . Then BQ7 9 homotopy equivalence.
(i.e., objects A4, there is Tn(M) = 0 ; BQ.A4 is a
T h e o r e m (4.8) (D~vissage theorem). Let ~4 be an Abelian category, 13 a full A belian subcategory which is closed under taking subobjects, quotients and finite products in .4. Suppose each object M E .A has a finite filtration in A O= Mo C M1 C ... c Mn = M
with M i / M i - 1 E 13 for all i > 1. Then B Q B equivalence, so that Ki(13) ~ K i ( A ) .
; BQr
is a homotopy
Finally, we state the localization theorem, which under certain circumstances gives us long exact sequences of K-groups. We recall t h a t if .4 is an Abelian category, B C ,4 a full Abelian subcategory closed under taking subobjects, quotients and extensions in .A, then B is an Abelian category, and is called a Serre subcategory of ,4. Under these conditions one can construct a quotient Abelian category .A/B (see Appendix B), which, in various concrete cases of interest to us, is naturally equivalent to a suitable "localization" of the category ,4; indeed the construction of .A/B is a generalization of the construction of the localization of a ring, and of modules over the ring, with respect to a central multiplicative set. T h e o r e m (4.9) (Localization). Let 13 be a Serre subcategory of the Abelian category .4, and let C be the quotient Abelian category .A/B. Let s" B ,4, p" .A ~.C be the natural exact functors. Then
BQB
~ BQA
BQ8
~ BQC
BQp
4. Exact Categories and QuiUen's Q-Construction
43
is a homotopy fibration (i.e., the natural map B Q B : F ( B Q p ) is a homotopy equivalence, where F ( B Q p ) is the homotopy fiber (see (A.27) of B Qp)). Hence there is a long exact sequence 9..----,
K (B)
,
-,
, Ko(C)
,...
,K,(c)
,g0(.a)
o.
It is possible to give a "naive" construction of the map Ko(C) ) 7rl (BQC, {0}) of T h e o r e m (4.0); we discuss this below. This may give t h e reader a little practice in thinking about the Q-construction. Example
( 4 . 1 0 ) . As noted above, the isomorphism O" Ko(C)
~-, Irl(BQC,{O})
is explicitly given by associating to each M E C a certain loop rM based at 0, such t h a t r = [rM] e lrl(BQC, {0}). There is a canonical admissible mono iM : 0 ~ M and an admissible epi qM : M --~ O, associated to any object M E C. Given any admissible mono i 9 M1 ~ M2, there is an arrow it : M1 ~ M2 in QC, corresponding to the (class of the) d i a g r a m 1
i
M1 ~-- M1 ~-, M2. Similarly, given any admissible epi q ' M i --~ M2, there is an arrow q ! : M2 --* M1 in QC, corresponding to the diagram 1
M2 q M1 ~-* M1. One sees at once t h a t if M q M ~ i : N is a diagram representing an arrow u : M - - ~ N in QC, then from the definition of composition of morphisms, u - i! o qt. T h e two arrows 0 ) M in QC given by i M! and q ~ give 2 p a t h s {0} , { M } in BQC, denoted (iM!) and (q~4)- Let rM = (iM!)o (q~t4) - 1 where the inverse and composition are the usual operations on paths; thus r M is the oriented loop obtained by first following (iM!) and then following ( q ~ ) in reverse. To see t h a t [M] ~ [rM] defines a homomorphism Ko(C) , IrI(BQC, {0}), we must show t h a t if (E)...
0
:M'--~M
q;M"--40
is an exact sequence in C, then [rMl [rM!]" [rM,,] in lrl(BQC) where denotes the group operation in 7rl. From the split exact sequences =
M I ----, M l @ M "
0
"
0
~ M"
) MI@M"
) M"
; MI
-----, 0, )0
one sees i m m e d i a t e l y t h a t the classes [rM,], [rM,,] commute, so the homomorphism r Ko(C) ~ n l ( B Q C , {0}) will be well defined. We note t h a t from the sequence (E) above, and the facts t h a t i M = I i o iM', qM = qM" o q in C, giving iMt = i! o iM,! and q~M = q! o q'M" in QC,
44
Algebraic K-Theory
we have a d i a g r a m where the shaded triangles c o m m u t e :
i
9
!
?iiiii!iiiiiiT}
!
!i',:,!',
I
:~iiii!ii!
!i};iliiiiii!ii? i
i?ili
2:iii:::?
.
0 T h e s h a d e d triangles give 2-simplices in B Q C . F r o m t h e d i a g r a m , we also notice 2 m o r e arrows 0 ; M in QC, n a m e l y i! o q"M" and q! o iM,,!. We claim t h a t in fact i! o q"M' = q! o iM,,!. By definition t h e composite i! o q"M" corresponding to the d i a g r a m 0 q_M, M ' --, i M in C, represents an arrow u 90 ~ M in QC. On the other hand, t h e c o m p o s i t i o n law gives t h a t q: o i M,,! is represented by
0
0 •
T T
iM,I r
B u t t h e r e is an isomorphism 0 x
M"
~
p
q
)M"
r" M >
1->M
M - - ~ M t such t h a t
OXM,,I~
M' c o m m u t e s , since 0 .. M~ i M q ~ M " - - * 0 is exact. T h u s u i! o q~4, a n d q! o i M,,! are represented by isomorphic d i a g r a m s , a n d so give equal a r r o w s in QC. Thus, in t h e earlier d i a g r a m w i t h 2 s h a d e d triangles, we can a d d on a t h i r d arrow u 9 0 ~ M , a n d a d d 2 s h a d e d triangles
4. Exact Categories and Quillen's Q-Construction
45
(corresponding to 2-simplices in B Q C ) , from the diagram M l
i~
~ M
"~
q~
M"
0
We do not draw the resulting diagram with 4 shaded triangles, but the reader can imagine it as yielding a CW-complex homeomorphic to a 2sphere with 3 holes, such that the boundary circles have a common point {0}, and (properly oriented) are just the 3 loops r M , r M , , r M , , . Taking the orientations into account one checks that r M is homotopic to rM, 9rM,,.
5. T h e K - T h e o r y of Rings and Schemes
If R is a ring, let P(R) denote the category of finitely generated projective (left) R-modules. This is a full subcategory of the Abelian category of left R-modules, so that 7~(R) is an exact category where all exact sequences are split. We will prove the following result, comparing the plus and Q constructions, in Chapter 7. T h e o r e m (5.1). There is a homotopy equivalence
BGL(R) +
, (I~BQ'P(R)) ~ which is natural up to homotopy.
(where ~ denotes the loop space, and the superscript 0 denotes the connected component of the trivial loop at {0} E BQT~(R)). Hence there are natural isomorphisms K i ( P ( R ) ) ~ lri(BGL(R)+), i >_ 1. From Theorem (5.1), we see that Quillen's new definition K i ( R ) Ki(Ta(R)) agrees, for i > 1, with the definition given by the plus construction; in particular, K0, K1 and Ks agree with the earlier definitions. This fact will be needed in a couple of places for certain computations. We will assume in the following discussion that all rings are left Noetherian, unless specified otherwise. If A is a Noetherian ring, let fld(A) denote the category of finitely generated (left) A-modules. Then AA(A) is equivalent to a small full subcategory, and any two such full subcategories are naturally equivalent to their union. Hence we can define Ki(~4(A)). A similar convention is used for all other exact categories which we will deal with, and has already been tacitly used in the definition of K~(Ta(R)) above. We define Gi(A) - Ki(AzI(A)); this is also sometimes denoted by K~(A). The inclusion P(A) C fld(A) induces a natural map K i ( A ) ~. Gi(A). By the resolution theorem (4.6), if A is (left) regular, then K i ( A ) ~Gi(A) (recall that a Noetherian ring A is left regular if every finitely generated A-module has a finite resolution by finitely generated projective A-modules).
5. The K-Theory
of Rings and Schemes
47
Our first goal is to prove: T h e o r e m (5.2). Let phisms for all i > 0 (i)
G,(A) ~
A
be
Noetherian.
Then there are natural isomor-
a,(A[t]), induced by change of rings;
(ii) Gi(A[t,t-1]) ~- Gi(A) r Gi-I(A).
(in (ii), for i = 0 we define G_ 1(A) = 0). P r o o f (ii). We first prove (ii), assuming (i). Let B C ~t(A[t]) be the Serre subcategory consisting of modules annihilated by a power of t (strictly, we must first replace M(A[t]) by an equivalent small full subcategory, and let B be the Serre subcategory of this small Abelian category consisting of modules annihilated by a power of t--we will in future suppress such points, leaving it to the careful reader to make the necessary modifications). The quotient Abelian category .M(A[t])/B is naturally equivalent to Ad(A[t,t-1]). Hence the localization theorem (4.9) yields an exact sequence 9"
~ ai+i
(A[t,
t-l])
, K~(13)
, a,(A[t])
, a,(A[t,
f;--1])
.....
Now j~4(A) C B as the full subcategory of modules annihilated by t. Hence by d6vissage (Theorem (4.8)), Ki(B) ~- Gi(A). Hence the localization sequence can be rewritten as
9. . - - - . Gi(A)
~ G,(A[t])
f~ Gi(A[t,t-x])
, Gi-1 (A) ---* -...
From (i), the fiat change of rings A ~ A[t] gives an exact functor A4(A) --. M(A[t]), M ~ A[t] | M, which gives an isomorphism Gi(A) ~ Gi(A[t]). Since A ~ A[t,t -1] also gives an exact functor i" .h4(A) ~ M(A[t,t-~]) we have a commutative diagram Gi(A[t])
I
G,(A[t,t-'l)
J'"
C,(A) It suffices to prove that i. is a split inclusion; then the same holds for f , and (ii) follows immediately. Let .Ml(A[t,t-1]) C .M(A[t,t-1]) be the full subcategory of modules A) = 0, where we let A be the A[t, t-1]-module M satisfying w"'A[t't-t](M, -'-"'l A [ t , t - l ] / ( t - 1). Then by the resolution theorem (or rather, Cor. (4.7)), K~(~l(A[t,t-~]) ~- G,(A[t,t-1]). The functor i" .h/l(A) ~ fl4(A[t,t-1])
48
Algebraic K-Theory
clearly factors through ~41(A[t,t-1]). Also, there is an exact functor j 9 .A41(A[t,t-1]) } jg4(A)given by M ~ A | M (where as above, A is the cyclic A[t,t-1]-module with annihilator ( t - 1)). Clearly j o i is isomorphic to the identity functor. Hence j . o i . - Gi(A) } Gi([t, t - 1]) } Gi (A) is the identity. P r o o f (i). We begin by proving a lemma. Let B = A[tl,...,tn] be the polynomial ring in n variables over A, with its usual grading (degti = 1 for all i, deg a = 0 for all a E A). Let MgrB denote the category of positively graded, finite B-modules. Thus each N E .MgrB can be written as N = (~p>0 Np where the Np are finite A-modules. Then N has a finite increasing filtration {FvN}v>0, where FpN is the B-submodule generated by No ~ . - - ~ 9 Np. Regard A as a graded B-module annihilated by t 1 , . . . , tn and concentrated in degree 0. Then A | N is a graded B-module with
(A
g).
+ . . . + B go
as A-modules, where B~ C B is the A-submodule of homogeneous polynomials of degree i. Thus
(.)
(A |
FpN)rn =
0
(A |
N)m
ifp<m ifp>m.
For any N ~_ .MgrB, and any p > O, let N ( - p ) E .M,qrB be the graded module with underlying B-module N and grading
Nm-v
N(-p)m-
0
ifm >p ifm
L e m m a (5.3). Let N e MgrB such that Tor~ (A, N) = 0. Then for each p > 0 the natural map of graded B-modules
Cn " B ( - p ) |
(A |
N)v
, FpN/Fp_IN
is an isomorphism. P r o o f . Since (A | N)v "~ (FpN/Fp-IN)v, and FpN/Fv_IN is generated over B by its A-submodule of elements of degree p, there is a natural map Cv as in the statement of the lemma, which is surjective. We prove t h a t under the condition Tor~(A,N) = 0, Cv is an injection for each p. We work by descending induction on p; for large p, N = FpN = Fv_l N so t h a t both sides vanish. We also have WorB (A, F p - I N ) = 0. Assume that r is an isomorphism, and that Tor B (A, FvN ) = 0; we will deduce that Cv is an isomorphism, and that Tor~(A, Fv_IN ) = O.
5. The K-Theory of Rings and Schemes From the sequence 0 have an exact sequence
, Fp_ 1N
, FpN
Tor~(A, FpN) --4 Tor~(A, FpN/Fp_IN) ---. A | ~ A|
l[
49
~ FpN/Fp_ 1N Fp_IN
', A |
, 0 we F~,N--,
(FpN/Fp_,N) ---, O.
0 From (.), A @B Fp_ 1N , A | B Fp N is an isomorphism in degrees _< p - 1, and A| Fp_ 1N vanishes in all higher degrees. Hence the map i is injective, and so T o r f (A, FpN/Fp_ I N) = O. Now consider the exact sequence (which defines M)
0 Tens|
;M
~B ( - p ) |
(A |
g)p r
FpN/Fp_IN
, O.
with the B-module A, we obtain
0--, Wor~(A, F p N / F p _ l g ) ~ A |
M --, (A |
N)n --'
~ (A | F p N / F , _ I N ) p - 4 0 .
[1 0
Since r is an isomorphism in degree p, we obtain A | M - 0. Since M is a finite graded B-module, this forces M = 0. Hence r is an isomorphism. Since B is a polynomial ring, Wor~(A, B ( - p ) | K) = 0 for all i > 0, for any A-module K. Hence, using the isomorphism Cp,
Tor~(A, FpN/Fp_IN) = 0 for a l l / > 0. Hence Wor~(A, F p g ) = 0 ~ Wor~(A, F p _ l g ) = O. This completes the inductive step, and the proof of the lemma. The operation N , N ( - 1 ) gives an exact functor from A4grB to itself. Hence there is a natural Z[t]-module structure on Ki(A4grB), where t acts by shifting the grading by - 1 . The change of rings map gives a homomorphism Gi(A) ~ Ki(A/Igr B), and hence a homomorphism 0 : G,(A)| ; Ki(.A4grB), where r =) = Cn(x), and Cn is induced by the functor fl4(A) ; A4grB, P ~-~ B ( - n ) | P. P r o p o s i t i o n (5.4). r gives an isomorphism G,(A) |
X[t] ~= Ki(A4grB).
P r o o f . Let Af C fl4grB denote the full subcategory consisting of all modules N such that T o r ~ ( A , N ) = 0 for all j > 0. Since the functors Tj = WorB ( A , - ) vanish for all j > n (the number of polynomial variables), Cor. (4.7) to the resolution theorem implies that Ki(Af) ~ Ki(A4gr~). Let Afp C At" be the full subcategory of modules N satisfying FpN = N i.e., N is generated by elements of degree < p. There are exact functors
. ~ ( a ) p+I
,,
~.Afp b ,
. M ( A ) P +~
50
Algebraic K-Theory
given by
p
a ( M 0 , . . . , Mp) = @ B ( - j ) |
Mj
j=O
and
b(N) = ((A |
N ) o , . . ., (A |
N)p).
Clearly b o a : M ( A ) p+I , M ( A ) p+I is the identity. Now the identity functor Ip :Afp , Alp has an admissible filtration by additive subfunctors Fq : N ~ F q N , O<_q<_p. I f 0 , N' , N , N" ,0isanexact sequence in Af, then 0
;A |
N'
*A |
N
, A |
N"
~0
is exact. Hence, from Lemma (5.3), the sequences
0
~FqY'/Fq_lg'
, FqN/Fq_IN
~Fqg"/Fq_lN"
~0
are exact, for 0 < q < p. Hence by Cor. (4.4), p
E ( Fq / Fq_ I ) . = ( Ip ) . " K i (./kfp)
, K , (.N'p) .
q=l
But by Lemma (5.3), the map ~ ( F q / F q _ l ) . is precisely the map on Ki(Afp) induced by the composite functor a o b. Hence the functor a induces an isomorphism a . : G,(A) p+I , g,(A/'p). Taking the direct limit over p (Lemma (3.8)), the proposition follows. P r o o f o f T h e o r e m (5.2) (i). Let B = A[t, u] be the polynomial ring, C - A[t, u, u -1] the localization with respect to powers of u. As above let A/IsrB denote the category of finite positively graded B-modules. Let A4grC denote the category of finite Z-graded C-modules. Then A4grC is naturally equivalent to the quotient of A/[grB by the Serre subcategory ,4 of modules annihilated by a power of u. Clearly A/fgr(A[t]) C .A as the full subcategory of modules annihilated by u; by d6vissage, K,(J~d3r(A[t]) ) ~ K , ( A ) . Also, we have a natural equivalence of categories Ad3rC -~; A/f(A[x]), where x = ~t E C, given by N ~-, No (the inverse functor is given by change of rings). Hence from the localization theorem we have an exact sequence
--, K,(M2p(A[t]) ) *, K , ( M g r B )
, G,(A[x])
, Ki_I(MSr(A[t]) ) ---~.
5. The K-Theory of Rings and Schemes Now
K~(~gr(A[t])) ~- G,(A) |
Z[y]
g,(MgrB) u G,(A)|
Z[y]
51
where multiplication by y corresponds to a shift in grading by - 1 in both cases. Hence r is Z[y]-linear, so to compute it, it suffices to compute For any A-module M we have an exact sequence of graded B-modules 0
, B ( -- 1 ) |
u ~B |
M
~ A[t] |
M
~0
where Air] = B / u B . Thus the composite functor i" M ( A )
, Mg(A[t])
M ~-, A[t] |
M
fits into an exact sequence of functors j~4(A)
o
,j(-1)
, MgrB , A/lgrB
,j:
,o,
where j ( - n ) " .M(A) , .M,qr B is j ( - n ) ( M ) = B ( - n ) | M. Hence i, ( 1 - y ) . j , 9 G,(A) ~ K~(MgrB). Thus r is identified with multiplication by 1 - y . In particular r is injective with cokernel isomorphic to Gi(A), where the isomorphism is given by
j , " G,(A) Hence G,(A)
, g~(.MgrB)
, g , ( . M g r S ) / ( 1 - y ) . g , ( M g r B).
; G,(A[x]), given by change of rings, is an isomorphism.
C o r o l l a r y (5.5). Let A be a regular ring. Then
(i)
K,(A)
~- g , ( A [ t ] )
(ii) Ki(A[t,t-1]) ~- Ki(A) @ K i _ I ( A ) (where if i = 0, K _ I ( A ) is defined to be 0). (5.6) K - T h e o r y of S c h e m e s . For terminology from algebraic geometry, we refer to Appendix D. If X is an arbitrary scheme, let P ( X ) denote the category of locally free sheaves of finite rank, which is a full subcategory of the Abelian category of quasi-coherent sheaves of Ox-modules on X (see (D.10), (D.17), (D.43)). Define K ~ ( X ) = K~(T'(X)). If X is a Noetherian scheme (D.24), let A4(X) be the Abelian category of coherent sheaves (see (D.44)) on X; define G i ( X ) - Ki(.A4(X)) (sometimes this is also denoted by K ' ( X ) ) . Since we will mainly study Gi at this point, we assume until further notice that all schemes under consideration are Noetherian and separated (D.33), unless explicitly mentioned otherwise.
52
Algebraic K-Theory
There is a natural homomorphism Ki(X) , a i ( x ) induced by 7)(X) C fl4(X). If X is regular (D.25), then (since X is Noetherian, hence quasi-compact (D.23)) every coherent sheaf on X is a quotient of a locally free sheaf of finite rank (see (D.54), (D.55)), and hence has a finite resolution by locally free sheaves of finite rank. Hence by the resolution theorem, Ki(X) ~ Gi(X). If E is a locally free sheaf on X , then the assignment (see (D.8) for tensor products)
c
.r, .r
M(X)
gives an exact functor [s176: M ( X ) , M(X), and hence a map [t:]. : Gi(X) ~, G,(X) for each i. If 0 ~ E' , E , C" ) 0 is an exact sequence in 7~(X), then [E]. - [E~]. + [C']., since we have an exact sequence of functors M(X) , M ( X )
0
,re'].
,[el.
,[e"].
,0
corresponding to the exact sequences in M ( X ) 0
~ ,~' |
~E |
, g" |
tO.
Thus, C ~-~ [g]. extends to a homomorphism Ko(X) , End(G,(X)), i.e., a pairing Ko(X)| G,(X) , G,(X). Since Ko(X) is a ring with the multiplication induced by tensor products of locally free sheaves, the above pairing in fact makes G,(X) a module over the commutative ring Ko(X). Similarly one can define K0 (X )-module structures on all the Ki(X), such that K~(X) ~G,(X) is a K0(X)-module homomorphism. If X - Spec A is affine (see (D.15)), then we have natural equivalences of categories P ( X ) -~ "P(A), .M(X) ~'~ .M(A). Hence K~(X) ~- K~(A), Gi(X) '~ Gi(A), and we identify these groups under these natural isomorphisms. R e m a r k (5.7). From Theorem (5.1), Ki(A) defined using the Q-construction agrees with the earlier definition in Chapter 3 as 7r~(BGL(A)+), for i >_ 1. The latter groups have the product K~(A)| Kj(A) ~ K~+j(A) defined in Chapter 3 for i,j >_ 1; one can show that this is K0(A)-bilinear, making {~)i>0 Ki(A) into a graded anti-commutative ring. More generally, F. Waldhausen (Algebraic K - T h e o r y of Generalized Free Products, Ann. Math. 108 (1978), 135-256) has shown t h a t if ,4, B, C w
are exact categories , and ,4x B F .~ C is bi-exact (i.e., F ( M , - ) and F ( - , N) are exact functors for each M E .4, N E B) then there is a product (for each i, j > 0)
g,(A) | Kj(B)
, gi+j(C)
5. The K-Theory of Rings and Schemes
53
satisfying certain naturality properties, and which agrees with Loday's product if ~4 - B - C - P(A). This new product will make ~ _ > 0 K , ( X ) into a graded anti-commutative ring, and ~ i > 0 G~(X) into a graded (~),>0 K~(X)-module (if X is Noetherian) such that
, Oa,(x) ~>_o
,>o
is a module homomorphism. (5.8) F u n c t o r i a l i t y P r o p e r t i e s . If f : X ~ Y is a morphism (see (D.16), (D.17)) of schemes, the exact functor f * : P ( Y ) .~ T'(X) (see (D.41) for f*)gives homomorphisms f* : K~(Y) ~ K~(X). Clearly K~ becomes a contravariant functor from schemes to Abelian groups. If f : X , Y is a flat morphism (D.40) of Noetherian schemes, we have an exact functor f* : A4(Y) ; A4(X), and hence a map on K-groups G i ( Y ) ~ G~(X). Thus G~ is a contravariant functor on the category of Noetherian schemes and flat morphisms. L e m m a (5.9). (i) Let i ~-, Xi be a filtered inverse system of schemes such that the transition morphisms X~ : ; Xj are affine (D.28), and let X = lim X,. Then Kq (X) = lim Kq (Xi). ---4
.---4
(ii) If in (i), X, X~ are all Noetherian, and the transition morphisms are also fiat, then Gq(X) - lim Gq(Xi). ---4
P r o o f . (i) We wish to apply Lemma (3.8), since :P(X) is essentially the direct limit of the P(X~). However i ~-, P(Xi) does not form a directed system of categories indexed by iop, since for morphisms i --, j --, k in I, (i --4 k)* " P ( X k )
, P(Xi)
(i -~ j ) * o ( j -~ k)* " P ( X ~ )
and ~P(X,)
are not equal, but only isomorphic (the tensor product involved in defining the pullback of a sheaf is not strictly associative, but only associative up to isomorphism). One gets around this problem by the following device. Let I' = ItJ {0} where 0 is an initial object, and let X0 = X. For i E I', let I ' / i be the full subcategory of I' consisting of objects j preceding i; then there is a unique morphism j ---, i for j e I~/i since iop, hence (It)Op, is directed. Let P~ denote the following category, for each i E / ' - - a n object in P~ is a family {(Pj,Oj) [ j e I ' / i } where Pj E :P(Xj), and Oj : (j --, i)*Pi , Pj is an isomorphism. An arrow f : { (Pj, 0j) } , { (P~, 0~) } is a collection of
54
Algebraic K-Theory
~rrow~ yj: Pj
) P; in P ( X j ) such that the diagram below commutes:
(j ---* i) *Pi
Oj
---.
(j ...4 i)*fi 1
0;
(J --. O'P"
~-
Pj
P~
, 7~(X,) given by { (Pj, Oj) } Pi is an equivalence Then 7~i egories for each i E I', and we claim (see the remark below) i a functor from (i,)op to small categories. Further, since (i,)op has object 0, the direct limit is just 79q~, which is equivalent to 79(X). the lemma follows from Lemma (3.8), and the fact t h a t
of cat, 7>/ is a final Hence
Q(lim 79i) = lim Q79i. --.4
(ii) The proof is similar and is left to the reader. R e m a r k . The transition functor T'i ~ Pj for a morphism j given by {(Pe, 0t) ]/~ e I ' / i } ~ {(Pt, Ct) ]/~ e ( I ' / j ) } , where so it makes sense to consider the subfamily of Pt with t E I ' / j family of Pc, e. E I'/i; the map Ce is the unique isomorphism following diagram commute: (e 9 j)* o ((j ~ i)*Pi)
(t -,j)'0j,
(e - , j ) * P j
(e ~ i)*Pi
ot
P,e
--~ i in I ' is I'/j C I'/i, of the given making the
where r/it is the "natural" isomorphism
(e - , j)* o ((j -~ 0*p~) -~ (e - , j)*P,, given by
O x , | (Oxj | Pi)
' O x t | Pi ,
which is locally given by a | (b | p) ~ ab | p. So one has to verify that with the above transition functors,
(e
,~)*=(e
.~j).o(j
~)*:p~
,7~,
for e ,j , i in I'. Both operations take a family {(Pro, 0,,,) [ m e I ' l l } to the same family of objects {P,,., [ m e I'/s and possibly two sets of isomorphisms Xm, X " " (m ---, e)*Pt ~ Pro; we have to verify that Xm = X " for every m E I ' / L This is a local question, so we are reduced to the affine case. Now the situation is as follows: we are given a direct system of rings {A/}iej, where g = (it)op is a filtering category. Let 79j = {(P~,O~)IP~ e
7)(Aj), Oj" Aj@A, Pi ~-} Pj}. The transition functors (i < j)*" 79i
; :Pj
5. The K-Theory of Rings and Schemes
55
are given by (i < j)* {(Pt, Oe)} = {(Pe,r where for j < t, Ce is the unique isomorphism making the following diagram commute:
Ae | (Aj | P,)
lt|
At | Pi
At|
ot
Pt
i.e., Ct = Oe o rio o (lt | 0#)- 1. Here rio is induced by a | (b | p) ~ ab | p. Now suppose i < j < g < m; then
(j < g)* o(i < j)*
{(Pro, Ore)} = { ( P m , xm)}
where Xm is defined by the commutativity of
Am | (At | Pj)
1.,,| ,
'1
Am|
1Xm
Vlmt
am | Pj
~",
I'm
(and ~7~e(a | (b | p) ) -- ab | p). On the other hand (i < g)"
{(Pm,0m)} = { ( P m , x ' ) } ,
where X" is the unique map making the following diagram commute:
A|174
1,,,|
A.-,|
Om
Am|
~
I'm.
Since Cm = Om o v/mj o ( l m | 0j)-x, the desired equality Xm = X" follows from the commutativity of the diagram:
A,~ | Pt (l,,,,@Ot Am|174
(x,,.|
Am|174174 ,. | (lt | 0,)
m|174 X,., = X "
qmt
l Illmt
Am| Pm
". e,,.
Am | P~
(
.,,.,
A,n |
@ P~)
56
Algebraic K-Theory
The commutativity of the outer border, with left vertical arrow Xm, is by the definition of Xm, while the commutativity of the left rectangle, with left vertical arrow X'm, is by definition of X ' - Hence, the commutativity of the hexagon proves Xm = X ' . To prove commutativity of the hexagon, let a E Am, b E At, c E Aj, p E Pi; then a | (b | (c | p)) E Am | (At | (Aj | Pi)) yields the hexagon | (b~ ~ p)(
~ ~ (b | ( ~ p))
~|174 Oj(~|
1
ab|174 abc | p
(
ab | (c@/p)
We note that the commutativity of the hexagon reflects a basic coherence property of the tensor product (see MacLane's book Categories for the Working Mathematician, Grad. Texts, No. 5, Springer-Verlag (1972)). (5.10) Let f : X , Y be a morphism (of Noetherian schemes) of finite Tor-dimension, i.e., there is an integer N > 0 such t h a t the Ox-modules Tor ~ (Ox, ~) -- 0 for all i > N for any ~" e A~t(Y). Let Ad (Y, f ) C j~4(Y) denote the full subcategory of sheaves ~- satisfying Tar~ v (Ox, ~ ) = 0 for all i > 0; then 7~(Y) C A/[(Y, f). Hence, if every coherent sheaf on Y is a quotient of a vector bundle, then by Cor. (4.7) of the resolution theorem, K,(.Ad(Y,f)) "~ G~(Y) for all i. Also, f * : fl/[(Y, f ) , Ad(X), ~ ~-. f*~-, is an exact functor in any case, yielding a map f* : Ki(.hd(Y, f)) Gi(X), so t h a t in the above situation, we have a map f* : Gi(Y) Gi(X). The condition that every coherent sheaf is a quotient of a vector bundle holds if either (i) Y is regular (D.54), or (ii) if Y supports an ample invertible sheaf (D.57) (e.g., Y is quasi-projective over a Noetherian ring (D.52)). Finally the formula (fog)* = g* of* is valid for maps f : X ." Y, g :Y , Z, both of which have finite Tot-dimension, such that coherent sheaves are quotients of vector bundles on Y and Z. (5.11) Let f : X ~. Y be a proper morphism (see (D.34), (D.38)), so that for any ~" E AA(X), Rif..T" E .Ad(Y) for each i, and the functors Rif. vanish for sufficiently large i on Ad(X) (see (D.12), (0.61), (0.62)). Let .A,4(X,f) C ~A(X) be the full subcategory of sheaves ~" satisfying Rif..T" - 0 for all i > 0. Then if every coherent sheaf on X is a subsheaf of a sheaf in M ( X , f ) , then K i ( M ( X , f)) ~ K i ( M ( X , f ) o p ) _~ Ki(.h~(X)Op) ~_
5. The K-Theory of Rings and Schemes
57
Ki(.M(X)), where the middle isomorphism follows from Corollary (4.7) of the resolution theorem, and we have used the fact that QC ~ QC ~ for any small exact category C (this is proved later, Cor. (6.3)). Hence, under this hypothesis on A4(X), there is a direct image map f . : G i ( X ) ---4 Gi(Y); and for appropriate maps f, g we have the formula ( f o g). = f . o g.. The hypothesis above on A//(X) (namely, that any coherent sheaf is a subsheaf of an object of A4(X, f)) is satisfied if either (i) f is finite, in which case A4(X) = A4(X, f), or (ii) X has an ample line bundle s In the latter case, s174 is generated by a finite number of global sections (as an Ox-module) for all sufficiently large n. For any coherent sheaf 3r on X, let ~'(n) denote 9r | s174 Then for all sufficiently large n, Rif.3r(n) = 0 for all i > 0 i.e., ~'(n) e A 4 ( X , f ) (this follows from (D.57), combined with (D.62)-(D.65)). If we fix such a value of n, which is large enough so that O x ( n ) is also generated by global sections, then we can find N global sections which generate Ox(n) for some N > 0, i.e., a surjection O~xI~ ~ O x ( n ) from a trivial bundle of rank N, whose kernel is a vector bundle on X. Applying the functor ~Omox ( - , O x ( n ) ) (i.e., dualizing and tensoring with O x ( n ) ) we obtain an exact sequence of vector bundles O --.
Ox
:
Ox(n)
,
-: , O,
and hence on tensoring with ~- an exact sequence in A4(X) o ---,
y(n)
--,
&, |
J:
o,
which gives an inclusion of ~" into a sheaf in A4(X, f). Note that in this construction, if j c e P ( X ) , then 3C(n)r162e 7~(X,f) = 7~(X)N A4(X, f ) . This is used below, in the proof of Prop. (5.12). P r o p o s i t i o n (5.12) (Projection formula). Let f : X ". Y be a proper morphism of finite Tot-dimension between schemes supporting ample line bundles, so that f* : G,(Y) ~ G,(X), f . : G,(X) ~ G,(Y) are both defined. Then (i) there is a well defined map f . : K i ( X ) ---* K i ( Y ) giving a commutative diagram
O,(X)
,-! K,(r)
l,-
---,
a,(y).
(ii) for any x e K o ( X ) , y e Gi(Y), we have the formula f . ( x ) . y f , ( z " f*y)
=
(iii) for any x e K o ( X ) , y e Ki(Y), we have the formula f . ( x ) . y
-
f,(x" f'y)
58
(iv)
Algebraic K-Theory for any y e Ko(Y), x e G i ( X ) , we have the formula f , ( f * ( y ) , x) -y. f.(~).
(in (ii), (iii), (iv) above,-denotes the K0-module structure; in (iv), we do not need the hypothesis that Y supports an ample line bundle).
Proof. The projection formula for sheaves (see (D.41)) f.(f*~ |
.7:') ~ ~ |
f..T,
for ~ y e e ~'(Y), J= e M ( x ) , immediately yidds Ov). We prove (i) ~nd (ii) below; the proof of (iii) is similar and is left to the reader. Let :P(X, f) = 7)(X) N Ad(X, f ) be the category of vector bundles C on X satisfying Rif.,f. = 0 for all i > 0. As noted at the end of (5.11), since X supports an ample line bundle, for any E E P ( X ) there is an admissible monomorphism E .~ s with E' E 7~(X, f ) (i.e., such that 0 ; C .~ E' .~ s ~ 0 is an exact sequence in 79(X)). Hence Ki(79(X, f)) ~ Ki(7~(X)) = Ki(X), as in (5.11). Let g" E P(X, f); then we claim that f . E E 7/(Y), the full subcategory of fld(Y) consisting of sheaves of finite homological dimension (see (D.56)); since Y supports an ample line bundle, these are precisely the sheaves which have finite resolutions by vector bundles (so that Ki(Y) ~ Ki(7"I(Y)) by the resolution theorem). Since being of finite homological dimension is a local property, our claim (that f . E E 7~(Y)) is local on Y. Hence for proving the claim we may assume Y = Spec A is affine. Now for any affine open subset U = Spec B in X, we know that T o r A ( B , - ) = 0 for all i > N (for some N independent of U), since f has finite Tor-dimension. Let Ui be an affine open cover of X by a finite number of open sets, such that C[u' is trivial; since f is separated, all the intersections Uil N - . . fq Ui.~ are affine (see (D.33)), and we can compute the cohomology groups Hi(X, E) from the (~ech complex associated to this affine cover (D.61). But Rif.E is just the sheaf on the affine scheme Y = SpecA associated to the A-module Hi(X,E), for each i _> 0 (see ( D . a 3 ) ( i v ) ) . Since ~ e P(X, f), we h a v e H'(X,C) = 0 for i > 0, and an exact sequence (with a finite number of non-zero terms)
0
' H~163
~
H H~163 i
' n
S~
fq Uj,s
--,....
i,j
Each germ in this sequence except the first is an A-module M which is given to have the property T o r ~ ( M , - ) - - 0 for i >_ N. Hence the functors Tor~(H~ C ) , - ) also vanish for i _> N, from repeated applications of the long exact sequence of Tors. But f.~" is the coherent Ov-module associated to the finite A-module H~ ~r which satisfies T o r f ( H ~ ~ ) , - ) -- 0 for
5. The K-Theory of Rings and Schemes
59
i > N. Hence H~ s has projective dimension < N over A, i.e., f . s has finite homological dimension. This completes the proof of (i). (ii) Let s E P(X, f) be fixed, and let
0
,Cm---~Cm-~
,'-----*Co
,f.E--,O
be a finite resolution by vector bundles on Y (which exists, as f , s E ~ ( Y ) ) . T h e n by definition, the action of [f,E] E Ko(Y) on G,(Y) is given by rn Y ~ ~-d=o(-1)Jxj.y, where y e Gi(Y), and xj = [E/]. Let Af C Ad(Y, f ) be the full subcategory of sheaves satisfying T o t OY ( / . E , ~') -- 0 for all i > 0 (this Tor sheaf is an Or-module). T h e n Ki(A/') ~ Gi(Y) by the resolution theorem. For any jc E Af, we have an exact sequence, natural in ~',
0
,c.~|174
,..---~c0|
,f.$|
,0
which we interpret as an exact sequence of functors Af , fld(Y). Hence the functor Af = ; fld(Y) given by tensoring with ].s represents the action of [f.E] E Ko(Y). Hence if x = [s E Ko(X), y H f. ( x ) . y is represented by the functor Af ; fld(Y) given by tensoring with f . s Next, we want a similar representation of y ~-. f. (x. f* (y)) by a functor Af ; Ad(Y). We claim t h a t for any ~- E Af, we have (a) R~f.(E| c) = 0 for all i > 0, (b) there is a natural isomorphism f . ( E | ~ (f.E)| In view of the claimed naturality in (b), both statements are local on Y, so to prove them, we may assume t h a t Y = Spec A is affine. T h e n ~" = M is the sheaf associated to a finite A-module M. Since T o t ~ ( f . s ~-) = 0 for i > O, T o r # ( H ~ 1 6 3 = 0 for i > 0 (as f.E is just the sheaf associated to H~163 Further, since ~ E Af C Ad(Y, f), if U = S p e c B is any affine open subset of X , then TorA(B, M) = 0 for all i > 0. Thus, if {Ui} is an affine open cover of X by a finite number of open sets such t h a t E Iu~ is trivial, then the exact sequence (obtained from the (~ech complex)
(,)
...0-
, H~ (X, s ) ~
H H~ ( Ui ' E ) --* H H~ ( Vi n Uj , s ) i i,j
,...
remains exact on tensoring with the A-module M, since for each t e r m in the sequence, TorA( - , M ) = 0 for all i > 0. For any affine open subset U = Spec B C X, f*.T'lv is the sheaf associated to the finite B - m o d u l e B | M , and so (E | f*.T')l v is the sheaf associated to the B - m o d u l e g~ F_.)| (B | M) ~ g ~ F_.)| M. Thus the sequence ( , ) , tensored with M (over A), is the analogous complex obtained from the (~ech complex for E | f * ~ by adjoining H~ | M at the beginning. Since this complex is exact, we conclude t h a t H~(X,F_. | f*.T') = 0 for i > 0, and
60
Algebraic K-Theory
H~ | f*:F) ~- H ~ | M as A-modules. This means precisely that R * f . g | f * ~ = 0 for i > 0, and f . ( g | f - j r ) ~ (f.E) | jr, proving the claims (a) and (b). Now the exact functor iV" , M ( X , f ) given by 5v ~-. g | f*~" represents G , ( Y ) ~.G~(X), y ~ x. f*y. Hence the exact functor Af ~ A d ( Y ) given by ~" H f . ( g | f*.T') represents G i ( Y ) ~ Gi(Y), y ~-* f . ( x . f ' y ) . Since we have an isomorphism of functors Af , Ad(Y) given by the natural isomorphism f . ( s | f ' J : ) "~ ( f . g ) | Jr, we obtain the formula ( f . x ) . y = f . ( x . f ' y ) for any y E Gi(Y) and x = [g], with g E 7~(X, f). But such classes x generate K o ( X ) , and both sides are additive in x. Proposition
(5.13). Let X !
y'
gt }
X
,
Y
g
be a fiber product diagram of schemes with ample line bundles. Assume that f is proper and g has finite Tor dimension, and that O x , Or', are Tot-independent over O r . Then g* o y. = $" o g'* in Hom(a,(X), G , ( Y ' ) ) . P r o o f . We leave the proof to the reader (see Quillen's paper, Prop. (2.11) for a proof using the analogous formula in the derived category). The point is to prove that if ~ E j~d(X, f) n 2vI(X, g~), then f . ~ E Ad(Y, g), g~.jc- E M ( X ~, f~), and we have an isomorphism g*f..T ~ ].g ; the interested reader can give a direct proof of this using a suitable (~ech complex, along the lines of the previous proof. C l o s e d S u b s c h e m e s . Let i : Z ~ X be a closed subscheme, j : U ,X the open complement. Let Iz denote the (coherent) sheaf of ideals of Z in O x . We can identify M ( Z ) with the full subcategory of A d ( X ) consisting of sheaves annihilated by Iz, via the functor i. (see (D.29), (D.43)). Proposition
(5.14). If I z is nilpotent, then Gi(Z) ~ G , ( X ) .
P r o o f . Immediate from the d(~vissage theorem (4.8). Proposition
(5.15). There is a long exact sequence
) Gi+l(V)
' Gi(Z) i . Gi(X) J ' ; ,...
,Go(X)
Gi(U)
,Go(U)
9 a~_ ~( Z ) ~0.
5. The K-Theory of Rings and Schemes
61
P r o o f . Let B c A4(X) be the Serre subcategory of sheaves ~" with Jc[u = 0; then A4(Z) c B is a homotopy equivalence by d6vissage, and the quotient . A 4 ( X ) / B is equivalent to A4(U). The result follows from the localization theorem (4.9). (5.16) N a t u r a l i t y . The exact sequence of Prop. (5.15) has certain naturality properties. For example, if Z
i
;
Z l
,}
i
X
are closed subschemes, we have a diagram 9. . - - ,
G~+~(X-Z)
-.
G~(Z)
9. . - 4
G~+~(X-Z')
-.
G~(Z')
(i' o i).
G~(X)
--.
~-L
a~(x-
z)
-~...
II
it
G~(x)
-~
(J~
G~(X-Z')--,...
where X-
Z'
j,
} X-Z
.}
3
X
are the corresponding open immersions. Also, a fiat map f : X ' , X induces a map from the sequence for (X, Z) to that for (X', f - 1 Z ) . Finally, if U, V c X are open subschemes, there is a Mayer-Vietoris sequence
G,(U)@G,(V)
,G,(Uuv)
9
, c,~(unv)
~....
Indeed, we may assume without loss of generality that U U V = X; then ifY =X-U, Z =X-V, then Y N Z = O a n d X - ( Y O Z ) -- U A V . Since A4(Y O Z) ~ Ad(Y) x A4(Z), as Y and Z are disjoint, we have G , ( Y U Z) ~- G , ( Y ) @ G , ( Z ) . The Mayer-Vietoris sequence follows by a standard diagram chase from G~(Y) ~9 Gi(Z)---.Gi(X) @ Ci(X)--~ G~(U) ~ G~(V) ---~G~-I(Y) ~ g G i - l ( Z )
II ---~G,(YUZ)
1 --~
Gi(X)
II
l ~
G~(UAV)
---,
Gi-I(YuZ)
(where the top row is the direct sum of the localization sequences for
Y, Z).
P r o p o s i t i o n (5.17) (Homotopy property). Let f : P ~ X be a fiat map whose fibers (see (D.31)) are al~ne spaces (e.g., a geometric vector bundle (D.59)). Then f * : G,(X)
-, G , ( P )
is an isomorphism for all
i.
62
Algebraic K-Theory
ProoL
For any morphism T
, X let PT = P x x T, so t h a t all fibers of
PT f r T are also affine spaces and f T is flat. In particular, if Z C X is a closed subscheme, U = X - Z, we have a diagram 9. . ~
c,(z)
c,(Pz)
9 . . - ~
~
c,(x)~
~
c,(P)
By the 5-1emma, the proposition to hold for the other two.
-~ holds
c,(v)
~
c,(Pv)
~
G,_,(z)
--,...
C,_a(Pz)
- ~ . . .
for any one of X, Z, U ifit
is known
By Noetherian induction, we may assume t h a t the proposition is valid for all proper closed subschemes Z c X , and we will prove it for X . If X is non-reduced, we can take Z = Xred, SO t h a t P z = Pred, and the proposition holds for X if it holds for Z, by Prop. (5.14). Next, if X is reducible, say X = Z1U Z2, then the proposition holds for Z1 and X - Z 1 = Z 2 - ( Z 1 N Z 2 ) (since it holds for Z2 and Z1 N Z2), hence it holds for X. Thus, we are reduced to the case when X is reduced and irreducible. Now we take the direct limit of the above diagrams as Z runs over all proper closed subschemes of X (ordered by inclusion; use the functoriality properties (5.16)). Since direct limits are exact, we obtain a d i a g r a m (where k ( X ) is the function field of X ) 9-.--~ l i m O i ( Z )
I
9-. --. l i m G i ( P z )
--*
Oi(X)
--*
Gi(P)
I
---*
Oi(k(X))
-4
Gi(Pk(x))
I
~
limGi_l(Z) --,---
--*
limai_l(Pz)
I
.=-b
-*'"
where we have identified lim G~(U) with G~(k(X)) (and similarly c o m p u t e d lim G , ( P u ) ) by Lemma (5.9). Thus we are reduced to the case X = Spec k, where k is a field, and P -~ Spec k [ t l , . . . , tn] is affine space of some dimension over k; this follows from T h e o r e m (5.2).
P r o p o s i t i o n (5.18) (Projective Bundles). Let s be a vector bundle of rank r over X, P ( E ) = Proj(S(E)) the associated projective b~ndt~, wh~,~ S ( ~ ) is the symmetric algebra of E. Let f : P ( C ) ~ X be the structure map. Then we have an isomorphism of Ko(P(s
go(P(e)) |
c,(x) -% c,(~(e))
for each i > O, given by y | x H y . f * x . Further, if z E K0(]F(C)) is the
5. The K-Theory of Rings and Schemes
63
class of Or(z)(-1), then the above isomorphism can be rewritten as
a,(xy r--1
i=0
P r o o f . One knows that K0(P(s is a free Ko(X) module of rank r with basis 1, z , . . . ,z r-1 (we prove a stronger result, Theorem (5.29), later; but this fact is "classical"-- see e.g., Manin, Lectures on the K-functor in algebraic geometry, Russian Math. Surveys 24 (5) (1969) 1-89). Hence it suffices to prove the second formulation of the result. Using the localization sequence and Noetherian induction as above, we easily reduce to the case X = Spec k, where k is a field, and ~: corresponds to a vector space E of dimension r over k. Then S(s corresponds to S(E) ~- k[to,..., tr-l], the polynomial ring in r variables, and P(s ~ p~-l. The standard correspondence (see (D.46)) between finite graded S(E)modules and coherent sheaves on P(E) identifies A4(P(E)) with the quotient of the Abelian category of finite positively graded S(E)-modules A4grS(E), by the Serre subcategory A4f,grS(E ) of modules of finite length. By d6vissage, the Serre subcategory has the same K-groups as the subcategory A42p(k) of positively graded finite dimensional k-vector spaces, considered as S(E)-modules annihilated by E . S(E). The K-groups of A4grS(E) are computed by Prop. (5.4) as
K~(A4gr(S(E)) ~- Gi(k) |
Z[t]
where t acts by shifting the grading by - 1 . Clearly K~(A42?rk) "~ G~(k) | Z[t] since fl4grk ~ limA4(k) n, where A4(k) n C A4grk is identified with the subcategory of graded finite dimensional vector spaces which vanish in degrees > n; again t E Z[t] acts by shifting the grading. Hence the localization sequence gives a diagram (whose commutativity defines h)
9..
,
K~(fl4grk)
l
a (k) |
Z[t]
where h is Z[t]-lineax.
-----, K~(A4grS(E))
h
l a (k) |
~. G, (P(E))
,..-
Z[t]
Hence to compute h it suffices to compute the composite G~(k) , Ki(A4,qrk) i. K~(A4grS(E)), which is induced 'by the functor associating to a k-vector space W the S(E)-module obtained by letting E . S(E) act trivially on W; this is graded by assuming it to be concentrated in degree 0. The Koszul resolution for the S(E)-mcdule k
64
Algebraic K-Theory
gives an exact sequence 7"
0
, S(E)(-r) | , S(E)|
W
AE| , W
W ,
,...
, S(E)(-1) |
E|
W
O.
Regarding this as an exact sequence of functors ~4(k) , A4grS(E), we see t h a t h is just multiplication by X - t ( E ) e Z[t], where A_t(E) = ~ ( - 1 ) ~d i m ( ~ ' E ) - P . Since A - t ( E ) is a monic polynomial in t of degree r-1 r, h is injective and ~ j = o Gi(k) tj maps isomorphically onto the cokernel. Thus the map
a~(k) r
, G,(F(E)), (xo,. . . , x r - 1 ) H ~
z'x,
is an isomorphism, since the operation of shifting the grading by - 1 corresponds to tensoring with Op(E)(--1), i.e., to the action of z e K0(IP(E)) under the module structure. (5.19) F i l t r a t i o n b y C o d i m e n s i o n o f S u p p o r t . (See (D.30), (D.45).) Let X be a Noetherian scheme, Z c X a closed subscheme; we define the codimension of Z in X to be c o d i m ( Z , X ) = i n f z e z ( d i m O z , X ) . If E 2vl(X), the support of ~" is the subscheme of X with ideal sheaf A n n ~ ' , the annihilator of ~" in O x . Let M P ( X ) C j ~ ( X ) be the Serre subcategory consisting of those coherent sheaves ~ whose support is a subscheme of codimension ~_ p in X. From Lemma (3.8) and Prop. (5.14), K,(j~/IP(X)) = lim Gi(Z), where Z runs over closed subsets of codimension --t
> p in X. If f " X ' , X is flat, then f*.A/ln(X) C J~4n(X'); indeed it suffices to note that if Z C X is a closed subscheme with codim(Z, X) > p, then c o d i m ( f - l ( Z ) , X ') > p, which follows from the following statement about local r i n g s - if R ~ S is a flat homomorphism of local rings such t h a t the maximal ideal of R generates an ideal in S primary to the maximal ideal of S, then dim R = dim S. (Compare with (D.40).) If i ~-~ Xi is a filtered inverse system of Noetherian schemes with affine, flat transition morphisms, such that the inverse limit X is Noetherian, then Kq(.A4P(X)) - - l i m Kq(.A4P(X,)). Indeed, since Kq(.A4P(X)) - lim Gq(Z), ---t
---t
where Z runs over subschemes of codimension _> p in X, and a similar result holds for Kq(.MP(X~)) for each i, it suffices to prove t h a t if Z C X is a closed subscherne of codimension >_ p, and fi " X , Xi is the canonical map, then for some i and some subscheme Z' C Xi of codimension > p in Xi, we have f i - l ( z ') = Z. If i E I is fixed (where I is the indexing category), and Ui c Xi is an affine open subset, then for any j E I \ i, (j --, i ) - x ( u i ) = Uj C X j is affine open, and if Uj = SpecAj, then f - 1 (U~) = U = Spec A c X is affine open, with A = lim Aj. The ideal of ---t
5. The K-Theory of Rings and Schemes
65
l(z~)
Z N U is finitely generated, since X is Noetherian, so t h a t Z fq U = fjfor some closed subscheme Z~ C Uj. Since Xi has a finite cover by affine open subsets, the claim follows. Define a decreasing filtration on G~(X) = Ki(A4(X)) by
FPG,(X) =
image(g,(fl4n(X))
, K,(M(X))).
This is called the filtration by codimension of support, and is a finite filtration provided X has finite Krull dimension. ( 5 . 2 0 ) . Let X n C X be the set of points of codimension p in X. There is a spectral sequence (of cohomological type)
Theorem
Ef'q = E['q(X) = H
K-n-q(k(x)) ~
G-n-q(X),
xEXn
which is convergent when X has finite Krull dimension, such that the induced filtration of Gn(X) is the filtration by codimension of support. The spectral sequence is contravariant for fiat morphisms; further, if i ~ Xi is a filtered inverse system of Noetherian schemes with affine, fiat transition morphisms whose inverse limit X is Noetherian, then the spectral sequence for X is the direct limit of the spectral sequences for the X~. R e m a r k . This spectral sequence is sometimes referred to in the literature as the BGQ (Brown-Gersten-Quillen) spectral sequence. In the statement of the above theorem, we interpret the notation to mean Kn = 0 for n < 0. Thus the spectral sequence is concentrated in degrees p _> 0, p + q _< 0; in particular it is a 4th quadrant spectral sequence of cohomological type. Proof of Theorem
(5.20). Consider the filtration by Serre subcategories M
(x) . . . .
There is an equivalence of categories I_[
x6_.X,
where .A(Ox,x) is the category of O.,x-modules of finite length. By d~vissage, if k(x) is the residue field of Ox,x, then
Kq(k(x)) ~ Gq(k(x)) ~ Kq(.A(O.,x)), for any x E X. Hence we have localization sequences
, Ki(MP+'(X))--~ K,(A,ff(X))
K,(k(x))
' H xEXp
, . . .
66
Algebraic K-Theory
giving rise to a spectral sequence by the method of exact couples (see Appendix C). The functoriality assertions follow immediately from the functorial properties of the filtration {AdP} noted earlier. (5.21). The following conditions on a Noetherian scheme X are equivalent:
Lemma
(i) Vp ~ 0, .MP+I(X)
, M P ( X ) induces 0 on K-groups.
(ii) V q < 0, EP'q(x) = 0 for p ~ O, and the edge homomorphism a _ q ( X ) - - , E~ is an isomorphism. (iii)
Vn
>_ O, the sequence of Abelian groups 0
,On(X)
e
H
Kn(k(x)) d ,
xEX o
H
K n - l ( k ( x ) ) d , ...
xEX 1
is exact, where dl is the differential on the E1 terms of the spectral sequence, and e is obtained by functoriality from the (fiat) morphisms Spec Ox,x : ; X, for x e X ~ and the isomorphism Gq(Ox,x) ""= Gq(k(x)) ~= Kq(k(x)). P r o o f . This follows easily from the construction of the spectral sequence. For any p _> 0, i > 1, the differential dl on I_Ixexp K~(k(x)) is obtained from the following diagram whose rows are exact localization sequences:
--* I_Ixex~' K~(k(x))
--'
' Ki-1 (AdP+I(X)) ~
'
Ki_I(fldP(X))--.
' ILex,§
Now if (i) holds, the various localization sequences break up into short exact sequences 0
' K'(MP(X))
' H
K,(k(x)) --~ K , _ I ( M p + I ( x ) )
,0.
zEXP
These sequences splice together to give the exact sequences in (iii). Since the sequences in (iii) are constructed from the complexes of E1 terms, whose cohomology groups are precisely the E2 terms, clearly (ii) is a reformulation of (iii). Hence (i) =~ (iii) r (ii). Now assume (iii). We prove by induction on p _> 0 that
K,(.AdP+ I (X)) ~ K,(.MP (X) ) is 0, for all i. To start the induction, we note that the injections e "G,(X)
' H xEX o
K,(k(x))
5. The K-Theory of Rings and Schemes
67
fit into the localization sequence
9..---, g,(.h,41(X))---,
g,(M(X))--~I_[g,(k(x))--, K , _ I ( . M I ( X ) ) 4 . . . xEX o
II
a (x) Hence this breaks up into short exact sequences 0 ~
Ki(.A,4(X))
e;
n
Ki(k(x))
, Ki_I(M'(X))
, O,
zEX o
and Ki(.MI(X)) ~ Ki(.M(X)) is 0 for all i _> 0. Suppose we already know, by induction, that for 0 < p' < p, the localization sequence of AAP'+I(X) c M r' (X) splits into short exact sequences 0
' H K,(k(x))
' Ki(M/"(X))
, K,_,(MI"'+I(x))
, O,
xEXP'
so that Ki(.MV+I(X))
dl"
. Ki(.MV(X)) is 0 for all i. The differential
H
K,+l(k(x))
'
xiF.X I , - 2
H
K,(k(x))
xEX t,- t
factors as the composite
H
g,+,(k(x))
", K i ( M P - ' ( X ) )
xeXv-2
I I K,(k(x)), zEXI,- t
whose cokernel is
,,
H
zEXv-t
Thus, in the factorization
dl"
H
K,(k(x))
, K,_I(MP(X))
' II
:r.EXP
xEXP-l
we see that by the exactness of the sequence in the hypothesis (iii) of the lemma, K'-I(MP(X))
' H gi_l(k(x)) zEXv
must be injective, for all i. Thus the localization sequence for AdP+I(X) c AdP(X) breaks up into short exact sequences, and Ki(aMP+I(X)) Ki(.MP(X)) is 0 for all i. We have assumed p > 1 in the above argument, but a minor variant works for p = 0.
68
Algebraic K-Theory
P r o p o s i t i o n (5.22). Let Gn,x denote the (Zariski)sheaf on X associated to the presheafU ~-. Gn(U). Assume that SpecOx,x satisfies the equivalent conditions of Lemma (5.21) for each x E X . Then there are canonical isomorphisms E~ 'q ~ H P ( X , G-q,X) (where the E2 groups are those obtained from the spectral sequence of Theorem (5.20)). P r o o f . For each open set U C X, form the complex given in Lemma (5.21) (iii); as U runs over all open subsets of X, we may view these complexes as defining a complex of presheaves for the Zariski topology on X. The associated complexes of sheaves have the form 0
} Gn,X
} H
(ix).gn(k(x))
' H
xEX o
(ix).gn_l(k(x))~
...
xEX 1
for n > 0, where ix " Spec k(x) } X is the canonical map, and for x E X n, K n _ n ( k ( x ) ) is regarded as a constant sheaf on Speck(x). The stalk of the above complex (for a given n) at x E X is just the corresponding complex for the scheme Spec Ox,x, since lim Gn(U) - Gn(Ox,x), and the xEU
spectral sequence commutes with filtered inverse limits of schemes (with affine flat transition maps). By hypothesis, this complex of stalks is exact for each x c X and n > 0. Hence the above complex of sheaves gives a resolution of Gn,x by flasque sheaves, which are known to be acyclic for the Zariski topology. Hence the associated complex of global sections, which is a complex of E1 terms, computes the cohomology groups H i ( X , Gn,x). But the cohomology groups of the complexes of E1 terms are precisely the E2 terms of the spectral sequence. G e r s t e n ' s C o n j e c t u r e . The equivalent conditions of Lemma (5.21) are valid if X = Spec R, where R is a regular local ring. T h e o r e m (5.23). Gersten's conjecture holds for R = k [ [ k l , . . . , x , ] ] , the ring of formal power series in n variables, and for R equal to the ring of convergent power series in n variables over a field k complete with respect to a non-trivial valuation. P r o o f . First consider the case n -
k [ [ x l , . . . ,xn]]. We prove that , M
(R)
induces 0 on K-groups (where 2vln(R) stands for Adn(Spec R)). Clearly K , ( M n + ' ( R ) ) = lirn K , ( M n ( R / t R ) ) t
5. The K-Theory of Rings and Schemes
69
where t runs over non-zero non-units of R. Hence it suffices to prove t h a t fl4n(R/tR) ~, fl4P(R) induces 0 on K-groups for any such t. By the Weierstrass preparation theorem, after a change of coordinates, we can assume t h a t A = k [ [ X l , . . . , xn-1]] is such that the composite A } R ~ R / t R is injective, and R / t R is a finite A-module. Let B = R | R/tR; since R -~ A[[x.]], B ~ (R/tR)[[~.}l. There is a natural surjection of (R/tR)-algebras r B = R | R/tn , R/tR; if r = a E R/tR, then the kernel of r is generated by ( x n - a) (here xn E R C R | R/tR). Thus, given any (R/tR)-module M we have an exact sequence 0
;B |
M (x.-~) B |
M
~M
;0
of B-modules, where ( x n - a ) denotes multiplication by x n - a . Considering these as R-modules, if M E A4P(R/tR), then the above sequence yields an exact sequence of exact functors
A4n(R/tR) ---. A/In(R). Since the first two terms correspond to isomorphic functors, which yield the same map on K-groups, the last term, corresponding to the inclusion functor fl4n(R/tR) } fl4n(R), induces 0 on K-groups. The above argument also works when R is a convergent power series ring, since the Weierstrass preparation theorem holds in t h a t case too. Theorem
(5.24) (Quillen). Let R be a regular semi-local ring, which is a
localization of a finitely generated algebra over a field k. Then the equivalent conditions of Lemma (5.21) hold for R. P r o o f . We will only prove the result in the special case when Spec R is smooth over an infinite field k (see (D.67)). We refer the reader to Quillen's paper Higher Algebraic K-Theory I for the proof in the general case. Let A be a finitely generated k-algebra, and S a finite set of primes of A, such t h a t R is the semi-local ring of S on X = Spec A. Since R is smooth, we may assume t h a t A is smooth over k. Without loss of generality we may take R, A to be domains. We want to prove t h a t A4 n+ I(R) ---, .hAP(R) induces 0 on K-groups. Clearly K , ( M n+l (R)) = lim K,(.Mn+I(AI)), where f runs over all elements of A which do not vanish at S. Replacing A by any one such AI, it suffices to prove that A,In+I(A) } .A4n(R) induces 0 on K-groups. Now
K,(~4n+' (A)) = lim K,(~4P(A/tA)) t
Algebraic
70
K-Theory
where t runs over non-zero divisors in A. Hence it suffices to prove t h a t for each non-zero divisor t E A, there exists f E A such t h a t f does not vanish on S, and fl4V(A/tA) ,
~ fl4P(AI),
M ~ M f,
induces 0 on K-groups. We now use: N o r m a l i z a t i o n L e m m a ( 5 . 2 5 ) . Let A be a smooth, finitely generated k algebra o f dimension r; let S C Spec A be a finite set, and t E A a non-zero divisor. T h e n there is a polynomial subring B = k [ x l , . . . , Xr-1] C A such that (i) A / t A is finite over B (ii) A is smooth over B at the points o f S. P r o o L Let X = Spec A; then there is an embedding X C AN as a smooth, closed subvariety. Let Y = S p e c ( A / t A ) c X; then dim Y = r - 1. Since k is infinite, the "general" linear projection A N ; A t - 1 restricts to a finite morphism Y , A r - l , by the Noether normalization lemma (see (D.69)). Since X is smooth, the "general" linear projection X , A r-1 is s m o o t h at S, since k is infinite, by Bertini's t h e o r e m (see (D.68)). This proves t h e lemma. Now let B ' = A / t A , A' = A | B ' , so t h a t there is a m a p of B ' algebras s 9 A ' ~ B ' , giving a diagram A'
(
(
"
"A
J~
9 B
Let S ' - v - 1 (S) be the set of primes of A' lying over S. Since A is s m o o t h over B on S, A' is smooth over B ~ on S'. Since the relative dimension of A ~ over B ' is 1, if I = ker s, then I is locally principal at the points of S ~ (if x ~_ S ' C SpecA t, y E Spec B ' the image under (u~) * 9 Spec A' ; Spec B ~, t h e n to see that I is principal near x we may replace A ~, B ~ by their respective complete local rings at x, y, by N a k a y a m a ' s lemma; now we are reduced to the situation ( A ' ) ^ ~ (B~)A[[z]], the ring of formal power series, where the claim is obvious--see the proof of (5.23)). Thus I is principal in a neighborhood of S'. Since A' is finite over A, and S ' = v - l ( S ) , we can find f E A such t h a t f does not vanish on S, and I f = I . A~ is principal. Since A' is smooth over B ' on S', it is s m o o t h in a neighborhood of S ~, and we m a y assume f above has been chosen so t h a t in addition A~ is s m o o t h (hence flat) over B'.
5. The K-Theory of Rings and Schemes
71
Given any finite ( A / t A ) = B~-module M we have an exact sequence of finite A~ modules 0
~ II |
M
~A~I |
M
~M f
, O.
Since A~ is flat over S', if M E A/IP(B'), then A~ | M - ~ / / | M lies in j~cff(A~). Since A~ is finite over A! we can then regard the above sequence as an exact sequence in .MP(AI) , giving an exact sequence of exact functors ./vff( A I t A )
, .h/ff ( A l ) .
Since b is principal, b ~ A~, and so M ~ - , / / | M, M ~-, A~ | M yield isomorphic functors, giving the same map on K-groups. Hence M ~ My induces 0 on K-groups. P r o p o s i t i o n (5.26). Let X be a scheme of finite type over a field k. Then the image of the differential dl "
H
K,(k(x))
a:EXp -t
' H Ko(k(x)) ~- H Z xEXP
x6.XP
in the spectral sequence of Theorem (5.20), consists precisely of the group of codimension p cycles rationally equivalent to 0 (in the sense of Fulton's book Intersection Theory). Hence E~ '-p ~ CHP(X), the Chow group of codimension p cycles modulo rational equivalence. C o r o l l a r y (5.27) (Bloch's formula). Let X be a regular scheme of finite type over a field k. Then there are natural isomorphisms HP(X, ICp,X) "~ CHP(X),
p>0
where lCv,x is the sheaf associated to the presheaf (for the Zariski topology) U ~ Kp(U). We have a flasque resolution xEX 1
xEX o
'""
H
xEXP
(izl.KoCk(x))
,0
and isomorphisms E~ 'q ~= H P ( X , ~ _ q , X ) for the terms of the spectral sequence (5.20). P r o o f of C o r o l l a r y (5.27). have isomorphisms
By Theorem (5.24)and Prop. (5.22), we
J~'q ~ H p ( X , ~ - q , X ),
and from the proof of (5.22) we have a flasque resolution as above for ~p,x. But Gp(U) "~ Kp(U) for every open set U C X, since X is regular; hence Gp,x ~/Cp,x. The formula for CHP(X) now follows from Prop. (5.26).
72
Algebraic K-Theory
P r o o f o f (5.26). For any y e X p-1 and x e X p such t h a t x E {y}, we have a natural map ordx~ : k(y)* , Z, defined as follows: let Y = {y} with the reduced structure, and let R = Ox,y. T h e n R is a 1-dimensional Noetherian local domain with quotient a field k(y). Given a E k(y)*, choose a, b E R - {0} with ~ - a, and define ordxu(~x) = e(R/aR) - e(R/bR),
e = length.
One sees (cf. Fulton, Intersection Theory, Appendix A) t h a t this gives a well-defined homomorphism ordxu with integer values. Combining the maps ordxu for all x, y we obtain a m a p
H
ord"
k(y)"
' H
yEXp-1
Z.
xEXp
By definition, the cokernel of 'ord' is the Chow group CHP(X) of codimension p cycles modulo rational equivalence. Hence, we need to show t h a t 'ord' and dl have the same image. Let (dl)xy " k(y)* , Z be the (xy)-component of the differential dl, for each y E X p - l , x E X p. Fix y E X p - 1 , and let Y = {y}. T h e closed immersion Y , X gives an exact functor Azs ~ A4(X), such t h a t ~ i ~ ( Y ) c fl4P-I+~(X) for all i. Hence we have a map of spectral sequences
(5.20)
Eir'i(Y)
, Er
1J+'-P(X),
which increases the filtration degree by p diagram
1; in particular we have a
E~-"-P(X)
,
E~' ' - p ( X ) ~
~
'
E1
T
0,--1
gl(k(vl) = El
(Y)
T
1,--1
(Y)
I.Ixex,, Z
]
ILo,, z
Thus (dl)~u = 0 unless x e Y. Next, if we fix x0 E Y, and let R - Oxo,y, then the fiat map Spec R --, Y induces a contravariant map of spectral sequences, yielding a diagram
Kl(k(y)) ~- E~
--I
Kl(k(y)) ~- E~
d,
I
d,
E~'-'(y) ~
I
H Z xEY 1
TM
E~,-I(R)_
where P0 is projection onto the s u m m a n d corresponding to x0. Hence we are reduced to proving:
5. The K-Theory of Rings and Schemes
73
( 5 . 2 8 ) . Let R be an equicharacteristic Noetherian local domain of dimension 1 with quotient field F and residue field k, and let
Lemma
---, GI(R) --~ g l ( f )
o Ko(k)
~ Go(R)
be the localization sequence associated to the closed immersion Spec k Spec R. Then O: g l ( F ) ~ Ko(k) is isomorphic to o r d : F* L , Z (i.e., there are functorial isomorphisms K I ( F ) ~ F*, Ko(k) ~- Z under which c0 corresponds to ord). P r o o f . By T h e o r e m (5.1), there is a functorial isomorphism
F* ~ ~1 (BGL(F) +) '~ K1 (F), while by T h e o r e m (4.0), there is a functorial isomorphism
Ko(k) ~- g. W i t h respect to these isomorphisms, we show a = i o r d up to a universal choice of signs. We have an isomorphism
R* "~ lrl (BGL(R) +) "~ K1 (R) such t h a t
K,(R) R*
commutes.
K,(F)
~.
l
~
F*
K1 (F) factors through GI(R), we see t h a t
Since K I ( R ) ~
o ( x ) = 0 for x e R" c F * ; ~1~o o r d ( ~ )
= e(R/~:R)
= 0. So it ~ u m ~
to
show O(x) = + o r d ( x ) for all x e R - (R* U {0}), for a universal choice of the sign; we fix such an x. Let ko be the prime field; then there is a homomorphism ko[t] --~ R mapping t to x, where ko[t] is the polynomial ring. Since x -r 0 and x is a non-unit, this is flat. By the naturality of the localization sequence for fiat maps, we have a diagram
,
K,(ko[t])
1
-.
G,(R)
K,(ko[t,~_,])
--~
----*
KI(F)
o ~
Ko(ko)
o,
Ko(k)
--~
and a diagram
K, (ko[t, t-'l) g,(FI
lrl(BGL(ko[t,t-l]) +) ~ ko[t,t-l] *
~1 (BGL(F)*) ~ F*
,
74
Algebraic K-Theory
such that u(t) = x. The map v is induced by the functor sending a k0-vector space V to the R-module of finite length ( R / x R ) | V, and using d~vissage to identify the K-groups of the categories of finite torsion R-modules and of finite dimensional k-vector spaces. Hence under the identifications Ko(ko) - Z, Ko(k) = Z, v is just multiplication by ~ ( R / x R ) = ord(x). Hence it suffices to prove that O(t) - 4-1 in the top row. But from Cor. (5.5) (ii), g l (k0[t, t - l ] ) -~ g l (k0[t]) (9 Ko(ko) with the latter summand being identified with ker(Kl(k0[$,t-1]) ---* Ki(k0)),
t ~ 1:
Under Kl(ko[t,t-1]) ~- ko[t,t-1] *, the summand K0(k0) is thus the cyclic subgroup generated by t, i.e., O(t) is a generator of Ko(ko) -- Z. Hence O(t) -- +1. To check that the sign is universal, we compare with the localization sequence for Z[t] ~-, Z[t,t -1] (which is possible because k0[t] has finite Tor-dimension over Z[t]).
Projective B u n d l e s a n d S e v e r i - B r a u e r S c h e m e s . Let S be an arbitrary scheme, s a vector bundle on S (i.e., locally free Os-module) of rank r, and X = P(~) - Proj(S(s where S(s is the symmetric algebra of s over Os (see (D.58), (D.59)). Let O x ( 1 ) be the canonically defined line bundle on X. We had earlier computed G.(F(E)); our present goal is to prove: T h e o r e m (5.29). If S is quasi-compact, then one has isomorphisms
g (s) for all q > O, given by
r-1 (ai)o~i
f*ai'
where z E Ko(X) is the class of O x ( - 1 ) , and f " X morphism. Equivalently,
Kq(X) Ko(X)|
Kq(S),
S is the structure
aU q >_O.
In the following discussion, let "Ox-module" stand for "quasi-coherent Ox-module", and similarly interpret "Os-module".
Lemma (5.30). a) For any Ox-module J~, Rq f.cF is an Os-module, and isOif q>r.
5. The K-Theory of Rings and Schemes
75
b) For any Ox-module jz and any vector bundle C' on S, we have isomorphisms
(Rq f . Yz) |
~' "~ Rq f . ( y: |
f*g'), for all q > O.
c) For any Os-module Af we have
n~ y.(Ox(~) |
f*~f)
0,
=
Sn(E) | S-"-r(s
if q ~ 0, r -
Af, A r *| s |
1
if q = o
iV', if q = r - 1
(where 9 denotes the dual vector bundle, and we define SIn(C) = 0 for
m<0).
d) If yz is an Ox-module of finite type, and S is al]ine, then yc is a quotient of O x ( - n ) Sk for some n, k > 0 (~" is of finite type if it is locally finitely generated). P r o o f . These are all standard facts about projective bundles. Clearly a), b), c) are local on the base, so we may assume S - SpecA, and E, s ~ are trivial bundles. T h e n R q f . ~ is the Os-module associated to the A-module Hq(x,.T'), since the cohomology of ~- on X can be computed using the (~ech complex corresponding to the standard affine open cover of IF~-1 (see the proof of (D.63)), and the formation of this (~ech complex commutes with localization on S = Spec A (i.e., the (~ech complex for ~r on X ! = X • s Spec A I is obtained from the (~ech complex for 5r on X by tensoring with AI). This proves a), and b) follows since E7 is trivial. To prove c), we note t h a t if ~" = Ox(n), the terms of the (~ech complex are fiat A-modules; hence if Af is the Os-module associated to the A-module N, then
H q ( X , O x ( n ) ~ f*JV') ~ Hq(X, O x ( n ) ) |
N.
To prove d), since S = Spec A is affine, E is associated to a finitely generated projective A-module, so t h a t there is a surjection O~sr --~ s giving an embedding X c IP~- 1 which satisfies Op..-, (1)Ix "~ O x (1) Hence we are reduced to the case X = IP~-1 = IP~-1. If {Ui}0
0. For example, O x ( n ) | f * Af is regular for n >_ 0, and any O s - m o d u l e Af.
76
Algebraic K-Theory
The idea of the proof of (5.29) is to show that any regular vector bundle on X has a canonical resolution by twisted pullbacks of vector bundles on S. L e m m a (5.31). Let 0 of Ox-modules.
~ if:'
" 3z ~
J:"
~ 0 be an exact sequence
a) If :7:'(n) and 3:"(n) are regular, then so is 3Z(n). b) If 3Z(n) and ~r'(n + 1) are regular, then so is ~ " ( n ) . c) If 3:(n + 1) and 3z"(n) are regular, and if f . ~ r ( n ) - - ~ f . 3 : " ( n ) , then 3z'(n + 1) is regular. P r o o f . Immediate from the definition of regularity and the long exact sequence of higher direct images. L e m m a (5.32). If jr is regular, JZ(n) is regular for n > O. P r o o f . It suffices to prove that ~-(1) is regular, by induction. We have the Koszul exact sequence of vector bundles on X T
0
; Ox(-r) | Af*s
~""
-+ O x ( - 1 ) | f * s
,
Ox
,
O.
Tensoring with ~" we obtain the exact sequence r
(.)
0
, ~ ' ( - r ) | A f*E - , - . . - . ~ ' ( - 1 ) | f * e
, Jr
9 0---.
Since j r is regular, Lemma (5.30)(b) (projection formula) yields P
P
so that ~ @ A p f * c -~ ( ~ ( - p ) @/~P f*~)(p) is regular. Splitting (.) into short exact sequences P
0
,z _a
,o,
(with Z~ = O, Zo = ~') we see, by descending induction on p and Lemma (5.31)(b), that Zp(p + 1) is regular for each p > O. In particular Zo(1) = ~(1) is regular. L e m m a (5.33). I f 3 z is regular, the natural map f * f . Y
; ~" is onto.
P r o o f . From the proof of (5.32) above, we have an exact sequence 0
; Z1 ~
~'(-1)
|
f*~"
' ~ " ----4 0
5. The K-Theory of Rings and Schemes
77
where Z1(2) is regular; hence Zl(n) is regular for all n > 2, and R l f . Z x ( n ) = 0 for all n > 1. Hence we have exact sequences
0
,A
(n-1)|
, y.y(n)
,o,
n > l.
Thus the natural map of graded S(E)-modules
f , jr |
S(F.)
~.( ~ f , Jr(n) n>_O
is onto. The lemma follows by taking the associated sheaves on lP(E) (in fact, the lemma would follow from the weaker statement that the above map of graded S(E)-modules is a surjection in sufficiently large degrees). L e m m a (5.34). Any regular Ox-module Y: has a resolution
; O x ( 1 - r) | f*Zr-l(.~) , f*To(5r) , jF ,0
-'-*''"
~
O x ( - 1 ) | f*Zl(ff~)
where Ti(.~') are Os-modules determined up to isomorphism by .~'. Further, ; Ti(~r) is an exact functor from the category of regular Ox-modules to the category of O s-modules. P r o o f . We first prove uniqueness. Given a resolution as in the lemma, since ~ is regular, the sequence obtained by twisting by O x ( n ) ,
0
~.O x ( n + l - r ) ~ f * T r _ l ( f f ~) ---~...--~ Ox(n)@f*To(J] ~)
, ff~(n)
,0
consists of O x - m o d u l e s which are acyclic for R i f , , i > 1, provided n >_ O. Applying f , thus yields exact sequences 0
~" s n + l - r ( ~ )
~ T r - l ( ~ ) --* --" --* Sn(s | T0(~)
; f , JcCn)
;0
(as before we use the convention Sn(E) = 0 for n < 0). In particular, for 0 < n < r we have 0
' Tn(ff~')
~" ~ @ O s
Tn-l(~')
"'r
" " " -'r
f.~'(n)
~ O.
This shows t h a t Tn (Jr) is uniquely determined, by induction, up to isomorphism. This also tells us how to define Tn(Jz). We define inductively a sequence Zn = Zn(2F) of Ox-modules, and Tn = Tn(Jc) a sequence of Osmodules, by taking =
T. = y.(Z._l(n)),
and letting Zn = ker(Ox ( - n ) | f ' T n ' Zn- 1). Clearly Zn, Tn are additive functors. We will prove, by induction on n, that Zn(n + 1) is regular, this being given when n = - 1 since ~ is regular.
78
Algebraic K-Theory If Zn-1 (n) is regular, then by (5.33)
f*Tn = f * ( f . Z n - l ( n ) )
, Zn-~(n),
and by definition the kernel of this m a p is Zn (n). T h u s we have an exact sequence
0
(*)n
' Zn(n)
; f*Tn
; Zn-l(n)
; 0 ....
By (5.31)c) and (5.32), Zn(n + 1) is regular, giving the inductive step; the sequences O x ( - n ) | (*)n splice t o g e t h e r to give the resolution in the s t a t e m e n t of the lemma (modulo the injectivity of the first m a p on the left). We also obtain
f . Z n ( n ) = 0 for all n > O,since f . f * T n = Tn = f . Z n - l ( n ) . F r o m the sequence Ox(1)| we see t h a t if Zn-1 and Tn are already known to be exact functors on the category of regular O x - m o d u l e s , t h e n so is Zn(n + 1) (and hence also Zn); since f . is exact on the category of regular O x - m o d u l e s , Tn+l = f . Z n ( n + 1) is also an exact functor. Lastly, we show Z r - 1 = 0. F r o m O x ( n ) | ( . ) , + q we have exact sequences
Rq-l f.Zn+q-l(n)
' Rq f.z,.,+q(n)
, Rq f . O x ( - q ) | f*Tn+q.
from f.Z,.,(n) = O, these sequences inductively show t h a t Rqf.Zn+q(n) = 0 for all q,n > O. T h u s R q f . Z r _ l ( r - 1 - q) = 0 for all q > 0 (this is a special case of the previous s t a t e m e n t if r - 1 - q = n _> 0, but is trivial if r - 1 - q < 0, i.e., if q > r - 1). This means precisely t h a t Z r _ l ( r - 1 ) is regular. But f . Z r - l ( r - 1 ) = 0, so by (5.33), Z r - l ( r - 1) = 0, i.e., Z r - 1 = 0. Starting
Assume that S is quasi-compact. Then for any vector bundle jr on X , there exists an integer no such that for all Os-modules Af and n >_ no, we have
Lemma
(5.35).
(a) nqf.(JZ(n) | f*Af) = 0 for all q > O. (b) f . (.T'Cn) | f*JV') ~ (f,.T'Cn)) |
(r
jV"
,, v , a o , - b ,ndt, on S.
P r o o L B y quasi-compactness, the existence of no is local on the base S, so we m a y assume S is affine. Then by (5.30)(d), ~ is a quotient of a vector bundle s -- O x ( - m ) r for some m, k > 0, and we have an exact sequence of vector bundles 0 ,~'---*s -~" ' , 0 .
5. The K-Theory of Rings and Schemes Further, (5.30) implies the lemma for s 0
,
|
79
Since
|
,
,0
is exact, we have a sequence
Rq f,(L(n)
@ f*.N')
Hence (a) follows
; Rq f,(.T'(n)
by descending
| I*N')
,
induction
Rq+I f,(.T"(n)
on q, being
| f*.N').
trivially
valid
for
q > r (note t h a t 9r' also satisfies the hypothesis of the lemma). Using (a), if n > no, we have a d i a g r a m with exact rows (for any O s - m o d u l e A/') (f..T" (n)) | .Af
0 --, f . ( ~ ' ( n ) | f*AO
v
(f.E(n)) |
,
f.(s
~
@ f*AO
-*
(f..~'(n)) | .Af
~
0
/.(Jr(n) | f*JV') --* O.
Hence u is onto; a similar argument applied to ~ t shows t h a t u t is onto, so t h a t u is an isomorphism. Again, the same argument applied to 9~' shows u' is an isomorphism. T h u s ker v = Tar ~ ( f . i F ( n ) , N ' ) = O, for any Af, i.e., f.ffZ(n) is a flat O s - m o d u l e . Since it is a quotient of f . E ( n ) , it is of finite type; applying a similar argument to ~ , we see t h a t f . ~ ( n ) is finitely presented. Since a finitely presented flat module over a ring is projective, f.ff:(n) is a vector bundle. L e m m a ( 5 . 3 6 ) . / f ~- is a vector bundle on X with R q f . J : ( n ) = 0 for all q > O, n > O, then f . J r ( n ) is a vector bundle on S f o r all n >_ O. P r o o f . Since the assertion is local on S, we may assume S is affine; now by (5.35)c) the result holds if n > no. Applying the functor 7-lOmox ( - , 3~(n)) to the Koszul exact sequence in the proof of (5.32) yields an exact sequence , .T'(n)
0
----, .T'(n 4- 1) C~ f ' C * |
~, .T'(n + 2) --+ ... --, ~'(n + r) ~ A f*~*
.~0.
For n > 0 all of these sheaves are acyclic for R q f . , q > O; hence on applying f , we have an exact sequence r
0 --, f..T'Cn) --4 (f.JrCn + 1)) |
E* - , . - .
--, (f.:FCn + r)) |
A E* --, O.
Hence by descending induction on n, f . ~ : ( n ) is a vector bundle on S for n>0. m
L e m m a ( 5 . 3 7 ) . /jr ~" is a regular vector bundle on X , then Ti(J:), 0 < i < r are vector bundles on S.
80
Algebraic K-Theory
P r o o f . As in the proof of uniqueness in (5.34) we have sequences o ----, T ,
, C|
T,_~ --,...
--, y , J r ( n ) ~
O,
0 <_ n < r.
Hence the result follows from (5.36), by induction on n. Recall t h a t P ( X ) denotes the category of vector bundles on X , and K q ( X ) = Kq(7)(X)) by definition. Let 7~n C 79(X) be the full subcategory consisting of bundles ~" satisfying R~f..T(k) = 0 V i > 0 and for all k > n. Let T~n C 79(X) denote the full subcategory of bundles ~ such t h a t j r ( n ) is regular. Both :Pn, 7~n are closed under extensions in T~(X), and so are exact categories; we have inclusions :Pn-1 C :Pn, T~n-1 C 7~n and 7~n C Pn and further tAT~n = lim 79n = 7~(X) = tAT~ = lim 7Zn. ---4
----r
( 5 . 3 8 ) . For all n, the inclusions 7Zn C 7),, C T'(X) induce iso-
Lemma
morphisms Kq(T~n) "~ Kq(Tan) ~- K q ( P ( X ) ) = K q ( X ) . P r o o f . For any vector bundle ~', we have an exact sequence, which is functorial in jc,
o
, .r
, ~-(i) | I'E* - , . . .
~ .r(,-) | I* ] ~ E"
, o.
For each p > 0, Y ~-~ ~'(p) | f* Av E* gives exact functors Cp : ~)n "---~ P n - - i ,
~)p : T~n 9 ; T ~ n - 1 .
If i : ~Pn-1 - - ' ~Pn is the inclusion, then we have an exact sequence of functors (id = identity) 79n ~ 7~n, 0
*ida, --~ior
,ior
and an exact sequence of functors T~n_1 0 ----* idr=_l ---+ r
o i
~-'"
'ioCr---*0,
* 79n- 1, , . . . ---+ C r o i
Thus i. " Kq(79n_l) , Kq(P,.,) and ~-~p>0(-1)p-l(r Kq (Pn_ 1) are inverse maps. Similarly, if j : T~n-1 ~- T~n is the inclusion, then
j . : gq(nn-1) and
:~(-1)~-~(r
.K~(n.)
*0.
" Kq(Pn)
'
". gq(n,.,)
: K~(n._i)
p>O are inverse maps. T h u s Kq('Pn) ~- lim Kq('Pn) "~ Kq('P(X)) = K q ( X ) , and similarly
K q ( n . ) ~ aim K~(nn) ~ K~(~'(X)) = Kq(X).
5. The KoTheory of Rings and Schemes
81
P r o o f o f ( 5 . 2 9 ) . Let un " K q ( S ) ' Kq(PO) ~- K q ( X ) be the homomorphism induced by the exact functor Af H O x ( - n ) | where 0 < n < r (these inequalities ensure t h a t the functor has values in Po, by (5.30)). Let
u" Kq (S) $r
; Kq (X) be given by r--1
9
n---0
T h e n Theorem (5.29) states precisely t h a t u is an isomorphism. If 9r E P0, then f.:F(n) is a vector bundle on S for any n > 0, and R~f.Y:(n) = 0 V i > 0, by (5.36). Hence we have maps vn " Kq(7~o) Kq(S), induced by Y ~-* f.:F(n), for each n ___0. The composite vn o u m 9 Kq(S) ---, Kq(S) is induced by the functor
.hf ~-~ ( f . O x ( n - m)) | Af { = 0 =Af
if m > n i f m = n.
Thus if v = ( v o , . . . , V r _ l ) ' K q ( 7 9 o ) , Kq(S) ~ , then v o u is described by a triangular m a t r i x with all diagonal entries equal to 1; hence v o u is invertible, and so u is injective. On the other hand, by (5.34) and (5.37) we have exact functors 71, 9 T~ ----4 79(S), for 0 < n < r, which induce maps t , 9 K q ( 7 ~ ) ". Kq(S). Let t = (..., ( - 1 ) ~ t , , . . . ) " K q ( X ) ~. Kq(S) ~ . The exact sequence
0
, O x (1 - r) | f * T r - 1 --*"" --* f ' T o ~
~"
;0
can be interpreted as an exact sequence of functors T~ --~ P0, so t h a t u o t . Kq(RO) } Kq(79o) equals the isomorphism induced by the inclusion T~ C 790. Hence u is surjective. II~1 o v e r a R i n g . Let A be a not necessarily commutative ring, t a (commuting) indeterminate and let i + : A[t]
, Air, t - ' l ,
i _ : A[t-1]
, Air, t - ' ]
be the natural inclusions of the polynomial ring in the Laurent polynomials. We define ~A(IP~) to be the Abelian category of triples M = (M+, M_,O) where M+ is a left A[t]-module, M_ a left A[t-']-module, and 0 : i ; M+ --. i*_ M_ is an isomorphism of A[t, t-1]-modules. Similarly define P ( P ~ ) to be the full subcategory of j~4(IP~) consisting of triples M with M+ e P(A[t]), M_ e P(A[t-']).
Theorem (5.39). Let hn " 7~(A) P
(Air] |
; ~(P1A) be given by
P, Air
|
P, 0.)
82
Algebraic K-Theory
where On = multiplication by t" on A[t, t -1] | phisms gq(A) .2
P. Then we have isomor-
~-, Kq(V(IPIA)), (x, y) H (ho).X + ( h , ) . y ,
and we have the relations ( h , _ l ) . - 2 ( h , ) . + ( h n + l ) . = 0 for all n.
P r o o f . For any M = (M+, M _ , 0) define M ( n ) = (M+, M _ , t-nO), V n.
T h u s hn is induced by P ----. h o ( P ) ( - n ) . For any triple M = (M+, M _ , 0) define X0, X1 e Hom(M, M (1)) by XolM+ = 1 =
XolM_ =t
t, x, IM_ =
t
t
(here 't' denotes multiplication by t). Then one checks easily that there is an exact sequence, functorial in M, 0
$2
,M(m)~M(m+l)
~,M(m+2)~O
where ~ = (Xo, X1) and 13(x,y) = X l x - Xoy. In particular taking m - 1 - n and M = ho(P), P e TO(A), we have an exact sequence of functors V(A)
,
0
} hn+l ~
h~n2 ---"* hn-1 -
}0
giving the relation (hn-1). - 2(hn). + (hn+l). = O. Next, for any triple M = ( M + , M _ , O ) , we define f . M , RI f . M A4(A) to be the cohomology modules of the 2-term complex 0
.~ M+ ~ M _
~.~ i*_ M _
E
-. 0,
where r y) = 0(1 | x ) - 1 | y. We now define a triple M to be regular if R l f . M ( - 1 ) = O. With this definition, one can check t h a t the various steps in the proof of (5.29) go through in the given situation also. T h e details are left to the reader. S e v e r i - B r a u e r S c h e m e s . Let S be a scheme, and let X be a SeveriBrauer scheme over S of relative dimension r - 1. By definition X is locally isomorphic to IP~-1 as an S-scheme, in the ~tale topology on S, i.e., for some faithfully flat ~tale morphism S' ~ S, if X ' = X x s S ', t h e n X ' ~ IP~, 1 as S'-schemes. Let f 9X , S be the structure map. If there exists a line bundle s E P i c X which restricts to O ( - 1 ) on each geometric fiber, then X = IP(C), where C = f . s In general, s will exist only ~tale locally on S. However, there is a canonically defined vector
5. The K-Theory of Rings and Schemes
83
bundle ff of rank r on X , which restricts to O ( - 1 ) r on each geometric fiber of f. T h e idea is as follows (compare (D.70)). Let Y = IP~-,, and let GLr,s act on O sr in the usual way. T h e n the induced action on Y = IP(OsS t ) factors through the quotient group scheme G = PGLr, s = GLr, s / G m , s . Since Gm,s acts trivially on O y ( - 1) | f * f . O y ( 1 ) ~- O y ( - 1 ) Cr, G acts on this vector bundle, compatibly with its action on Y. Since X / S is 4tale locally on S isomorphic to Y / S , and G is the group scheme of automorphisms of Y / S , we know t h a t X = Y x c T, where T is a torsor (principal G-bundle) for G over S which is 4tale locally trivial, and Y x G T is the associated fiber space over S with fiber Y (i.e., X = (Y x T ) / G where G acts by g(y, t) = (gy, gt)). At any rate, by descent theory, O y ( - 1) | f* f . O y (1) descends to a vector bundle ff of rank r on X. The construction of , f is compatible with base change; further, ff = O x ( - 1 ) | f*E if X = ]P(E) for a vector bundle E on S. In general, there is a faithfully fiat 4tale morphism g" S' ~ S giving a Cartesian square
X'
S'
g' '~
X
,
S
g
such t h a t X ' "~ = P~71 , and further g'*ff ~ O x , ( - 1 ) | f'* f~Ox,(1). Let .A be the sheaf of non-commutative Os-algebras
.,4 = f . End o x ("7)op (where the superscript "op" denotes the opposite algebra). g* f . =" I- ,. g ,. 9 Hence
As g is flat,
g*~4~ ~ ft. Endox , ( O x , ( - 1 ) | f*.f, Ox,(1)) "- f :'.M (Ox )= where M r denotes the algebra of r x r matrices. Hence j t is an A z u m a y a algebra of rank r 2. Further, the natural map
y*.a = l * Y.
__, end(Y)oo
is an isomorphism, as is easily seen by applying g'* (note t h a t g' is also a faithfully fiat 6tale map). Conversely, if ,4 is an Azumaya algebra over S of degree r, so t h a t the underlying sheaf of O x - m o d u l e s of ~4 is locally free of rank r 2, let f :X ~ S be the scheme over S parametrizing locally free quotient O smodules of rank r of ,4, whose kernels are right ideals of J[, so t h a t X is a closed subscheme of the Grassmannian of locally free quotients of rank r of Jr. T h e n X is a Severi-Brauer scheme, and the vector bundle ff on
84
Algebraic K-Theory
X is identified with the dual of the restriction of the universal quotient bundle on the Grassmannian. These facts are checked by passing to S', where g" S' ~ S is faithfully flat and &ale, and g*,4 ~ .A4r(OS,). Let fin = fl| An = A | so that J n is a vector bundle of rank r n on X, and An is an Azumaya algebra on S of rank(m) 2. Then .An f . ~ n d o x (fin) ~ and f*3", ~- s (fin) ~ (as can be verified by pulling back to X ' ~ P~s-;1). Let 79(.An) be the exact category of vector bundles on S which are left .A,-modules. Since J , is a right f*(.4,)-module, which is locally (on X) a direct summand of f * ( A , ) , we have an exact functor
Cn "79(.An)
, "P(X),
M ~-, ..Tn |
f* (M) .
Define K , ( A n ) = K,(79(An)). T h e o r e m (5.40). If S is quasi-compact, we have isomorphisms r--1
r--1
IIK,(A.)
K , ( X ) , (xn)o<,
n=0
n=0
P r o o f (Sketch). This is again just a modification of the proof of (5.29). Fix g 9 S' ~ S, a faithfully flat &ale morphism such that X ' - X • s S' IP~-I and let g " X ' ~X f " X ' ~ S' be the resulting maps We define an Ox-module Jr to be regular if g ' * ~ is regular on X ' . For regular ~', define inductively 9
'
~
Z _ I ( ~ ) = :7:, Tn(]:) = f . TlOmox (Jn, Zn-,(.7:)), and
Zn(.T') = ker(ffn | f*Tn(.T')
' Zn-I(.T')) (where if0 = O x ) .
Then we obtain a resolution (~)
0
--* "'" ~ f*To(.T')
' Jr-1 |
, .T" ,
0...
where Tn (~-) has the natural left A~-module structure; this resolution pulls back to that of (5.34) on X ' for g'*~. Similarly, the canonical epimorphism ,.7" --~ O x gives rise to a Koszul complex T
o
,AJ---,...-,y
,Ox
,o
which pulls back to the Koszul resolution on X ' used in (5.32), (5.36), (5.38). In particular, the analogue of (5.38) is valid: let T~n (X) C 79, (X) C P ( X ) be the full subcategories of bundles ~" such that g'*~" lies in the corresponding subcategory of P ( X ' ) ; then
K , ( T ~ ( X ) ) ~- K,(T'n)) ~ K,(79(X)) = K { ( X ) .
5. The K-Theory of Rings and Schemes
85
Now let un : K~(.An) , K~(X) be given by un = ( r where C n ( M ) = fin | Let u 9 IJ0
N ~ f . 7"lom(flm, J ~ | B u t we c o m p u t e t h a t
7"lOmox(ffm, ffn |
N) = ?'lOmox(ffm, ffn ) |
f~ N
{ =bO yf , aN p pi f m l =
(. . ., ( - ) " t , , .
. .) : K , ( X )
;
II
K,(A.),
O
then the resolution (a) obtained above shows t h a t u o t is the isomorphism K i (T~) , Ki (7)0) induced by T~ C P0. Hence u is surjective. There is a different technique for proving lemma (5.34), which is perhaps longer t h a n t h a t given earlier, but yields an explicit description of the functors Ti. This point of view is closely related to the following paper of A. A. Beilinson: "Coherent sheaves on pn and problems of linear algebra", J. Funct. Anal. Appl. 12 (1978), pp. 214-215. T h e idea is t h a t on X = P(s we have an exact sequence of locally free O x - m o d u l e s 0 --~ S ~ f * e - - 4 0 x ( 1 ) --~ 0 and a dual exact sequence 0 --~ O N ( - 1 ) ---* f*~* --* S* ~ O. Hence if pi " X Xs X -4 X , i = 1, 2 are the projections, then one has a surjection
P~f*s | P2J c
--, p l O x ( 1 ) | p2 8 .
Now if q : X • s X -~ S is the structure morphism, then there is a n a t u r a l isomorphism
q* End(E) "~ = pl*f* E | p2*fiE* , and the identity endomorphism of E yields a global section E F(X •
X,p~Ox(1)@p~S*).
86
Algebraic K-Theory
We claim that the scheme of zeroes of ~ is the diagonal A x C X x s X. This is easily proved locally on S using homogeneous coordinates, for example. Or else, it can be deduced from the universal property of a projective bundle (for any S-scheme g 9T --* S, there is a bijection between S-morphisms T --* ]P(s and isomorphism classes of invertible quotient sheaves of g'C). We leave the details to the reader. Granting this, we deduce that the structure sheaf of the diagonal has a resolution by a Koszul complex i
9. . - - , A ( P ; O x ( - 1 ) |
p~S) - - , . . . p ~ O x ( - 1 ) |
p~S ---, O X x s X --' O a x ~ O.
Further, if Lr is the ideal sheaf of the diagonal, then Z / Z 2 ~ (p~ O x ( - 1) | p~S) | O a x
(this is a general property of ideals resolved by a Koszul complex). But Z / Z 2 ~ f~lx/s, so that S ~ f~c/s(1), and our Koszul complex assumes the form *' (i)--,.-. "'" ~ Pl9 O x ( - i ) | p2f~x/s 9
9
---' p l O x ( - 1 )
1
| P2f~X/S(1) ~ O X x s X ~ O A x ---~O.
Now (5.34) follows from: P r o p o s i t i o n (5.41) Let f " X --, S be as above, Y= an O x - m o d u l e . (a) There is a convergent spectral sequence of O x - m o d u l e s (the Beilinson spectral sequence), concentrated in the range p < O, q > O,
with E~q --
(b)
f 0 I. .1"
if (p, q) # (o, o) if (p, q) = (o, o).
Suppose R' f . (f~Jx/s(J) | :7=) = 0 for aU i > O, j > O. Then there is resolution of ~ given by the E1 terms along the line q = 0
a
9..--+ O x ( - i ) 0 f*Ti(.7~) --*"" --* f*To(fl:) ~ if" ~ O, where
~,(~=) =
~:).
s. (a.~/s(O |
(~) %
i > O,j > O, and Ti is an exact functor on the category of regular O x - m o d u l e s for each i > O.
5. The K-Theory of Rings and Schemes
87
P r o o f . (Sketch) We have a canonical identification p~.(O~ x | =~ jc, and R i p l . ( ( 9 ~ x | ~ ) = 0 for i > 0. On the other hand, the resolution of the diagonal yields a resolution on X x s X
9 .. -~
.~o~(-~) |
(~'x/s(~) |
9.. --* , ~ O x ( - 1 )
| p~
~) -~'"
(a~/s(1) | Y) -* p~-
(O,,x
| p.~.T')
O.
Hence there is a spectral sequence for the hyperderived functors of f . , concentrated in degrees - r <_ p _< 0, r > q >_ 0 (i.e., set p = - i in the resolution)
E~ 'q = R q p l . ( p ; O x ( p ) | p~(gtxP/s(-p) | ~ ) )
=~ RP+qpl. ( O a x | p ~ )
.
Now Pl " X x s X --, X is itself a projective bundle, associated to f ' E , so t h a t by (5.30)(b), we may rewrite the E l terms as
E~ ,q
= Ox(-p)| n~p,.p~ (nx%(-p)|
~).
Further, since f is fiat, we have a canonical isomorphism
This gives the expression for the E1 terms in the proposition. The vanishing condition in (b) means that E[ 'q the spectral sequence is concentrated on the line q - 0. E~om'~ : 0 for all p > 0, so that the complex E~-p'~
statement of the
0 for q > 0, and Further, E ~ p'~ is a resolution of
E~ '~ = --ooE~176 = ~'. Now we prove (c). One first proves t h a t if ~" is regular, then
nJs. ( ~ / ~ ( ~
+ ~ - J)|
.*) = o v ~ _> o, ~ > o.
This is done by descending induction on i; the case i - r is the definition of regularity, since fffx/s ~ O x ( - r - 1). By taking exterior powers of the Euler exact sequence 0 --~ f ~ / s ( 1 ) - , f * E ---, O x ( 1 ) --* 0 (which we constructed earlier, when we identified 8 with f ~ / s ( 1 ) ) , obtain exact sequences of locally free Ox-modules
we
i
--4
---,
f~XlS(Z) ---, O.
Tensoring with ~ ' ( - j ) and applying R J f , , we obtain an exact sequence of Os-modules
(
'
)
88
Algebraic K-Theory
where of ~',
RJf. ( f * ( A ' e) | ~ ( - j ) ) ~ A' F-.| and RJ+lf. (fl~/s(i) | 3 r ( - j ) ) = 0
= o by the regularity by induction.
In particular, setting i = 0, we see that 3r(1) is regular, and hence ~'(i) is regular for all/_> 0 (compare (5.32)). Hence RJf. ( f ~ / s | JZ(k)). = 0 for i >_ 0, j > 0, and k >_ i+ 1 - j . This includes the case k = i, which is what was needed. The exactness of Ti on the category of regular Ox-modules is immediate from the vanishing of Rlf.Q~/s(i) | 3~ for regular ~'. The above argument for (c) was shown to me by Donu Arapura. Resolutions for the diagonal are known, which yield computations of the K-groups, in certain other cases (Grassmannians, their twisted forms, and quadrics); this point of view is seen in papers of Kapranov, Panin and a joint paper of mine with Levine and Weyman. Panin noted that in fact for K-theoretic consequences, one only needs that the class of the diagonal in K0 has a certain form, which would be implied by the existence of an appropriate resolution of the structure sheaf of the diagonal. A recent reference putting much of this work in perspective is the following paper of Panin: A.I. Panin, On Algebraic K-Theory of Generalized Flag Fibre Bundles and Some of their Twisted Forms, in Advances in Soviet Mathematics, Vol. 4, ed. A. A. Suslin, Amer. Math. Soc. (1991).
6. P r o o f s o f t h e T h e o r e m s of C h a p t e r
4
We first prove:
There is a natural isomorphism K0(C) ~ 7rl(BQC, {0}) for any small exact category C and null object 0 E C. Theorem
(4.0).
Here, K0(C) is the quotient of the free Abelian group on the objects of C modulo the relations [M] = [M'] + [M"] for all exact sequences 0
~M'
,M
~M"
t0
in C. We begin by proving a lemma. We use the notation and terminology of Chapter 3. ( 6 . 1 . ) . The category of covering spaces of the classifying space BC of a small category C is naturally equivalent to the category of functors F : C ~. Set (where Set is the category of sets) such that F(u) is a bijection for any morphism u of C.
Lemma
ProoL
Let p - E
.~ BC be a covering space. For any object X E C, let
E ( X ) be the fiber over X E BC (where X is regarded as a 0-simplex in NC, and hence determines a 0-cell in BC). Given a morphism u : XI .~ X2, we may regard u as a 1-simplex in NC, which determines a p a t h B u in BC joining X1 to X2. Since p is a covering, it has the unique p a t h lifting property, which gives a bijection ( B u ) . : E(X1) ~ E ( X 2 ) , by associating to a point y E E ( X I ) the second end-point of the unique path in E which lifts B u and begins at y. Hence X ~ E ( X ) , u ~-. (Bu). determine a functor C .~ Set carrying all arrows of C into bijections. Conversely, if F : C ~. Set is a "morphism inverting" functor (i.e., F(u) is a bijection for each u E Mor(C)), let F \ C be the category of pairs ( X , x ) with X E ObC, x E F ( X ) , where a morphism ( X , x ) ( X ' , x ~) is a m o r p h i s m u : X .~X ~ such t h a t F(u)(x) = x'. T h e forgetful functor F \ C ~ C, (X, x) ~-~ X gives a map on classifying spaces PF :
90
Algebraic K-Theory
B ( F \ C) , BC, with fibers p F l ( X ) = F ( X ) for any object X e C. We claim t h a t PF is a covering space. If we prove this, then for any morphism u" X , X ' in C and any x e F ( X ) , if x' = F ( u ) ( x ) , then u determines a morphism ( X , x ) ~ ( X ' , x ' ) in F \ C, which gives the unique p a t h in B ( F \ C) lifting B u and beginning at x E p F I ( X ) . Thus the above two constructions are inverse to each other, and give the desired equivalence of categories. To check t h a t PF " B ( F \ C) : ; BC is a covering space, we use the criterion (A.48) of Appendix A. Thus, we m u s t show t h a t the map of simplicial sets N ( F \ C) ; NC is a simplicial covering, i.e., if A(n) is the simplicial set A(n)(p) = H o m a ( p , n ) , (so t h a t [A(n)[ = An, the s t a n d a r d n-simplex), then given any diagram of maps of simplicial sets A(O)
1
A(n)
;
----4
N ( F \ C)
1
NC
we must show t h a t there is a unique m a p A ( n ) : , N ( F \ C) (of simplicial sets) m a k i n g the diagram commute. Of course, a m a p a 9 A(n) ; NC is just an n-simplex a E NnC, so we must show t h a t if a E NnC is an n-simplex of NC, ao E No(F \ 17) a 0-simplex lying over the ith vertex of a, then there exists a unique n-simplex T E N n ( F \ C) which maps to a, such t h a t a0 is the ith vertex of T. Assume t h a t a is given by the diagram in C M0
ul ,
M1
u2
; - - - -u-. 4 Mn;
the ith vertex of cr is given by the object Mi, so t h a t a0 is given by an object (M~,xi) e F \ C, where x, e F(M~). We have bijections
F(Mo) F(u,) ; F(M1) F(u2) ~ F(M2)
; ... F(u.)~. F ( M n )
giving composite bijections f j " F ( M j ) ; F ( M i ) for each j (f~ is the identity) such t h a t f1 = f j + l o F ( u j + l ) . Let xj e F ( M j ) be the unique element satisfying f i ( x j ) = x,. Then clearly x, = f j ( x j ) = f j + l ( f ( u j + l ) ( X i ) ), so t h a t x j + l = F(uj+l)(Xj). Thus we have a d i a g r a m in F \ C, giving re Nn(F\C),
(Mo,xo) r,, , ( M l , x l ) a2 , ' "
- (Mn, xn) "",
where fii is the morphism induced by uj. One sees at once t h a t v is the unique n-simplex lifting a whose ith-vertex is ( M i, xi). This proves t h a t p F ' B ( F \ C) , BC is a covering, and finishes the proof of (6.1). Now let C be a (small) exact category. Recall t h a t if M, N E C, an arrow i 9M , N is called an admissible monomorphism if we have an
6. Proofs of the Theorems of Chapter 4
91
exact sequence in C (for some P E C) 0 Similarly, q : M sequence
;M
i.~N
,P
,0.
.~ N is an admissible epimorphism if t h e r e is a n e x a c t
0
;P
,M
q,N----,O
for some P E C. We write i : M ~-4 N to denote t h a t i is a n admissible m o n o m o r p h i s m , a n d q : M -~ N to denote t h a t q is an a d m i s s i b l e epimorphism. Next, we recall t h e construction of morphisms in t h e c a t e g o r y QC. A m o r p h i s m M --, N in QC is an equivalence class of d i a g r a m s M ~-- M ~ ~-, N , where M ~- M " ~ N is an equivalent diagram if a n d only if t h e r e is an isomorphism u : M ~ --, M " m a k i n g the following d i a g r a m c o m m u t e : M !
\/
N
M" C o m p o s i t i o n of m o r p h i s m s is defined as follows. Given d i a g r a m s M ~M ~ ~-, N , N ~- N ' ~-, P t h e composite morphism M --, P in QC is represented by the d i a g r a m M ~- M ~ x Iv N ~ ~-* P. We have t h e d i a g r a m (where t h e square is a pullback)
Nl
M t • jv N '
M'
~-~
~
P
N
M In particular, if i : M ~-, N , we have an associated arrow i! : M --, N in QC, given by M ~- M ~-, N , where 1M is the identity of M . Similarly, 1M
i
if q : M --~ N , we have an associated arrow qt : N --, M in QC, given by
N,--M~--, q
IM
M.
If f : M --, N is an a r b i t r a r y arrow in QC, given by t h e d i a g r a m M ~-- M ' ~-* N , t h e n in fact f = i! o qt as is i m m e d i a t e from t h e a b o v e q
i
description of composition of m o r p h i s m s in QC. We can form t h e p u s h o u t
92
Algebraic K-Theory
s q u a r e in C:
i ~-~
M t
N
it
M ~-* N' w h i c h will also be Cartesian (i.e., pullback); such a square is called bicartesian. T h e n we claim t h a t f = q a o i~ in QC, from the d i a g r a m giving t h e c o m p o s i t i o n on the right: M' ~ IN ~-~ N N
M
it ~
N p
M T h u s , t h e assigments i ~-, i,,9 q ~ q! (i) if i, i ~ are composable admissible d o m a i n i) t h e n (i oil)! = it oi~; similarly, e p i m o r p h i s m s , then (q o qt), = q,! o q~'. (ii) if i M' ~-,
q~
have t h e following properties: m o n o m o r p h i s m s (i.e., r a n g e i ~ = if q,q' are composable, admissible
N
~q'
M ~-+ N' is a b i e a r t e s i a n square, where i, i' are admissible m o n o m o r p h i s m s , q, q' are admissible epimorphisms, t h e n i! o q' - q': o i~. In fact, (i) and (ii) characterize QC, in the following sense. Lemma
( 6 . 2 ) . Let C be an exact category, I) a category. Assume that
(i) for each M e C, we are given an object F ( M ) e T), (ii) for each admissible mono i : M ' ~ M , we are given an arrow FI(i) : F(M') ~ F ( M ) , such that Fl (i o i') = Fl (i) o F1 (i') if i, i' are composable; similarly, for each admissible epi q : M --. N , we are given F2(q) : F ( N ) ~ F ( M ) , such that F2(q o q') = F2(q') o F2(q) if q, q' are composable (iii) if
i
M'
~-~
M
~-+
i'
N
N'
6. Proofs of the Theorems of Chapter 4
93
is a bicartesian square, then F1 (i) o F2(q) = F2(q') o F1 (i').
Then there is a well-defined functor F " QC ~ 1) given by M ~-~ F ( M ) , ( M ~-- M ' ~-, N ) H F1 (i) o F2(q). q
Proof.
If M
i
~ - M ~ ~-, N , M q
i
~- M "
~-~ N are equivalent d i a g r a m s
ql
ii
giving a m o r p h i s m M ---4 N in QC, we have an isomorphism u " M ~ M " such t h a t q - ql o u, i - i l o u. Regarding u as admissible epi, we get F~(q) = F 2 ( u ) o F2(ql), and regarding u as admissible mono, we get F1 (i) = F1 (il) o FI (u). F r o m the bicartesian square Ml
~-~
M"
M tt
we have Fl (u)oF2(u) -- F2(1M,, )oF: (1M,, ) -- 1F(M,,). Hence Fl (i)oF2(q) -F l ( i l ) o F2(ql). T h u s , Fl(i) o F2(q) d e p e n d s only on t h e arrow in QC, a n d not on t h e p a r t i c u l a r d i a g r a m which represents it. Next, if M ~- M ' ~-, N , N ~- N ~ >-4 P are given a n d M ~ql
~1
q2
is
q
M ~ x jv N ~ >-4 P represents the composite arrow in QC, we have a d i a g r a m i (where t h e s q u a r e is bicartesian) i
M I x N N'
~-~
M'
~-*
il
N'
i2
~-,
P
N
M and q = ql o q', i = i2 o i'. T h e n
F1 (i) o F2(q) = F1 (i2) o F1 (i') o F2(q') o F2(ql) = Fl(i2) o F2(q2) o El(J1) o F2(ql). T h i s proves t h a t ( M
~- M ' q
~-, N ) ~-~ F l ( i ) o F2(q) is c o m p a t i b l e w i t h i
composition in QC, and so yields a well-defined functor QC proves t h e l e m m a .
.- :O. T h i s
C o r o l l a r y ( 6 . 3 ) . For any exact category C, there is an i s o m o r p h i s m QC ~QC ~ .
94
Algebraic K-Theory
P r o o f . For any arrow f in C, let f be the corresponding arrow in C ~ T h e n if i 9M ~-, N is admissible mono, i 9N --~ M is admissible epi, and if q" M --~ N is admissible epi, q = N ~-, M is admissible mono. If i
M'
~
N
M
~-,
NI
i'
is bicartesian, then
q - I
N'
~-~
N
M
~-, q
MI
is bicartesian in C~ Thus i,9 o qt H q,. ~ o -i '! gives a functor (which is the identity on objects) QC ~ QC~ inducing a bijection Horn Qc (M, N) ---, n o m Qco, (M, g ) . This is the desired isomorphism. P r o o f o f (4.0). Let C be an exact category, 0 E C a null object. T h e category of covering spaces of B Q.C is equivalent to the category 3c of functors F ' Q C " Set such t h a t F(u) is a bijection for every arrow u of QC. Let ~ ' C ~ be the full subcategory consisting of functors F 9QC Set w i t h F ( M ) = F(O), F(i!) = 1F(0) for any admissible mono i " M ' ~-, M in C. If F E 5r is an a r b i t r a r y functor, let F E ~ ' be the functor given by /~(M) = F ( 0 ) , and if M q~- M ' ~-, ~ N represents an arrow u " M --, N in QC, let F ( u ) = F(iM,,.) -1 o f ( q !) o F(iM~) (where for any M e C, we have iM " 0 ~ M, qM " M ;~ 0). Clearly M H F(iM!) gives a n a t u r a l transformation F --, F which is an isomorphism of functors, since F(iMI) is an isomorphism in the category Set. Thus every object of ~" is isomorphic to an object of ~c,, so t h a t ~ is equivalent to ~ ' . It suffices to show t h a t ~ ' is equivalent to the category of Ko(C)sets (a K0(C)-set is a set on which Ko(C) acts t h r o u g h permutations). Indeed, by (6.1) and C h a p t e r 3, Ex. (3.10), t h e category of coverings of BKo(C) (the classifying space of the group Ko(C)) is clearly equivalent to _
,
,
the category of K0(C)-sets. But BKo(C), the universal cover of BKo(C), is an initial object in the category of pointed coverings of BKo(C), and the a u t o m o r p h i s m group of BKo(C) in the category of coverings is just Ko(C), the fundamental group of BKo(C). Hence the category of pointed
6. Proofs of the Theorems of Chapter 4
95
coverings of B Q C also has an initial object (which m u s t be the universal cover) whose a u t o m o r p h i s m group in the category of coverings is Ko(C). If S is a K0(C)-set, r : Ko(C) ; Aut(S) the p e r m u t a t i o n representation, let Fs : QC .~ Set be the functor defined by L e m m a (6.2) and the assignments F s ( M ) = S V M E C, (Fs)l(i~) = l s , the identity map, and (Fs)2(q !) = r q]) E Aut(S). If i
M'
~-4
N
q~
~q'
M
~
Nt
il
is a bicartesian square, then ker q ~ ker q', so that r q) = r q'), and the conditions of (6.2) hold. One sees at once t h a t Fs E ~r~. Conversely, if F E jr,, let C f : Ko(C) , A u t ( F ( 0 ) ) be given by CF([M]) = F(q"M). To see t h a t this gives a well-defined h o m o m o r p h i s m on K0(d), if we have an exact sequence in d 0
~- M '
~
i
M
~
q
M"
;0
we have a bicartesian square M'
~-*
M
qMt
0
M"
M" !
so t h a t q'o iM,,, = i, o qlM, SO that F ( q ~ , ) = F(q"); also q~t = q ' o qkx,,, so that F(q~M ) = F(q"M,, ) = F ( q ~ , ) o F(q~M ,, ). In particular, by considering the split exact sequences 0 0 ~
; M'
~M ' @ M "
M " ----* M I @ M "
; M"
t0
; M'
;0
we see t h a t F(qIM , ) , F(q'M,, ' ) e Aut(F(0)) commute. Hence CF is well defined. Clearly (5, r ~-~ Fs, F ~ (F(0), e l ) give the desired equivalence of categories. This proves Theorem (4.0). Our next goal is to prove two technical results on classifying spaces of (small) categories, which are the basic homotopy theoretic tools needed to prove the remaining results of Chapter 4. We begin with a result, called "Theorem A" by Quillen, which gives a criterion for a functor f :C ; :D to be a homotopy equivalence. For any functor f 9C ; / ) , and any object Y E T~, let Y \ f be the category whose objects are pairs (X, v), X E Ob C, v: Y ; f ( X ) an arrow in T), where a morphism (X, v) ; (X', v') is a
96
Algebraic K-Theory
morphism w : X
, X'
such that the triangle below commutes:
Y
"
, f(X)
f(X') Given an arrow u : Y' , Y in :D, we have a functor u* : Y \ f - , Y' \ f given by u*(X, v) = (X, v o u). Theorem
A . If f : C ~ ~ 1) is a functor, such that Y \ f is contractible,
for each Y E T~, then f is a homotopy equivalence. If f : C ~ T~ is a functor, the fiber f - l ( y ) over Y E T~ is the subcategory of C whose objects X satisfy f ( X ) = Y , with morphisms v :X , X ' being precisely those morphisms in C such t h a t f ( v ) = l v , the identity morphism. There is a naturally defined functor f - l ( y ) ~y \ f , for any Y E D, given by X H (X, 1y). We say t h a t f makes C preJibered over 1) if for each Y e D, f - l ( y ) __, y \ f has a right adjoint. If this is t o hold, t h e n for any (Z, v) e Y \ f , we have an object v*X e f - l ( y ) (so that v*: f - l ( f ( X ) ) ; f - l ( y ) is a functor) such t h a t H o r n y \ l ( - - , (X, v)) ~ Hom f - , ( v ) ( - - , v*X). T h u s if v : Y , Y~ is any arrow in :D, we have a functorial basechange v* : f - l ( Y ' ) , f-l(r). We say t h a t f : C , Z~ makes C
fibered over :D if for Y ~, Y' ' ' ; Y " in T~, the canonically defined n a t u r a l transformation v* o v'* , (v' o v)* is an isomorphism. Thus, we have the following corollary to Theorem A (recall (3.7) t h a t a functor which has a right adjoint is a homotopy equivalence). C o r o l l a r y (6.4). Let f : C ~ • make C prefibered over l). Suppose that for each Y E ~), f - l ( y ) is contractible; then f is a homotopy equivalence. Since BC is naturally homeomorphic to BC ~ we can deduce "dual" versions of Theorem A and Corollary (6.4). For any functor f : C , 1), and any Y e :D, let f \ Y denote the category of pairs (X, v), where X e C, v: f ( X ) .~ Y a morphism in T~, where a morphism (X, v) L, ( X ' , v ' ) i s a morphism w : X ~ X ' in C such t h a t the triangle below commutes:
f(X)
y(x')
6. Proofs of the Theorems of Chapter 4
97
T h e o r e m A (dual version). Let f 9C ~ l) be a f u n c t o r such that f l Y is contractible f o r all Y E ~9. Then f is a homotopy equivalence. Next, f :C , :/9 is said to make C pre-cofibered over :D if the functor f - l ( y ) .___, f / y , X ~-~ (X, 1y), has a left adjoint, for every Y E :/9. This gives functorial co-base change arrows v . : f - l ( y ) ___. f - 1( y , ) associated to morphism v : Y ~ Y'. f makes C cofibered over :D if the natural transformations (u o v). .~ u. o v. are isomorphisms, for any Y u y , v y . in D. C o r o l l a r y (6.4) (dual version). Let f " C ~ 1 ) m a k e C pre-cofibered over D. Suppose f - 1 ( y ) is contractible for each Y E C. Then f is a homotopy equivalence. (Note: Below, we may refer to Theorem A, Corollary (6.4), or to their dual versions, simply as Theorem A; it will be clear from the context as to which result we mean). We need two lemmas in the proof of Theorem A. L e m m a (6.5). Let i ~-~ X i be a functor from a small category I to the category of topological spaces, and let g 9X t ~. B1 be the space over B 1 obtained by realizing the simplicial space
P~
lI
X~o
(where io ~ . . . ---, ip ranges over p-simplices in the simplicial set N I ) . I f f o r every i --, i t in I, X i ~ X e is a homotopy equivalence, then g 9X I -~ B I is a quasi-fibration. (See Chapter 3 for the definition of a simplicial space; if 19 ~-~ Xp is a simplicial space, its realization is obtained by putting the quotient topology on (H,>_0 • a , ) / ~ , where ,~ is the equivalence relation used to define the realization of the underlying simplicial set. See Appendix A, (A.29) for the definition and some properties of quasi-fibrations). ProoL
Since B I = lim Fp, where Fp c B I is the p-skeleton (the re-
alization of the simplicial subset of N I generated by the non-degenerate simplices of dimension < p), it suffices to prove that g - l ( F p ) ---, Fp is a quasi-fibration for each p > 0, by (A.35). We have a map of pushout
98
Algebraic K-Theory
squares, induced by g,
llX,o•
N~,I
I
g-l(Fp-1)
, HX,o• N~,I
,
I
g-1 (Fp)
H0
N~,I
I
F,-1
,
' II
N~,I
,
,
I
F,
where N~I C NpI is the set of non-degenerate p-simplices in N I . Let U C Fp be the open set obtained by removing the barycenters of the p-cells (indexed by N~I), and let Y = F p - Fp-1. By (A.30), it suffices to prove t h a t U, V and U N V are distinguished for g (as U U V - Fp); this is clear for V and for U N V, since g is a product over V. We may assume by induction t h a t Fp_ 1 is distinguished for g. There is a fiber-preserving deformation Dt of g - l ( u ) into g-l(Fp_l) induced by the radial deformation of Ap, with its barycenter removed, onto 0Ap. Let dt be the induced deformation of U onto Fp-1. To apply (A.34), we need only show t h a t if x e U, x ' = dz(x), then Dz 9 g - Z ( x ) ~ g-Z(x') induces an isomorphism on homotopy groups. Since dz is the identity when restricted to Fp-1, we may assume x E U N V. Suppose x lies in the interior of the simplex i0 --* il -* .-- --* ip, and suppose dl (x) lies in the interior of the q-face with vertices ijo , i j l , . . . , ijq (the interior of Aq is Aq --OAq, where OAq = 0 if q : 0). Then g - l ( x ) : Xio, and g-l(dl(x)) = g - l ( x t ) - Xk, where k - ijo. The map D1 " X~o * Xk is the one induced by the edge i0 --~ ijo of i0 --* il --* -.. --* ip. Indeed, Dt is induced by the deformation of Ap \ {b} onto 0Ap, where b E Ap is the barycenter, by first taking the product deformation of Xio • (Ap -- {b}) onto Xio • 0Ap, for each simplex in N~I. If x' lies in the interior of a q-face of 0Ap whose first vertex is k, then we identify Xio x {x'} C X~,, • 0Ap with the image of Xio in Xk, where Xio * Xk is induced by the arrow in I corresponding to the edge joining i0 to k in OAp. (The geometric realization is constructed using such identifications.) Now X~o - . Xk is a homotopy equivalence, by hypothesis. Hence D t 9 g - l(x) . g - 1(x ~) induces isomorphisms on homotopy groups, proving
(6.7).
The next lemma involves the notion of a bisimplicial space. A bisimplicial space is defined to be a functor A ~ x A ~ - Top where Top denotes the category of topological spaces. Thus for each ordered pair (p, q) of objects of A, we are given a topological space Tpq, such that given any pair of morphisms p ; if, q ; q' in A, we have a corresponding map of topological spaces Tp, q, ~ Tpq. In particular, for each fixed p E A, q ~-~ Tpq is a simplicial space; similarly for each fixed q E A, p_ H T ~ is a simplicial space. Thus, we can form the geometric realizations Iq ~-* Tpql
6. Proofs of the Theorems of Chapter 4
99
and [q ~-, Tpq[, and obtain simplicial spaces ;2_ ~
lq ~
Tpql,
Ip ~ Tpql,
q ~
and hence their geometric realizations (which are topological spaces)
Ip ~ Iq ~ T~.II,
Iq ~ le ~ T..II.
Finally, we have the diagonal simplicial space p ~ realization ]p ~-~ Tpp]. Lemma
Tpp, with geometric
(6.8). There are natural homeomorphisms
P r o o f . Suppose first that T is of the form A(r) x A(s) x S for a given topological space S, i.e., T is given by the functor A ~ x A ~ --, Top, h r,s x S" (p, q) ; H o m a (p, r) x Homa (q, s) x S, where the Horn-sets are regarded as discrete spaces. Then we claim that the diagonal realization has a natural homeomorphism (1) . . .
[p ~-+ Hom~ (p, r) x Hom~ (p_,_s) x S I ~ A r x As x S.
Indeed, given any simplicial set F " A ~ ~ Set and a topological space S, if F x S " A ~ ; Top, is the simplicial space p ~-. F(p) x S, where F ( p ) is regarded as a discrete space, then from the definition of the geometric realization of a simplicial space, there is a continuous bijection
IF x sl
(2)...
, IFI
x S
which is a homeomorphism if [F[ is compact. Thus, it suffices to verify that there is a natural homeomorphism
lh""l
, A ~ x A..
But h ~'s 9 _ 4 Set is just the product simplicial set A(r) x A(s), so t h a t the projections h r,8 --, A(r), h r,8 ~ A(s) (which are maps of simplicial sets) induce maps Ihr'Sl --, At, ]hr'Sl --' As and hence a map Ihr'Sl Ar x As. By Appendix A, (A.55), this is a homeomorphism. This gives the homeomorphism in (1). Next, we have homeomorphisms (applying (2)) IP_~-* Iq ~-* Homh (p, r) x H o m a (_q,S) X Sll --~ IP_H HomA (p_,_r) x As x Sl u Ar x A s x S, and similarly
lq ~
Ip_. ~
HomA (p, r) x
[q ~ A r x ~A~xAsxS.
HomA (_q,_s) x S[
H o m ~ (q, s) x S[
100
Algebraic K-Theory
Thus, L e m m a (6.8) holds for bisimplicial sets of the form h r's x S. Now let T = {T~s} be a general bisimplicial space. Given any arrow (r, s) , (if, s') in A x A, we have (i) a map of topological spaces Tr,,, T~s, and (ii) a natural transformation of functors (i.e., a m a p of bisimplicial sets) h r,s --, h r''s'. Thus we have two maps of bisimplicial spaces (~,s)-~Cr,,s,)
(r,s)
such t h a t the direct limit of the above diagram is T, i.e., the direct limit in Top of the diagram H h r'8 (p, q) x Tr, s, :=~ H hr's (P' q) x Tr, (~_,,_)-~(_~',~_') (~,~_) is Tpq, for every (p, q) E A x A. We leave the proof of this claim to the reader (who may find it instructive to first prove the analogous claim for simplicial spaces; it is useful to observe t h a t there is at least a map g
H
,q_) • T .
(~,s)
,
since if f " (p__,q_) ) (r__,s) E hr'S(p, q), we have a m a p f * 9 Trs , Tvq as part of the data defining the bisimplicial space T. One verifies t h a t g induces a map from the direct limit of the diagram to Tpq, which one proves is a homeomorphism). Now Lemma (6.8) follows from the special case dealt with earlier, and the observation that all three realization functors commute with direct limits. P r o o f o f T h e o r e m A. Let S ( f ) be the category of triples (X, Y, v) with X E C, Y E l:), v : Y ~. f ( X ) an arrow in :D; a morphism of triples (X, Y, v) ; (X', Y', v') is defined to be a pair of arrows u : X , X', w : Y' , Y in C, 1:) respectively, such t h a t
Y yl
f(x) I Ut )
f(x')
commutes, i.e., v' -- f ( u ) o v o w. We have functors pl : S ( f ) ~. C, , :D~ given by pl ( X , Y, v) -- X , p 2 ( X , Y, v) -- Y . Let T ( f ) be the bisimplicial set given by Tvq = set of pairs of diagrams (Yp ~ " " ~ Yo --* f ( X o ) , X o ~ . . . ~ X q ) where Yp --, ..- --. Y0 is a p-simplex in N D ~ and X0 --* ..- --* Xq is a q-simplex in NC. The bisimplicial structure of T ( f ) is induced by the simplicial structures of NC
P2 : S ( f )
6. Proofs of the Theorems of Chapter 4
101
and NT) ~ in the obvious way. We may regard NC as a bisimplicial set with (gC)pq - NqC. Then (Y,,
. . . - - , Yo
f(Xo),Xo
--,...--,
(Xo --,...--,
yields a m a p of bisimplicial sets
(,)...
T(f)r ~
~.(gc)pq = gqc..
The diagonal simplicial set of T ( f ) is just g s ( f ) , the nerve of S ( f ) , and the map N S ( f ) , N ( of simplicial sets given by (,) is just the natural m a p on nerves induced by the functor pl. Hence the realization of (,) (in any of the equivalent senses of Lemma (6.8)) is the map Bpl " B S ( f ) , BC. On the other hand, we may compute the realization of (,) by first realizing in the p-direction, to obtain a map of simplicial spaces in the variable q, and then forming the associated map between the realizations of these simplicial spaces. Realizing (,) in the p-direction yields the m a p of simplicial spaces which, on the spaces of q-simplices, is
H
B(:D/f(X~176
(Xo. . . . --,X, )e N~C
'
H
(point) = NqC.
(Xo--,. . . . X,)e N~C
Here NC is regarded as a simplicial space with the discrete space NqC of qsimplices; for any Y E D, D / Y is the category of pairs (Y', v) where Y' E T~, v" Y' --, Y is a morphism in T), with morphisms (Y', v) , (Y", w) being arrows u- Y' , Y" such that yl
y" commutes (thus, ~ / Y = I ~ / Y , 1~ 9~ .~ :D the identity functor). Now X ~ B ( D / f ( X ) ) ~ is a functor C , Top from the small category C to the category of topological spaces, where if u 9X ~ X ~ is an arrow in C, composition with $(u)'l(x) , f ( x ' ) yields a functor D / f (X) , :D/f(X'), and hence a map of topological spaces
B(:D/f(X)) ~
, B(:D/f(X')) ~
The simplicial space obtained by realizing T ( f ) = (Tpq} in the p-direction is clearly just the simplicial space associated to C ; Top by L e m m a (6.7), with I = C. Clearly B(iD/Y) is contractible for any Y e T), since (Y, l v ) is a final object of Z)/Y. Hence the map
B(V/J'(X))
~
, BC
I/CX'))
102
Algebraic K-Theory
induced by any given arrow X ::; X', is a homotopy equivalence. Hence the hypotheses of (6.7) are satisfied, and Bpl " B S ( f ) , BC is a quasifibration. Clearly, p ~ l ( X ) = (:D/f(X)) ~ for any X e C, so that (Bpl) -1 {X} - B ( : D / f ( X ) ) ~ (where X is regarded on the left as a 0-cell of BC), which is a contractible space. Thus Bpl induces isomorphisms on homotopy groups, from the definition of a quasi-fibration; since B S ( f ) , BC. are CW-complexes, Bpl is a homotopy equivalence, from the Whitehead theorem (see Appendix A, (A.9)). We note here that so far, the arguments given are valid for any functor f " C ---, :D between small categories. Next, let N / ) ~ be regarded as a bisimplicial set with (NT)~ NpT) ~ Then
(Yp ---, " . ' ~ Yo --, f ( X o ) , Xo - - , . . . - - , X q ) ~ (Yp - - 4 . . . - , Yo) gives a map of bisimplicial sets
T ( f ) vq
,,
(NT)~
-- N p T ) ~
whose diagonal realization is just Bp2 " B S ( f ) , B D ~ On the other hand, if we first realize in the q-direction, we obtain a map of simplicial spaces, which is given on the space of p-simplices by
H Yo. . . . .
S(Yo\f)-----* Yv
H Yo. . . . .
(p~
= NP:D~
Yv
Since by the hypothesis of Theorem A, B ( Y \ . f ) is contractible for every Y E Z), Lemma (6.7) applies again, showing (as above) t h a t Bp2" B S ( f ) =, BZ) ~ is a quasi-fibration with the contractible fibers ( B p 2 ) - I ( Y ) = B ( Y \ f ) , and is hence a homotopy equivalence. Finally, let f ' S ( f ) , S(lz)) be the functor f ( X , Y , v ) = ( f ( X ) , Y, v). Then we have a commutative diagram of categories and functors ~:)op
P2
73op
,
S(f)
pl
S(1~) P2
C
;
7)
Pt
where all the arrows except f, fl are known to be homotopy equivalences. This finishes the proof of Theorem A. Given a functor f : C ; :D between categories, we have the homotopy fiber F ( B f ) of the map B f : BC ----, B:D (over any given base point of
6. Proofs of the Theorems of Chapter 4
103
B7)), giving rise to a long exact homotopy sequence (we omit the base points) 9..---. ~ ( F ( B f ) )
~
7r,(BC) ~
~,(BT))
,7ri_l(F(Bf)) -*....
However, F ( B f ) is not in general the classifying space of a category naturally associated to f . Suppose t h a t we are given a category C', together with a functor g : C' 9 C, and a natural transformation from the composite functor f o g : C' ; :D to the constant functor C' ; {Y}, for some object Y E D. T h e n we have the map B g : BC' ~ BC, together with a homotopy from B f o g : BC' ; B1) to the constant map BC' ; {Y}, where we regard Y as a 0-cell in BiD. Thus we have an induced factorization of B g through the h o m o t o p y fiber over Y, i.e., a map h : B C ' , F ( B f , {Y}). If this is known to be a homotopy equivalence, then we would have a long exact h o m o t o p y sequence
9. . ~
7q(BC') ag. ~,(BC) ~
~r,(BI))
,1r,_I(BC')
~....
Given a d i a g r a m of topological spaces and maps EI
B'
h'
h
E
"- B
we say t h a t the d i a g r a m is homotopy Cartesian (or a h o m o t o p y fiber product) if the induced m a p E 1 ----- B 1 x B F(I, B) x B E is a homotopy equivalence. Here F ( I , B ) is the path space of B, and B ' xB F ( I , B) X a E = {(bl,7, e) e B ' x F ( I , B ) x E [ h ( b ' )
= 7(0),
g(e) = 3'(1) } is the h o m o t o p y fiber product; the map E ' , B ' x B F ( I , B ) XB E is e' ~ (g'(e'), "/o, h'(e')) where ~/0: I ; {g o h'(e')} is the constant path. The h o m o t o p y fiber p r o d u c t is characterized by the p r o p e r t y t h a t to give a map X ---, B 1 x B F ( I , B) x B E is equivalent to giving a pair of m a p s gx :X ; E , h x : X ------, B ' together with a homotopy H : X x I ~B between h o g x and g o h x . In particular, the identity m a p of B 1 x B F ( I , B ) X B E is given by the natural projections to B 1 and E , and the homotopy ( B ' x B F ( I , B ) x B E) x I) ; B, ((b', 7, e), t) " 7(t). Suppose B 1 is contractible; let H 1 : B 1 x I ; B I be a h o m o t o p y from the identity m a p of B 1 to the constant map B I ; {b~}, ,or a fixed base point b~ E B ' . Let b0 = g(b'o) E B. Let H - ( B ' x B F ( I , B) x s E) x I - - , B' XB F ( I , B ) XB E
104
Algebraic K-Theory
be given by
~I((b',7, e),t) = (H'(b',t),Tt, e) where 7t E F(I, B) is the path 7 (s) =
g ~ H ' ( b ' , t - (1 + t ) s )
if o _< s _<
7((1 + t)s - t)
if T~t <_ s <_ 1.
Then one verifies immediately that H is a deformation retraction of B' XB F ( I , B ) xB E onto {b~} xB F ( I , B ) XB E, which is naturally homeomorphic to the homotopy fiber F(h, bo). Thus, if the square is homotopy Cartesian, a n d B t is contractible, there is a homotopy equivalence E ~ F(h, bo), from E ~ to the homotopy fiber of h over any point
bo e g(B'). B. Let f 9C ; "19 be a functor between small categories such that for every arrow u 9Y ~ Y~ of T), the functor u* 9 Y' \ f ~Y \ f is a homotopy equivalence. Then for any object Y of l) the square
Theorem
BY \ f
Bj
BC
BS'~ BY \ D
~BS B~,' B D
is homotopy Cartesian, where j ( X , v) = X , j ' ( Y ' , v) = Y', f ' ( X , v) = ( f ( X ) , v). Since Y \ l) is contractible, for any X e f - l ( y ) we have an exact homotopy sequence ( X -- (X, 1y)) 9-. --* ni+l(B'D, {Y}) ~,(B:D, {Y})
,1ri(B(Y \ f ) , {X}) S j . ni(BC, {X}) et'~ , lr,_l ( B ( Y \ f ) , {X}) - - --..
As with Theorem A, there is a dual version of the above theorem, involving the categories f l Y ; further, if f is either prefibered or pre-cofibered, we have an analogous result with suitable hypothesis on the fibers f-1 ( y ) (if f is prefibered, u" Y~ ~ Y an arrow in :D, we assume u* 9 f-l(y) , f - 1 (y~) is a homotopy equivalence; then we deduce the result that
B f - I (Y)
l
{Y}
;
,
BC
1
B/)
is homotopy Cartesian). We leave it to the reader to formulate these statements in detail, and deduce them from the above form of Theorem B. P r o o f o f T h e o r e m B. As in the proof of Theorem A, we consider the category S ( f ) of triples (X, IF, v) with X E C, Y E l), v" Y ~f ( X ) an
6. Proofs of the Theorems of Chapter 4 arrow i n / ) , together with the diagram ~)op,
]O2
S(f)
Pl
105 : .~C.
As remarked
during the proof of Theorem A, pl is a homotopy equivalence for an arbitrary functor f. Since u* : y t \ f , Y \ f is a homotopy equivalence for any arrow u : Y ; Y' in :D, by hypothesis, Lemma (6.7) shows that Br, 2 : B S ( f ) , B D ~ is a quasi-fibration, as in the proof of Theorem A, i.e., any fiber of Bp2 is weak homotopy equivalent (and hence homotopy equivalent, since both spaces have the homotopy types of CW-complexes) to the corresponding homotopy fiber. If Y E D, p ~ l ( y ) = y \ f; hence
BY \ f
{Y}
, ,
BS(f)
~
BD~
is homotopy Cartesian. Consider now the diagram ( f ' is as in the proof of Theorem A) BpI BC BY \ f , BS(f)
l
BY \ D
|
,
BS(lv)
,
Bt)~
| {Y}
|
""1
Bpl
""I
BZ)
where the arrows marked Bpl are homotopy equivalences. The square O + @(with vertices B ( Y \ f ) , {Y}, B S ( f ) , B:D~ has been shown, above, to be homotopy Cartesian. The two vertical arrows marked ~ are homotopy equivalences. Hence (T)is homotopy Cartesian, and also (~)+ (~)(since the maps Bpl are homotopy equivalences). This is precisely the conclusion of the theorem. Our next goal is to prove Theorem (4.1) and its corollaries. We recall the statement below. Let C be an exact category, E the category of (short) exact sequences in C. If E, E ' , E " E s we say that the diagram 0 E~ ; E ~ E" ; 0 in g" is an exact sequence if the corresponding diagram in C has exact rows and columns. We have three exact functors s, t, q from E ; C and natural transformations s ---4 t, t ; q such t h a t for any E E s
0
, s(E) ----, t(E)
; q(E)
:0
is the corresponding exact sequence in C.
T h e o r e m (4.1). equivalence.
The functor (s, q) : Qs ~
QC • QC is a homotopy
106
Algebraic K-Theory
P r o o f . By T h e o r e m A, it suffices to prove t h a t the category (s, q ) / ( M , N ) is contractible for every ( M , N ) e QC x QC. Let 171 = ( s , q ) / ( M , N); by definition it is the category of triples (E, u, v) with E E QC, u " s(E) M , v" q(E) ~ N morphisms in QC. Let C2 c 171 be the full subcategory of triples (E, u, v) such t h a t u = j" for some admissible epimorphism j, and let Ca c 172 be the full s u b c a t e g o r y of triples (E, u, v) such t h a t v = i! for some admissible monomorphism i. L e m m a ( 6 . 9 ) . The inclusion functors C2 C C1 and C3 C C2 have left adjoints, and are thus homotopy equivalences. P r o o f . First consider (:2 c C1. If X = (E, u, v) E C1, we must show t h a t there is a universal arrow X , X in 171, with X E (:2. Factor u as u = j" o i!, where i 9 s ( E ) ~ M ' , j 9M ~ M ' are respectively an admissible mono and an admissible epi in C (by the discussion preceding (6.2), every arrow in QC has such a factorization which is unique up to isomorphism). Define an exact sequence i . E in t7 by pushout"
E'O
i.E'O
~ s(E)
~
,1
M'
,
t(E)
,
T
III
~ q(E)
,
q(E)
:*. 0
;
0
(since 17 is closed under extensions in an ambient Abelian category, T E (7, and so i . E E g). Let ~" = ( i . E , j ! v ) ; clearly )~ e (72, and there is an evident m a p X .* X given by the admissible monomorphism E ~ i . E in g;, described by the above diagram. We claim X , X is universal for maps X : ~- X ' with X ' E C2. Indeed, given X , X ' in C1 where X ' = ( E ' , j,!, v') lies in C2, we represent the corresponding morphism E ; E ' in QE by the diagram E ~-~ Eo *- E ' in C. By definition of the morphisms in (:1 = ( s , q ) / ( M , N ) , we have a d i a g r a m in QC
s(E)
) s(E') M
and hence a diagram in 17
ql
e/'
(Eo)
"-' M
M
6. Proofs of the Theorems of Chapter 4
107
Since the factorization u = q!o i~ is unique up to isomorphism (by the definition of a morphism in QC), there is a (unique) isomorphism s ( E o ) M making the diagram commute. Hence, without loss of generality we m a y assume s ( E o ) = M ' , i ~ = i, and q' 9 s ( E ~) --~ M ~ is the unique arrow in C such t h a t q' o j~ = q. By the universal property of the pushout, E --. Eo factors uniquely as E ~ i . E ~ Eo. This gives a map X ; X ' in C2, associated to the diagram i . E ~ Eo ~- E ~ in s such t h a t X , X ' factors as X ---, X ~. X ' . If E0 is replaced by an isomorphic exact sequence (so as to yield the same morphism X ---, X'), clearly we obtain the same morphism i . E ----. E ~ in Q E , and hence the same morphism X ---. X ~ in G2. Lastly, the morphism X , X ' , factoring the given morphism X ~. X ' , is unique; indeed, once we choose a diagram E ~-* Eo ~- E ' representing the underlying morphism E ---, E ' in Q~:, it suffices to show t h a t there is a unique arrow i . E ~ Eo in ~: factoring E ~-. Eo, which yields an arrow in C2. But by the universal property of the pushout, such morphisms i . E ~-. Eo are in bijection with morphisms s ( i . E ) ~ s ( E o ) factoring s ( E ) ~-, s ( E o ) . However, as seen above, since E ; E ' is an arrow in QE arising from X ~. X ' in C1, for the arrow s ( i . E ) , s(Eo) to arise from an arrow in C2, we must have a diagram
s( o)
J s(E)
\ I
M
where all the arrows except the vertical one are already given. Thus the vertical arrow is uniquely determined, and is an isomorphism. This completes the proof t h a t C2 C C1 has a left adjoint. Next, consider the inclusion C3 c C2, and let (E, u, v) E C2. We factor v" q(E) ~. N as v - i~ o j!, where j 9N ~ --~ q ( E ) , i" N t ~-, N , and define j * E by pullback: j*E " O
~
s(E)
---,
T
;
Nt
~
0
By an argument similar to the one above, we see that the map (E, u, v) ( j ' E , u, il), induced by the admissible epi j * E ~ E in s is universal, so t h a t ( E , u, v) ~ ( j ' E , u, il) is a left adjoint to the inclusion (:3 C C2. This proves (6.9).
108
Algebraic K-Theory
To finish the proof of (4.1), it suffices to prove t h a t C3 is contractible, for any M, N E C. Let ( E , j !, i!) E (33, and consider j M " M --~ O, iN " 0 N , giving an object (0, 3"M, " iN!) E C3. We claim there is a unique morphism (0, 3"M",iN,) ~ ( E , j !, i!) in C3, so t h a t (0, 3M," ira) is an initial object. A m o r p h i s m (0, 3M, " iN!) ' (E, j! , i!) consists of a morphism 0 , EinQE such t h a t the induced diagrams below are commutative: ) M
s(E)
q(E)
) N
"
0
0
, s(E) to equal qs(E), ' This forces 0 and 0 , q ( E ) to equal iq(E) ! and hence the morphism 0 E in Q s must be the morphism given by the diagram 0
0
0
~
0
'*
s(E)
,
T Jt
0
-"
0
',
0
s(E)
,
o
--,
o
---,
0
T I.
t(E)
,,
I
q(E)
(this is immediate from a factorization 0 ~-- E0 ~-~ E). Hence (0, 3"M, Qw.)" is an initial object of C3. Thus C3 is contractible. Next, we recall the statement of Corollary (4.2). C o r o l l a r y (4.2). Let C be an exact category, s the exact category of short exact sequences in C, and let s, t, q 9E ---, C be the exact f u n c t o r s defined above. Then t. = s. + q. 9 K i ( E ) ----. Ki(C) ~/i >_ 0. P r o o f . Let f 9C x C ---, E be the exact functor given on objects by f ( M , N ) = (the split exact sequence 0 , M , M (9 N ,N , 0 in E). Now the exact functor (s, q) : s , C x C has the property t h a t the induced functor QE ---. Q(C x C) = QC x QC is a homotopy equivalence, by (4.1) proved above. Hence
q),. K (E)
Ki(C x C) ~ K,(C)(9 K~(C)
is an isomorphism. The composite exact functor (s, q)o f 9C x C .~C x C is t h e identity; hence f . 9 K~(C x C) ~ K~(E) is an isomorphism ~/i > 0. Next, we have an exact functor (9 9C x C , C given by (9 (M, N ) = M ( g N ; if we fix a 0-object 0 E C, then the functors C ; C given by M ~-~ (9(M, 0) and M ~-~ (9(0, M ) are both isomorphic to the identity functor of C. Since
6. Proofs of the Theorems of Chapter 4
109
t h e functors C ~. C x C given by M ~-, (M, 0), M H (0, M ) represent K i ( C x C) as a direct s u m Ki(C) ~ Ki(C), we deduce t h a t ~,:
K,(C x C)
, Ki(C)
is identified w i t h the addition m a p K,(C) ~ K~(C)
~, K,(C),
(x, y) ~ x + y.
Thus ~.o(s,q). :Ki(E) , Ki(C) i s j u s t s . + q . . N o w t o f = ( 9 o ( s , q ) o f : C • C , C; hence t . o f . = (s. + q.) o f . . Since f . is an i s o m o r p h i s m , t . = s. + q., which is w h a t we wanted to prove. C o r o l l a r y ( 4 . 3 ) . Let C, T) be exact categories, F, G , H " T~ ~ C exact fanctors, F ; G, G ~ H natural transformations, such that f o r any object U E T~, 0 ~. F ( M ) ~. G ( M ) ~. H ( M ) , 0 is an exact sequence in C. Then G . = F. + H . 9 K i ( T 0 ~ Ki(C) V i > O. P r o o L Let ~ be the exact category of short exact sequences in C. T h e n we have a well-defined functor L : T) , C given by L ( M ) = (the e x a c t sequence 0 ; F(M) , G ( M ) - - , H ( M ) ---, 0). If s, t, q : E ---, C are the exact functors defined above, clearly F = s o L, G = t o L, H = g o L. Since t . = s . + q . , we deduce t h a t G . = ( t o L ) . = t. o L . = (s. + q . ) o L . = (s o L ) . + (q o L ) . = F . + H . . Corollaries (4.4), (4.5) follow easily from (4.3), and are left to t h e reader. Next, we prove the resolution theorem. Let AJ be an exact category, P C A4 a full additive s u b c a t e g o r y which is closed under extensions in A4, so t h a t :P is an exact category, and P c f14 an exact functor. Theorem that
( 4 . 6 ) (Resolution Theorem). Let 7) C A4 be as above. A s s u m e
(i) if 0 , M' ~ then M ' E P
M
: M"
~. 0 is exact in f14, and M , M "
e P,
(ii) f o r each object M e f14, there is a finite resolution in f14 0
= ;Pn
;Pn-l~'"-~Po
~M-~-O
with Pi e P (and n m a y depend on M ) . Then Q7) ~
Qfl4 is a homotopy equivalence; hence K i ( P ) ~- K i ( f l 4 ) Vi.
P r o o f . Let r C A4 be the full additive subcategory whose objects are M E A/I which have a resolution O - - ~ Pn - - * P,~-I ~ " " ~ Po
~. M
,0
Algebraic
110 w i t h Pj E P V j , A4 = lim A4i.
andn_<
Lemma
Let 0
(6.10).
K-Theory
i. Clearly P = A4o C A41 C . - . C A/I, a n d
". M '
; M
0 be an exact sequence
; M"
in .A4. Then (i),~ M E A4n, M " e J ~ n + l ""@- M ' E A4n (ii)n M ' , M "
~_ A4,.,+1 --> M ~_ .A~n+1
(iii)n M , M " E .h,4n+l ~
M ' E A4n+l.
P r o o f . We first prove t h e above s t a t e m e n t s for n = 0; t h e n we verify t h a t t h e t h r e e s t a t e m e n t s for n - 1 t o g e t h e r imply t h e t h r e e s t a t e m e n t s for n, if n > 1. W e use t h e following two constructions: given a n admissible epi P " --~ M " w i t h P " E P , we have t h e pullback d i a g r a m in jVt (with e x a c t rows a n d columns) 0
0
l 1 II 1
0
--~
0
--,
0
;
M'
---*
0
~ M'
1 1 l l
0
M[' M
--,
X M"
P"
M~'
--~
P"
,,
M"
M
0
1 l 1 l
---,
0
,
0
~
0 ~ 0
0
Next, if P ' ~ M ' , P " --~ M are admissible e p i m o r p h i s m s in A4 w i t h P ' , P " E P , then t h e c o m p o s i t e P " --- M " is a n admissible e p i m o r p h i s m . W e have a d i a g r a m w i t h e x a c t rows a n d c o l u m n s 0
1 l 1 l
0
1 l l l
0
l l 1 l
0
~
K'
,
K
,
K"
0
;
pI
~
p,q~p,,
.
p.
'
M'
:
M
,
M 'l
0
-
0
0
w h e r e t h e middle row is t h e split e x a c t sequence.
0
0
;
0 0
6. Proofs of the Theorems of Chapter 4
111
If M E ~40 = 7~, M " E f141, choose an exact sequence 0
;M~'
;P"
-~M"
;0
with M~', P " E P. Then from the middle column of the first (pullback) diagram above, M x M" p , , E P. Hence from the middle row of the same diagram, M ' E 7~. This proves (i)0. If M ' , M " E A41, then from (i)0, we see that K', K " E :P in the second diagram above; hence K E P, and so from the middle vertical sequence, M E A41. This proves (ii)0. If Mr, M t~ E .h41, then from (i)0, we have K, K n E P in the second diagram above; hence K ~ E P, and from the left hand vertical sequence, M ~E r This proves (iii)0. Now assume n > 1, and (i)n-1, (ii)n-1, (iii)n-1 hold. Suppose M E A/In+l, M " E A/In. Then from (i)n- 1, M~' e A/in- 1 in the first diagram above (as p u E :P c 2~4n-1). Hence from the middle row of the first diagram and (iii)n-1, M " e r This proves (i)n. Next, suppose M ~ , M H E A/In+l. Then from (i)n, K ' , K " E A/ln in the second diagram above. Hence from (ii)n-1, K E A/In, and so from the middle column of the same diagram, M E 2~4n+1. This proves (ii)n. Lastly, let M, M ~t E A/[n+1. Then from (i)n, K, K u E A/In in the second diagram above. Hence from the top row of that diagram, and (iii)n-1, K E A/In, so t h a t M ~ E ~ n + l . This proves (iii)n, and completes the proof of (6.10). Thus, it suffices to prove:
T h e o r e m ( 6 . 1 1 ) . Let P be a full additive subcategory of an exact category A4 which is closed under extensions, such that (i) f o r any exact sequence O P-~
, M t,
, M
, M"
~ 0 in A4, M E
M' EP
(ii) for any M H E A4, there is an exact sequence as in (i) udth M E P. Then the natural f u n c t o r Q79
~ QA4 is a homotopy equivalence.
P r o o f o f ( 4 . 6 ) . Assuming (6.11), we see that QJ~/In c QA/in+l is a homotopy equivalence for each n >__0. Thus we have homotopy equivalences ~=
lim
n ~=
Q(n
m A4n ) ~=
9
P r o o f o f ( 6 . 1 1 ) . Note that Q P C Qcg4 is a subcategory which is in general not full. Let C C QA4 be the full subcategory with the same objects as Q:p. Then the inclusion of Q P in QA/I factors as the composite Q p ~-4 C ~-* QA4; g f it suffices to prove that f, g are homotopy equivalences.
112
Algebraic K-Theory
To prove g is a homotopy equivalence, it suffices to prove that g/P is contractible V P e C, by Theorem A. Let (P1, u) ~_g/P, where P1 e P, u : /)I , P the morphism in C C Q ~ t given by the diagram P1 ~- P~ ~-* P; a morphism (P2, v) * (P1, u) in g/P is given by a diagram
I P2
I ""
P~
"~
"~ P
where the top row defines u, the bottom row defines v, and the left vertical column defines a morphism in QP. The commutativity of P1
u
P2 in QA4 is expressed by the existence of an isomorphism w making the diagram commute. Since P2 ~- P3 ~-* P1 is an arrow in Q7 ), P3 ~-' P1 and P3 --~ P2 are admissible mono and epi in 7), respectively. Given an object (pl, u) E g/P, with u represented by P1 ~- P~ ~-* P, we q
9
can associate to it the A/l-admissible layer (image i, i(ker q)) of subobjects of P, (where an M-admissible layer (P1, P2) of subobjects in P is a pair of subobjects P2 C P1 C P, such that P2 ~-' P1, P1 ~-* P are admissible monomorphisms in ,A/I). Given an arrow (P2,v) ; (P1, u), if (P~,P~') and (P~, P~') are the layers associated to (P1, u) and (P2, v), respectively, then there are inclusions P;~ C P~ C P~ C P; C P. Thus, g/P is equivalent to the partially ordered set (considered as a category in the usual way--see Appendix B) of M-admissible layers (pt, p , ) in P such that P'/P" E ~, i.e., P " r P ' is P-admissible. The partial order is given by (P;,P;~) < (P~,P~') ~ P~ C P~' C P; C P~ (so that there is a (unique) morphism (P;,P;') , (P~,P~) in J) and P~ r P~ is P-admissible (so that all the inclusions P~' ~-~ P;' ~-, P~ ~-, P~ are/)-admissible). We have inequalities (i.e., arrows) in J
(P',P") < (P', O) >_ (0, 0), since 0 ~-, P', P" ~-~ P ' are P-admissible. These can be viewed as natural transformations of functors J .~ J 1j
~h~
-0
6. Proofs of the Theorems of Chapter 4
113
where 1j is the identity, 0 is the constant functor with value (0, 0), and h(P', P") - (P', 0). T h u s the identity functor of J is homotopic to the constant functor 0, and so J is contractible. This proves t h a t g is a homotopy equivalence. To prove f is a homotopy equivalence, it suffices to prove t h a t M \ f is contractible VM E QA4. If ~" = M \ f, then 9r consists of pairs (P, u) with P E :P, u 9 M .~ P a morphism in QcVi. Let ~'~ be the full subcategory of pairs where u = q! for some admissible mono q 9P --- M . If X = ( P , u ) E ~ - , w r i t e u = i ! o q ~ whereq'tb~M,i'lb~P. Since P is closed under taking subobjects, t5 E P, so that .~ = (/5, q!) E ~", and i! gives a m a p X , X in jc. From the uniqueness (up to isomorphism) of the factorization u = i! o q" we verify easily that )~ ; X is universal among morphisms X ~ , X in with X ' E ~'~. Hence ~ ' r ~" has a right adjoint, and so is a homotopy equivalence. Now ~-~op is equivalent to the category of pairs (P, j) where P E 7~, j " P ~ M an admissible epi in A/I, where a morphism (P, j ) ; (P', j') is an admissible epi P --~ P~ in A4 making
P
,~
f(X)
M commute. There exists at least one admissible epi jo " Po ~ M, by the hypothesis of (6.11); fix one such. (liven any admissible epi j 9P ~ M in 3,t, we have a diagram of admissible epis Po
T
Po xM P
Jo ;;
M
T
",~ P
where ker(P0 x M P -~ P0) ~ k e r ( P --~ M) e P, so t h a t P0 x M P E :P. Let k : .~t , .T"t be the functor k ( P , j ) - ( P o x M P, jo o j l ) . T h e n the above can be viewed as giving natural transformations from k to the identity functor of ~'t, and to the constant functor with value (Po,jo). Hence the identity functor of ~'~ is homotopic to a constant functor, and so ~-t is contractible. This completes the proof of (6.11), and hence (4.6). The proof of Corollary (4.7) is left to the reader. Next, we prove the ddvissage theorem.
Theorem
Let .A be an Abelian category, B c A a full Abelian subcategory which is closed under taking subobjects, quotients and finite (4.8).
114
Algebraic K-Theory
products in .A. Suppose each object M E .A has a finite filtration 0 = M0 C M1 C .-. C Mn = M (where n may depend on M) such that Mi/M~-I E 13 V i > O. Then QB ; Q A is a homotopy equivalence; hence K,(B) ~ Ki(.A). P r o o f . Let f : QI3 ~ QA be the inclusion; by Theorem A, it suffices to prove t h a t f / M is contractible for each M E ~4. The category f / M consists of pairs (N, u) with N E B, u : N ~ M a morphism in Q~4. Let N --- M ' ~-o M be a diagram giving u; there is a unique such diagram with M ~ C M , and we then have an associated layer ( M ' , M ") in M, where M " = k e r ( M ~ ~ N). Thus f / M is equivalent to the partially ordered set J ( M ) of ~4-admissible layers (M', M ' ) in M with M ' / M " E 13. Since M has a finite filtration 0 = M0 c M1 c ..- c Mn = M with M~/Mi-1 e 13, it suffices to show that if M ' C M with M / M ' E B, then J(M') ; J(M) is a homotopy equivalence. If (MI, M2) E J(M), then (M1,M2 M M') e J ( M ) , and (M1 N M ' , M2 M M') e J(M'), since
(M1 M M')/(M2 MM') C M1/(M2 N M') C (M1/M2) @ ( M / M ' ) E 13. Let r : J ( M )
; J(M'), s: J(M)
" J ( M ) be the functors
r(Ml, M2) = (M1 M M', M2 N M'),
s(M1, M2) - (M1, M2 M M ' ) ,
and let i : J(M') ~ J(M) be the inclusion. T h e n r o i = 1j(M,). Also, we have inequalities in J(M) (M1 M M ' , M2 M M ' ) < (M1, M2 N M ' ) :> (M1, M2) giving natural transformations i o r ~ - - s b 1 j ( M ) . Hence i o r is homotopic to the identity, and so i is a homotopy equivalence. This proves
(4.s).
O o r o l l a r y (6.12). Let A be an Abelian category (with a set of isomorphism classes) such that every object of A has finite length. Let {Xj l j e J} be
a set of representatives for the isomorphism classes of simple objects, and let Dj = E n d ( X j ) ~ Then K~(A) ~ H j e J g i ( D j ) . P r o o f . Let B c A be the full Abelian subcategory of semi-simple objects. From (4.8), Ki(13) ~= K~(A). Since Ki commutes with finite products and filtered direct limits, we reduce to the case when vi = B has a single object X up to isomorphism. But then M ~-, H o m A ( X , M ) is an equivalence of j4 with P ( D ) , D = EndA(X) ~ where 7~(D) is the category of finite dimensional left D-vector spaces. This proves the corollary.
6. Proofs of the Theorems of Chapter 4
115
The last aim of the chapter is the proof of the localization theorem (see Appendix B for the construction of the quotient Abelian category of an Abelian category modulo a Serre subcategory). T h e o r e m (4.9). Let 13 be a Serve subcategory of an Abelian category .A, with quotient Abelian category .A/B; let e" B ; A, s" .4 = ~ .AlE be the natural functors. Then there is a long exact sequence 9 .. --~
K~+~(~/U)
,
K,(B)
"" ,
K,(.,4)
e. K o ( A )
"-
,
K,(A/B)
~" Ko(.AIB) ,
t~
, K~_,(U)...
~0.
Further, the sequence is functorial for exact functors (,4, B)
~=, (,4', B').
C o r o l l a r y (6.13). Let A be a Dedekind domain with quotient field F. Then there is a long exact sequence
9.. - , Ki+I(F)
' H Ki(AIA4) .h4
, K~(A)
, KI(F)---,...
, Ko(A)
, Ko(F)
, O,
where A4 runs over the maximal ideals of A. P r o o f . Let ,4 = A4(A), B C ~4 the full subcategory of torsion A-modules (i.e., modules M with F| M = 0). Then .A/13 is equivalent to the category P ( F ) of finite dimensional F-vector spaces, while Ki(B) ~ L I ~ Ki(A/.h4) from Corollary (6.12) above. Since A is a Dedekind domain, any M E ,4 has a projective dimension < 1 over A, so that Ki(A) ~ K i ( P ( A ) ) = Ki(A) by the resolution of Theorem (4.6). Hence the result follows from (4.9). P r o o f of T h e o r e m (4.9). Fix a zero object 0 E .A, and let 0 also denote its image in .A/B, which is a zero object in r Now B is the full subcategory of objects M E A such that s ( M ) ~ 0 in .A/B; since 0 is a zero object, there is a unique such isomorphism. Thus the composite functor QB Qe
Q A Q'~ Q ( A / B ) is isomorphic to the constant functor with value 0, and Qe factors as QB ~.0 \ Qs ~. Q A U ~ (M, 0 ~-, s ( i ) ) ,
(N, u ) ~
N.
By Theorem B, it suffices to prove the following claims, in order to prove (4.9): (~) for . ~ . r y . ~ - v '
: v in QCA/U), ~,* . V \ Q s
, V' \ Q s
116
Algebraic K-Theory is a homotopy equivalence, and
(b) the functor QB
~ 0 \ Qs is a h o m o t o p y equivalence.
Indeed, (a) together with Theorem B implies t h a t
0 \ Qs
,
1
QA
1
o \ QC IB) , QC.alB) is h o m o t o p y Cartesian, where 0 \ Q(.A/B) has 0 as an initial object and is hence contractible. By (b), we deduce t h a t the following square is homotopy Cartesian (which implies (4.9))
QB
l
o \ Q( t/B)
O~,
,
Q.A
1
To prove (a) for u = i! o qV where i is an admissible mono, and q is admissible epi, it suffices to prove it when u = i! or u = q!. Now by Corollary (6.3), we have isomorphisms
Q(.AIB) ~ Q ( A I B ) ~ ~- Q(,4~176 under which the roles of admissible monos and epis are interchanged. T h u s u = i! o q! gets transformed into q~ o i ~-wwhere qr, i ~ are the dual arrows in (Jr/B) ~ to q, i respectively (in .A/B). T h u s if we prove (a) for all u = i.~, for all quotients ~4/B of an Abelian category by a Serre subcategory, we deduce (a) for all u = q! by passing to the opposite category (,4/B) ~ Hence it suffices to prove (a) for all u = i~.. For any V E .A/B, let iv : 0 ~-~ V; t h e n if i : V ~ ~-~ V, i~. o iv,! = iv!. Hence it suffices to prove (a) for all u = iv!, for all V E .A/B. Let ~ v be the full subcategory of V \ Qs consisting of pairs (M, u) such t h a t u : V ~ s ( U ) is an isomorphism in Q(.A/B). Clearly ~-0 is isomorphic to QB; thus (b) is a particular case of: Lemma
(6.14). The inclusion ~ v
~ V \ Qs is a homotopy equivalence.
P r o o f . Let f : :7:v ". V \ Qs be the inclusion functor. By T h e o r e m A it suffices to prove t h a t f / ( U , u) is contractible for any ( i , u) in V \ Qs. T h e arrow u : V ~ s ( U ) in Q(.A/B) corresponds to a unique d i a g r a m V ~- V1 ~-+ s ( M ) in .A/B; let V0 = ker j. This gives an A/B-admissible layer j i (i(V1),i(Vo)) in s(M). Choosing i to be an inclusion (there is a unique d i a g r a m representing u with this property), (II1, V0) is an A / B - a d m i s s i b l e layer in s ( M ) .
6. Proofs of the Theorems of Chapter 4
117
An object of f / ( M , u) is a triple (N, v, w) consisting of (N, v) e ~'v and a morphism w 9 5/ 9 M in Q~4 such t h a t y
t$
s(M)
s(N) commutes in Q(.A/B) (thus w determines an arrow (N, v)
; (M, u) in
V \ Qs). S i n c e ( g , v ) e J:v, v " V ~, s(N). L e t N ~ - - M 1 ~ - * M b e t h e unique diagram representing w such t h a t Ml ; M is a subobject, and let M0 = ker(Mi -~ N); thus we have an A-admissible layer ( M : , M0) in M associated to w. Since v is an isomorphism, (s(Ml),s(Mo)) = (V1, V0) as layers in s(M). T h e layer (M1,M0) determines (N,v,w) up to isomorphism; hence f \ (M, u) is equivalent to the partially ordered set J of A-admissible layers (M1,Mo) in M such t h a t (s(M1),s(Mo)) = (111,Vo), with (M1, M0) <_ (M~, M~) ~ M~ C Mo C M1 C M~. We claim t h a t J is filtering (so t h a t by (3.9), J is contractible), i.e., J is non-empty, and is directed (i.e., any two elements of J have a common upper bound). Every subobject VI of s(M) is (up to isomorphism) of the form s(N) for some subobject N of M; indeed V1 - s(N) for some object N E O b , 4 = Ob(~4/B), and 111 :; s(M) is induced by a morphism N ~ - ; M ' in .A, where N ~ C N, M -~ M ' , and N/N', k e r ( M --~ M ' ) E B. T h e n N ' x M' M -~ N ' has kernel in B, so s(N) "~ s(N') ~- s(N' x M' M). So s(N) "~ V1 is the image of an arrow u 9N :: ; M in ,4, without loss of generality. Since s(u) is mono, k e r u e B, and N -~ i m u maps to an isomorphism in .A/B. Hence if we let N = im u, t h e n N C M and s(N) ~- VI. T h u s any layer (V1, V0) in s(M) is of the form (s(M1), s(Mo)) for some layer (/141, M0) in M . Hence J is non-empty. If (MiM0), (M~, M~) are layers in J, then clearly (MI +M~, MoOM~) e J is a common upper bound. Hence J is directed, and hence contractible, proving (6.14). To prove (a) above for u = iv.,., we must show t h a t (iv:)* 9V \ Qs 0 \ Qs is a homotopy equivalence. This is done in a number of steps. First, we introduce some notation. For any N E ~4, let s be the category whose objects are pairs (M,h) with M E A, h 9 M ; N a morphism in ,4 such t h a t s(h) is an isomorphism, i.e., ker h, coker h E B. A morphism (M, h) ; ( M ' , h') in s is defined to be a morphism u " M ~. M ' in Q.A such t h a t if u = i, o j!, then the diagram in A M1
i,
M'
M
h ----,
N
118
Algebraic K-Theory
commutes; this condition does not depend on the specific factorization u = i! o j!, since for an isomorphic factorization u = i~ o j'", there is an isomorphism w making the following diagram commute: M1
M
~1 w
M'
Associated to a morphism ( M , h ) ; ( M , h ' ) , we have a m o r p h i s m ker h ---4 ker h' in QB, given by the diagram in B ker h ~- j - ] ( k e r h) ~-, ker h' corresponding to u = i! o j"; again, the morphism in QB is independent of the factorization u = i., o j!. Thus, there is a well-defined functor kN 9 CN ----, QB, k N ( M , h ) = kerh. We first prove t h a t klv is a h o m o t o p y equivalence, in two steps. Let C~ C SN be the full subcategory of pairs (M, h) where h is an admissible epimorphism in j4, and let k~ 9 C~ *. QB be the functor obtained by restricting kN. Lemma
(6.15). k~ " C~
, QB is a homotopy equivalence.
P r o o f . By Theorem A, it suffices to prove k ~ / T is contractible for any T E QB. Let C = k ~ / T , so t h a t C is the category whose objects are triples (M, h, u) with ( M , h ) E ~ , and u 9 ker h ; T a m o r p h i s m in QB. Let C' c C be the full subcategory of triples (M, h, u) with u = q! for q" T -~ ker h in B. Given X = (M, h, u) E C, with u = jtoi!, i 9 ker h ~-, To, j 9T -~ To, let (i.M, h) E C~ be defined by pushout along i, kerh
;
; M
-----~ N
To
;
~ i.M
= ~" N h
,1
ltt
Let )( = (i.M, h, j") E C'; there is an evident map X , ,~ in C. We claim this is universal among arrows X ; X ' with X ~ E C'. This is easy to verify from the definitions of C, C' and the uniqueness up to isomorphism of the factorization u = j! o i!, and is analogous to the argument given in the proof of (6.9); the details are left to the reader.
6. Proofs of the Theorems of Chapter 4
119
Thus, the inclusion of C' in C has a left adjoint, and is hence a homotopy equivalence. But (N, 1N,q~-) (where 1~ : N ; N is the identity, qr : T --~ 0) is an initial object of C~, from the diagram (in ,4) T
"" kerh
0
; M h--~N
~
N
1N
-,
N
(note that we need h to be an admissible epimorphism, for this argument to work, since h must simultaneously give a morphism in ,4 and Q~4). Thus C', and hence C, are contractible. This proves the lemma. Lemma
(6.16). kN : ~N ~
QI3 is a homotopy equivalence.
P r o o f . From the previous lemma, it suffices to check that s ~-~ s is a homotopy equivalence. Let Z be the ordered set of subobjects I C N such that N / I ~. B. There is a functor f : EN ~ iT given by (M, h) * im h. We claim f makes s fibered over Z; indeed the fiber f - 1 (I) = E~, and for any arrow J , I in iT, corresponding to the inclusions J C I C N, the base change functor ~} ; S~ is given by J•
: (M ~ I ) ~
(J •
M--~ J).
We claim
el
J•
,
e5
commutes up to a natural transformation (so that the induced diagram of classifying spaces commutes up to homotopy). Indeed, there is a natural isomorphism ker(U ;; I) =~ ker(J • U ~ g). Since k~, k~r are homotopy equivalences, by (6.15), J x t? is also a homotopy equivalence. By Theorem B, eN
1
{N}
1, ,
z
is homotopy Cartesian. Since N E I is a final object, X is contractible. Hence ~v ~ ~N is a homotopy equivalence, proving the lemma.
120
Algebraic K-Theory
We now want to prove that if s(N) ~ V, then f v is homotopy equivalent to t~N; this will give homotopy equivalences V \ Qs ~ f v ~ 81v ~ QB, so that the homotopy type of V \ Qs is independent of V. L e m m a (6.17). Let g: N , N' be a map in ,4 such that s ( g ) : s(N) , s(N') is an isomorphism. Then g . : 8 n ~ ~:N,, (M, h) , ( M , g o h) is a homotopy equivalence. P r o o f . Given (M, h) E tf~v there is a natural admissible monomorphism ker h ~-~ ker g o h. This defines a natural transformation kN ) k~r o g. of functors ~r162 ) QB. Since kN, k~,, are homotopy equivalences, so is g.. This proves (6.17). Given V e A/B, let Zv be the category of pairs (N, r with N E ~4, r s(N) ~-; V an isomorphism in A / B , where a morphism (N, r ) (N', r is a morphism g : N ~ N ~ in ,4 such that the following diagram commutes:
sCN') V L e m m a (6.18). Zv is a filtering category for any V e A / B . P r o o f . We must show (i) Ob I v is non-empty. (ii) Given ( g , r (N',r e Zv, there exists ( g " , r diagram in I v
(g,r
E Zv and a
\ (N n, r
(N',r
7
(iii) Given by two arrows ( g , r ==t ( g ' , r in Zv, there is an arrow (N', r ~. (N", r in Zv such t h a t the two composite arrows (N, r ==$ (N n, Cn) are equal (in earlier situations where we encountered filtering categories, these were based on partially ordered sets, so t h a t there was at most one morphism between any two objects; hence (iii) was always satisfied and was not mentioned explicitly). Since Obai = 0b,4/13, clearly (i) holds; indeed there is a unique pair (N, ls(N)) E I v . Given any two objects (N1, r (N2, r in Zv, where r : s(Ni) ; V is an isomorphism, then r is induced by a map N" ". N / N " for subobjects N~ C Ni, N~' c N with Ni/N~, N~' e B.
6. Proofs of the Theorems of Chapter 4
121
Replacing N~', N6' by N " = N~' + N6', we may assume N{' = N6' = N " . Consider the pushout diagram N,
,
l
fr
l
N'
~. N / N "
(which defines Ni); then N / N " ~ Ni with cokernel contained in B, so t h a t s(N) TM s(1Vi). Finally, let N be the pushout
N/N"
;
1
N1
1
Then N ~ N induces s(N) ~ s(N); let r be the inverse isomorphism. By construction, we have a diagram
(N~, r
(N2, r i.e., the composites gi " Ni
~; Ni s(N,)
\
(~r, ~)
/
; N yield commutative diagrams s(a,)) s(/~) V
This follows from the diagrams
l
N~
----.
l
N/N"
where all arrows become isomorphisms on applying the functor s. This verifies (ii). Finally, let (N1, ~bl) ::t (N2, r be two arrows in Zv given by gi : Nl ; N2, i = 1,2. Let N3 = N 2 / i m ( g l - g2); then s(gl) = s(g2) = r o r so t h a t s(gl - g2) = 0, i.e., im(gl - g2) E B, and s(N2) -~ s(N3). Let r : s(N3) ~. V, induced by this isomorphism (and r Clearly the two composite arrows (N1, r =:$ (N3, r are equal. This proves (iii), and completes the proof of the lemma.
122
Algebraic K-Theory
We claim the s give a directed system of categories indexed by I v . Indeed, associate CN to (N, r e Zv; if g " (N, r ---. ( N ' , r is an arrow in I v , associate to it the functor g. 9 s ) EN, of (6.17), g . ( M , h ) -(M,g o h). Then (gl o g~). = gl. o g2. follows from the associativity of composition of morphisms in a category. Next, for each (N, r E Zv, we have a functor P(N,r = s L) ~'V given by p(N,r = ( M , s ( h ) - l o ~-1), where we have isomorphisms (in .A/B) s(M) s(h! s(N) ~-~ V. For any morphism g ' ( N , ~b) -~ ( g ' , r in Zv, we claim P(N',~') o g. -- P(N,r This follows from the identity. s(g o h) -1 o r
~.
s(h)-I o s(g) -1 o (ib'-1 -- s(h) -1 o r
Hence we obtain a functor on the direct limit category p "= lim{(N, r
~-. EN}
9 Zv.
2:1/
L e m m a (6.19). p is an isomorphism of categories (i.e., p is bijective on objects and on Horn-sets). P r o o f . If (M,8) E O b J r v , where 8 9 V ~ s(M) is an isomorphism, then (M, 8) - p(M,0-1)(M, 1M). Hence p is surjective on objects. If p(jv,r = p(N,r then clearly M = M ' , and s ( h ) - l o r __ s(h') -1 o r i.e., s(h) - s(ff). Hence if N' - N / i m ( h - h'), we obtain an arrow g ' ( N , r ~ (N', r in Zv (where r is induced by r and s ( g ) ~ s(N')) such that g.(M,h) - g.(M,h'). Hence p is injective on objects (we have used condition (ii), verified in (6.18) for Zv). Next, let (M, h) :=$ (M', h') be two morphisms in CN which yield the same morphism on applying P(N,r T h e n the two morphisms M :=$ M ' in Qj4 are the same, and so define the same arrow in EN. Thus P(N,r and hence p, are injective on arrows. Lastly, let (M, h), (M', h') E s and let
t " P(N,~)(M, h)
~P(N,r
h')
be an arrow in ~-v. Then t is given by an arrow w 9M t h a t the diagram below commutes in Q ( A / B ) :
s(M)
.~ M ~ in QA such
"(~)- s(M') V
In particular, s(w) is an isomorphism. Hence if we factor w -- i) o j! where j 9M ~' --~ M, i - M " ~-, M ' , then s(j), s(i) are isomorphisms. For w to
6. Proofs of the Theorems of Chapter 4 yield a m o r p h i s m in s
123
the diagram below must commute: M"
i: ~ M '
M
~ N h But s ( w ) o s(h) - I o r -1 = s ( h ' ) o r -1, since w yields the morphism t in gN. Hence s(M") s(,) s ( M ' )
9(J)~
l~(h')
s(U)
s(h)
s(N)
commutes, all arrows being isomorphisms. Thus im(h o j - h' o i) E B. Let g ' = N / i m ( h o j - h' o i), so that s ( N ) ~ s(N'), and let r : V ~ s ( N ' ) be induced by this isomorphism (and r Let g : (N, r - (N', r be the natural map. Then w : M ~ M ' yields an arrow ~b : (M, g o h) ; ( M ~, g o h') in ~:N,, such that p(N,,#,)(~) = t" p(N,,~,)(M, g o h)
, p(N,,~,)(M', g o h').
Hence p is surjective on arrows. This proves (6.19). L e m m a ( 6 . 2 0 ) . For any (N, r equivalence.
E :Tv, P(N,r
" g'N ~ ; .T'v is a homotopy
P r o o f . From (6.17), for any arrows g " ( N , r ~ (N',r s -~s is a homotopy equivalence. Hence for any (N, r CN
; lim{(N', r
in I v , g. 9
~ CN,}
TV
is a homotopy equivalence, by (3.8). Hence the lemma follows from (6.19). We now prove (ivl)* " g \ Qs ~ 0 \ Qs is a homotopy equivalence. Choose (N, r 6 Z v , and consider the diagram
l('~,)"
k~ 1 QB
~-.
Y=o~ o \ Qs.
This does not c o m m u t e in general; the two composite functors are
(M, h) -~~ (ker h, 0 - ~ ~(k~r h)) (M, h) (,v,)_~r
(M, is(M),." 0 ~ s ( M ) ) .
124
Algebraic K-Theory
If i : ker h ~-, M, then i! gives a natural transformation between these two functors, so the induced diagram of classifying spaces commutes up to homotopy. Since all sides except (iv!)* are known to be homotopy equivalences, (iv!)* is one too. This completes the proof of the localization theorem (4.9), modulo the naturality of the resulting long exact sequence of K-groups. If f : (A,B) .~ (A',B') is an exact functor, where B C ,4, B' c .4' are Serre subcategories, then we have a commutative diagram of categories and functors
QB
Qe
QA
Qs
QB'
~
QA'
Q"'~. Q(A'/B')
IQ.
IQ,
Q(.4/B)
IQ"(')
where e' " B' ==~.A', s " A' ~.A'/B', and g = fiB" From the right hand square, we have a diagram of topological spaces
F(BQs)
1
F(BQs')
~ BQA
;
BQ.4'
BQs
~
BQs' ---,
BQ(A/B)
BQ(.4'/U')
including a diagram of long exact sequences of homotopy groups; here we fix a zero object 0 E O b . 4 = O b Q A , and its images in Q(.A/B), Q.A', Q(.A'/B') as the respective base points, and compute the homotopy fibers over these points. The localization sequences are obtained from homotopy equivalences c~ :
BQB
~ F(BQs),
c~' : BQB'
~.F(BQs')
obtained using the functors Qe, Qe' respectively, so t h a t we need to check that
BQB
"Q'I BQU'
---, CI~t
F(BQs)
1 F(BQs')
commutes. The composite functor Q(s o e) : QB ~ Q(.A/B) maps every M E QB to an object isomorphic to 0 E Q(.A/B); hence there is a canonical isomorphism of functors between Q(soe) and the constant functor QB {0}. This gives a null-homotopy of BQ(s o e) = BQs o BQe. The data, consisting of the map BQe, together with the nuU-homotopy of BQso BQe to the constant map from BQB to the base point 0 e BQ(.A/B), determine the map c~. The map a ' is similarly determined. The composite map
BQB a F(BQs)
F(BQs')
6. Proofs of the Theorems of Chapter 4
125
is given by (i) the map B Q f o B Q e : B Q B : BQ.A', and (ii) the nullhomotopy of B Q s ' o ( B Q f o BQe) - B Q s ( f ) o (BQe o B Q s ) induced by the null-homotopy of B Q e o B Q s (used to determine a). On the other hand, the composite BQI3BQ~. BQI3' ?'~ F ( B Q s ' ) is induced by (i)' the map BQe' o BQg : B Q B : BQ.A', and (ii)" the nullhomotopy of B Q s ' o (BQe' o BQg) = (BQs' o BQe') o B Q g induced by the null-homotopy (used to define a') of BQs'oBQe'. However, f o e = e'og, so the maps in (i), (i)' are equal. The homotopies in (ii), (ii)' are both induced by natural transformations from the functors Q s ' o Q f o Q e = Qs'oQe' oQg : QB ". Q(.A'/B') to the constant functor QB ; {0} e Q(j4'/B'). These natural transformations are equal, since there is a unique isomorphism between any two zero objects in A'/B', and hence in Q(.A'/B'). This proves the claimed naturality.
7. C o m p a r i s o n of the Plus and Q - C o n s t r u c t i o n s
Let S be an Abelian monoid, i.e., S has a commutative, associative binary operation with a two-sided identity. We say t h a t S acts on a set X if there is a homomorphism of monoids S ; Homse__tt(X,X); if s E S, the corresponding map of sets X .~X is called translation by s. We say t h a t S acts invertibly on X if each translation is bijective. If X is a set on which S acts, let S - 1 X = (S x X ) / S where S acts diagonally on the product (S acts on itself by left translation; here (5' x X ) / S denotes the quotient of S x X by the equivalence relation (s,x) (t, y) r for some u, v E S, we have (us, ux) = (vt, vy)). Define a new action of S on S - I x by t(s,x) = (s, tx) for any s,t E S, x E X , and let X ~ S - 1 X be given by x ~-* (1, x). T h e n X ~ S - 1 X is a map of Ssets, and S acts invertibly on S - 1 X (the inverse of translation by t is given by (s, x) ~. (ts, x)), and X .~ S - 1 X is a universal arrow from X to a set on which S acts invertibly. If S acts on itself by left translation, then S - 1 S is a group under the product ( s , t ) . (u,v) - (su, tv); S ~. S - 1 S is a homomorphism of monoids that is universal for homomorphisms from 5' to groups. Thus, 8 - 1 5 , is the Grothendieck group of the commutative monoid S. T h e above notions are generalized to categories, as follows (we retain the convention that the categories under consideration are equivalent to full small sub-categories, and one such equivalence has been fixed). A monoidal category S is a category 3 together with a functor + 9S x 8 ~ S and an object 0 E S, such that there are natural isomorphisms (A + B) + C ~A + ( B + C ) , and 0 + A ~ A ~ A+0, for all A , B , C E 3. These isomorphisms are required to be "coherent", i.e., the following diagrams must commute, for all A, B, C, D E ~: A + (B + (C + D)) ~ (A + B) + (C + D) ~ ((A + B) + C) + D A + ((B + C) + D) ~ (A + (B + C)) + n A + ( 0 + C ) ~ (A + 0 ) + C
A+C
7. Comparison of the Plus and Q-Constructions
127
(Here, "natural isomorphism" means a natural transformation giving an isomorphism of functors. One knows that the commutativity of the above diagrams implies the commutativity of all similar diagrams; see MacLane's book Categories for the Working Mathematician). A (left) action of a monoidal category 5" on a category X is a functor + : S • X ; X, together with natural isomorphisms A + (B + F ) (A + B) + F, 0 + F ~ F for all A, B (5 S, F (5 X. Diagrams analogous to the above two diagrams must commute (e.g., in the pentagon, the diagram with D replaced by F (5 X must commute, for all A, B, C (5 S). A monoidal functor between two monoidal categories S, T is a functor f : S ; T, together with natural isomorphisms f (A + B) ~- f (A) + f (B), f (O) ~- O, such that the following diagrams commute:
/((A + B) + C)
~
f(A + B) + / ( c )
I(A + (B + C))
"~ I(A) + I(B + C) ~- f(A) + (f(B) +
(f(A) + f(B)) +
f(C) .f(C))
y(0 + A)
u
f(0) + y(A)
I ( A + 0)
u
y(A) + y(0)
I(A)
~-
0 + I(A)
y(A)
~-
I(A) + 0
A functor f 9X ; 3) between categories with S-actions preserves the action ff there is a natural isomorphism A + f ( F ) ~- f(A + F) for all A E S, F E X such t h a t appropriate diagrams commute. If S is a monoidal category acting on a category X, we say that S acts invertibly on X if each translation X ~ X, F ~ A + F , for A E S, is a homotopy equivalence. We try to imitate the construction of X ---+ S - 1 X for a commutative monoid, to obtain a functor f 9X ; 8 - 1 2 of categories with S-action such that f preserves the action, S acts invertibly on S - 1 X , and f is "universal", at least in some homotopy theoretic sense. If S is a monoidal category that acts on a category X, let (S, X) be the category with the same objects as X, such that an arrow F * G in (S, X) is an equivalence class of pairs (A, A + F , G), where A E S, and A+ F , G is an arrow in X;'(A, A + F , G), (A', A' + F ,~ G) are equivalent if there is an isomorphism u" A ~ A' in S such t h a t A+F
\
"+~ ; A ' + F
/
G commutes. (S, X) is somewhat analogous to the "quotient" of X modulo S.
128
Algebraic K-Theory
Let ~' be a category with S-action, and let S act diagonally on ,5 x X, where ,5 acts on itself by left translation. We define S - 1 ~ ' = (S, S x A'). Suppose 8 is commutative up to natural isomorphism, i.e., there are natural isomorphisms A + B ~ B + A for all A, B E S such that appropriate diagrams commute. Then the S action on ,~ • X given by A + (B, F ) ( B , A + F) induces an ,5-action on S - 1 A '. To see that commutativity is needed, we note that if (D, (D + B, D + F ) - - ~ (C, G)) represents a morphism in S - I X ', we would want a naturally associated morphism in 8 -1X' represented by (D, (D + B, D + (A + F ) ) ---, (C, A + G)). On the other hand, we have a morphism(D + B , A + (D + F)) ~ (C, A + G) in S x X'. Thus if 8 is commutative up to isomorphism, D + (A + F ) (D + A) + F ~ (A + D) + F ~ A + (D + F). We observe that if S is commutative, then the above S-action on ~' is invertible. Indeed, if A E S, then the functor (B,F) ~ (B,A + F) has homotopy inverse (B,F) ~ (A + B,F), since both composite functors equal (B,F) ~ (A + B , A + F), and the arrow in 8 - 1 X ' given by (A,A + (B, F ) ~ (A + B, A + F)) gives a natural transformation from the identity functor to the functor (B, F) , (A + B, A + F). If every arrow in S is an isomorphism, we claim t h a t 0 E (S, S) is an initial object, so that (8, S) is contractible. To see this, if A E S, there is an arrow (A,A + O---~A) from 0 to A in ( 8 , 8 ) ; if ( B , B + OJL-~A) is any arrow in (S, 8) then there is an arrow u : B , A given by
B+O~-
B
.4 By the naturality of A + 0 ~- A, the square below commutes: B +0
1+~
A+O
B
",
A
Thus we have a commutative triangle B+O
"+~ ) A + O .4
and since u is an isomorphism (since we assumed all arrows in 3 are isomorphisms), (B, B + 0 ~ ~A) and (A,A + 0 ~ A) define the same arrow. Hence 0 E (S, S) is an initial object, as claimed.
7. Comparison of the Plus and Q-Constructions
129
Next, suppose t h a t (in addition to all arrows being isomorphisms) all translations 8 ; 5', B , A + B, are faithful (one-one on morphisms in S). T h e n given any arrow B ~ B ' in ( S , S ) represented by ( A , A + B s ~B'), we claim t h a t A is determined up to a unique isomorphism. Indeed, in any case A is determined up to isomorphism; so we must show t h a t if u : A ~. A is an automorphism such that the diagram in 8
A+B
"+~s~ A + B BI
commutes, then u = 1A, the identity map on A. But since f is an isomorphism, u + 1B = 1A+B = 1A + l s ; hence u and 1A become equal after translation by B, so t h a t u = 1A. Thus, if p" S - 1 X
, ( S , S ) is given by p ( ( B , F ) ) = B on objects,
p((A, (A + B , A + F ) - - - - , ( B ' , F ' ) ) ) = ( A , A + B ~.B'), (l,g) f then p is a functor making , S - I X cofibered over ( S , S ) . Indeed, given an arrow in ( S , S ) represented by ( A , A + B : ; B') we associate the functor p - l ( B ) ~ p-l(B'), (B,F) ; ( B ' , A + F ) ; there is a natural transformation from the identity functor on p - l ( B ) to the above functor, which (as B varies) gives an equivalence p / B ' ~ p'(B') for any B ' E S. An arrow in p - l ( B ) is an arrow (B, F ) , (B, F ' ) in S - 1 X t h a t covers the identity arrow of B in (S, S). The identity arrow of B is given by (0, 0 + B B). Hence F ; (B, F ) gives an isomorphism of categories X , p-I(B), for any B E S. Under this identification, if (A, A + B , B ' ) is an arrow in ( S , S ) then the associated cobase-change arrow p - l ( B ) ; p-l(B') becomes translation by A on X. We now have the tools to prove
T h e o r e m ( 7 . 1 ) . Let 8 be a monoidal category, commutative up to isomorphism, such that all arrows in 8 are isomorphisms, and the functor S , 8, B ~ A + B , given by translation by A, is faithful, for each A E S. Let 8 act on X . Then 8 acts invertibly on X ~ X ~S - a X is a homotopy equivalence.
P r o o f . If X ~ S - 1 X is a homotopy equivalence, then since S acts invertibly on ~ - 1 X and the functor preserves the S-action, 8 must act invertibly on X. Conversely, if 8 acts invertibly on X, so t h a t all translations on X are homotopy equivalences, then the cobase-change arrows of p are
Algebraic
130
homotopy-equivalences.
K-Theory
Hence by Theorem B t~' ~ 8-1,~ '
i
pt.
-,
I
(s,s)
is homotopy Cartesian. Since (,9,3) is contractible, X , ~-1~, is a homotopy equivalence. This proves the theorem. If ,9 is a monoidal category as in (7.1) above, then 7r0(S) = 7r0(B8) is an Abelian monoid. If 3 acts on X, Ir0(S) acts on H p ( X , Z) = H p ( B X , Z) for each p >_ 0. Since S acts invertibly on S-Ix, ~r0(S) acts invertibly on H. (,9-1 X, Z). The natural map H. (S, Z) ---~ H. (S- 1X, Z) thus induces a m a p on localized modules ~r0(S)-IH.(X, Z) : H.(,~-IX, Z). Theorem
(7.2). Under the above conditions, the natural maps
71"0(S)- 1g p ( x , Z)
; Hp(S- 1,~', Z)
are isomorphisms V p ~ O. P r o o f . Consider the double complex
Oq:
II
(Bo .....
B,,)eN,,(,V,S)
IIz
N, (p\Bo)
(which can be regarded as the double complex associated to the bisimplicial set Tpq -- Tpq(p) introduced in the proof of Theorem A, with the functor f :C ~ iD replaced by p: 8 - I X : (8, 8)). The natural S-action on S-I X (via the S-action on X), together with the trivial 3-action on (S, 8), yield an "action" of 8 on the double complex, in the following sense. Each object ofS yields an automorphism of E~p,q for each p, q, compatible with the differentials; an arrow A ~B in 8 givesa chain homotopy between the automorphisms induced by A and B, respectively, with respect to the differential in the q-direction (in fact, for any given B0 E (S, ,5), ,5 "acts" on the (single) complex given in degree q by H z, in the above sense). t%(p\Bo) W e may rewrite E~ as
(Fo . . . . .
H
Fq)eNq(,~ -1.,~')
II
z.
IVp(p(Fq)\(S,,S))
Hence, the homology groups in the p-direction (i.e., the E 1 terms for one of the spectral sequences for the double complex) are direct sums of groups Hp(p(Fq) \ (S, S), Z). But p(Fq) \ (,S, ,S) has p(Fq) as an initial object, and
7. Comparison of the Plus and Q-Constructions
131
is hence contractible. Hence we have
E ,q =
H
Z
N,(s-~x) 0
ifp=0. ifp > 0.
Hence this spectral sequence degenerates at E 2, with E02 ,~ = Eoco,q
X,Z),
H (S
E P,q ~176 = 0 for p > 0. Also, the automorphism induced by any object of 3 is just the automorphism of H q ( S - I ~ ' , Z) induced by translation on S - 1 ~ ', i.e., the action of 8 is induced by the natural (invertible) action of the monoid lr0(,~) on H q ( $ - I X , Z). The other spectral sequence for the double complex has E 1 terms obtained by computing the homology in the q-direction of the E~ terms. Thus we get
E~,q =
H
nq(p \ Bo, Z) =~ H
Bo-'~"" ~ Bp
nq(p-l(Bo),Z)
Bo'*'""* B v
~=
H
II
(X,Z)
Bo-*'"--* Bp
from the cofibered structure of p, and the natural isomorphism X p-1(Bo). Let Hq(X) 9 (S,$) ; Ab denote the functor B ~-* Hq(X,Z), (A, A + B , B') , (endomorphism of Hq(X, Z) induced by translation by A) (here ,4b denotes the category of Abelian groups). For any functor F-C , jib, define Hp(C, F) to be the pth homology of the complex
Cp (C, F) =
H (Ao . . . . .
f ( Ao) A~)ENI, C
with the natural differential induced from the simplicial structure of NC. Then we may write the E 2 terms in the above spectral sequence as
E~,q = H,( (S, S>, Hq(X)). The action of ,~ on E~,q is given by the action on Hq(X, Z) induced by translation on X. Thus, the action of ,~ corresponds to the natural lr0(,~)module action as endomorphisms of the functor Hq(X). Hence, the spectral sequence
E~,q = H,(<,~, 8>, Hq(X)) --~ g p + q ( S - l x , Z) can be viewed as spectral sequence of 7r0(,~)-modules. Since localization of modules over a commutative ring with respect to a multiplicative set is an exact functor, we may invert the lr0(S)-action on
132
Algebraic K-Theory
the above spectral sequence to obtain a new spectral sequence of lr0(S)-1 7r0(S)-modules
Ei, = II
Np(S,S)
_--Hp+q(S-lx, Z). N o w 0 E (,~, ,~) is an initial object, so t h a t we have a unique arrow 0 - - , B in (8, 8) for any B E (S, 8), given by (B, B + 0 ~ B). The corresponding cobase-change functor p - l ( 0 ) ---, p-l(B) is identified with translation by B on X ~ p-1 (0) ~ p-I(B). Now translation by B induces an automorphism of 7ro(S)-lHq(X), so that we have a canonical isomorphism
7ro(S)-lHq(p-l(o),z) ~-lro(S)-lHq(p-l(B),Z) for any B E (8,,~). Thus we may rewrite the complex of
E~,q terms as
Elp,q-- ( H
Z)(~)zTro(~)-lSq(p-i(0),7]~). N, {,~,S)
Since (S, S) is contractible, the complex
C,( (S, S) , Z) = [ I Z N,(S,s) has homology groups Hp((S, 8), Z) = 0 unless p = 0, and H0((S, S), Z) = Z. Thus E~,q = 0 unless p = 0, and E2o,q = 7ro(S)-lHq(X). Hence the localized spectral sequence degenerates at E 2, giving isomorphisms
7ro(,S)-lHq(X, Z) '~ H q ( S - 1 X , Z). To see that this isomorphism is just the map induced by localizing
Hq(X,Z) ~
Ha(S-Ix, z)
given by X .~8 - 1 X, compare the spectral sequence above with the trivial spectral sequence E~,q = { HP(X'Z) i f q = O 0 ifq ~: 0
E~,q ==~ Hp+q (X, Z), which we regard as the analogous spectral sequence for the functor X .~ {0}. The comparison is done using the diagrams of categories and functors X
l
{0}
This completes the proof of (7.2).
~
,S-IX ,
1
(s,s)
7. Comparison of the Plus and Q-Constructions
133
(7.3). T h e Functorial Version of the Plus Construction. Let :P be an exact category in which all short exact sequences split, and let Iso(7 ~) be the subcategory with the same objects as :P, whose arrows are all the isomorphisms of :P. Then the direct sum (9 : Iso(P) x Iso(P) ~ IsoCP ) makes 8 = Iso(7 ~) into a monoidal category in a natural way. Then B S - 1 8 is an H-space, with the multiplication B 8 - 1 8 x B 8 - IS ~ B 3 - 1 S induced by the functor 8 - 1 8 x 8 - 1 S ---4 8 - 1 8 , ((A,B), (C,D)) H (A (9 C , B (g D). In particular, let R be a ring, P = P(R), the category of finitely generated projective (left) R-modules. One checks easily that ~r0(8-1~) Ko(R) (this follows from the fact that ~ 0 ( S - I ~ ) = ~o(F1B,S-1,.q), where Fx B S - l~q is the 1-skeleton of B S - 18). If A E P, let h u t ( A ) be the category with 1 object A, and arrows given by the group h u t ( A ) of automorphisms of A as an object of :P. Thus h u t ( A ) is the full subcategory of S with the single object A. There is a functot h u t ( A ) ~ 8 - 1 8 given by A , (A,A), u ~ ( 0 , 0 + ( A , A ) a-9--~(A,A)) for u E h u t ( A ) , where ~ : (O(gA, O(gA) , (A,A) is given by ~ = (Ul, u2), u l : 0 (9 A ; A the natural isomorphism, and u2 : 0 (9 A , A being the composite u2 = u o Ul. The arrow ( R , R + (A,A) ~ (A (9 R , A (9 R)) yields a natural transformation of functors hut(A) , S - 1 S , making the diagram B hut(A) " B Aut(A (9 R)
\
BS-IS
J
commute up to homotopy. Thus, there is a map, well defined up to homotopy, B G L ( R ) = lim B hut.(R en) , B S - 1S. In fact, the map factors through the identity component B ( S - 1 S ) 0 . The map B G L ( R ) ; B 8 - 1 3 can be concretely realized as follows. Let 3n be the connected component of ~ containing R an (so that ~ is the full subcategory of ~ with objects A ~ Ran). Let N be the totally ordered set of non-negative integers (regarded as a category, using the ordering). Let s denote the category of pairs (n,B) with n 6 N, B 6 3n, where an arrow ( n , B ) - - . (n + m, C) is an isomorphism TIn(B) ~ C, where To(B) - B, TI(B) - R (9 B, TIn(B) - T1 o Tm-I(B) for m > 1 (there are no arrows (n, B) ---. (n', C) with n' < n). Let f : E ; N be the functor f ( n , B ) - n. Then f - l ( n ) ~ ,gn, and f makes s cofibered over N with ,
+mi.
N. Clearly s ~ lim(f \ n), and f - l(n) ~-, (f \ n) is a homotopy equivalence
134
Algebraic K-Theory
for each n E N. Further Aut(R Sn) ~-Sn is a homotopy equivalence for each n E N, so that we have homotopy equivalences B G L , ( R ) ; B(f\n), such that the diagram
BGL,(R)
1
B ( f \ n)
~ BGLn+I(R)
,
1
B ( f \ n + l)
commutes up to homotopy. Thus lr,(Bs
= lira lri(B(f \ n)) = lim 1ri(BGLn(R))
i.e., 7rl (BE) ~ GL(R), l h ( S s = 0 for i ~ i. Thus B E is another model for the homotopy type of BGL(R). Finally, we note that (n, B) ; (R ~n, B) yields a functor g : E ~ ( s - t s ) , if we regard R Sn as Tn(O). We claim that Bg : BE , B ( S - l s ) 0 induces an isomorphism on integral homology groups. To prove the claim, we first note that if e E lro(BS) is the class of the component containing R, then 7ro(B,S)[e-1] '~ 7ro(SS-l,S) ~ Ko(R), since ~r0(BS) is the set of isomorphism classes of objects of 7~(R), made into a monoid by direct sum, and P, Q ~. T'(R) have the same image in K0(R) if and only if R $~ ~ P ~ R ~ @ Q for some n E N. Hence, from (7.2), H p ( 8 - 1 8 ) ~ 7r0(S-1)Hp(,~) ~- gp(S)[e-']. Now
go(R),
gp(~-ls) ~ gp((S-ls)0) • and
Hp(8) ~- H Hp(B Aut(P)) ~ H Hv(SP) [P]E~vo(B$) [P]EIro(B$) where ,~p is the connected component of S containing P. The map g p (S,)
, g p ( S - tS)
is induced by the composite functor 8 , c 8 ~ 8 - 1 8 , A H (0,A). If S , = f - 1 (n) C s ; ( 8 - IS)0 c ( S - 1 S ) is the functor induced by g, given by A ~-, (R e", A), then we have the functor obtained by translating this functor by R e" (which corresponds on homology to multiplication by
en)~
A ~ (R e", R ~ @ A). But we have an arrow in S-1 $
(R ~ R
(o, A) - - . (R
R
A))
7. Comparison of the Plus and Q-Constructions
which yieldsa natural have a commutative
transformation diagram
~(&)
1
Hp(s
.~
~
offunctors
$,,
, ~-I~.
H~(S)
,
H,(S-'S)
Hp(($-l,S)o)
,
Hp(,S-I,S)
135
Thus,
we
l-:
where .e n denotes multiplication by e n. Thus, the m a p
g,(&)
, H~((S-~S)o) c H , ( S ) [ : 1]
induced by g is x ~-, x - e - n . We claim t h a t in fact HP(($-15)~
~ E
Hp(,Sn)- e - n C HP($)[e-1].
Since H p ( ( $ - l ~ q ) o ) C Hp(~q-l~q) ~ g p ( s ) [ e - 1 ] , every y e H p ( ( ~ q - l $ ) o ) can be w r i t t e n as y = x . e - n with x E H p ( $ ) ; since
H.(S)= I_[ H.(S.). [Ple~o(S) we m u s t have x E E [ p ] Hp(,Sp) where [P] runs over the classes in 7ro(~q) such t h a t [ P ] - e - " = 0 in Ko(R), i.e., [P] ~ [R en] in Ko(R). Since x . e - " = ( x - e m) 9e -(m+") and x ~ x - e m is induced by the functor ,9 , ,9 given by t r a n s l a t i o n by R era, we can write y = x . e - " w i t h x ~. Hp($n), for some n. T h u s
H~((S-~S)o) ~ ~ H~(S.) . ~-" as claimed. B u t it is easy to see (from the definition of localization) t h a t
nEm
is canonically isomorphic to lim Hp(,~n), where the maps ---4
Hp(&)
, Hp(S.+~)
in t h e direct s y s t e m are all -e. Hence, from the diagrams
HAS.)
\
e > H~ (S.+I)
/
a~((s-~S)o)
136
Algebraic K-Theory
and
Hp(S,)
~ ~ H p ( f -1 (n))
" Hp(f_,)
Hp((8-18)o) we deduce that lim Hp(Sn)
-
....@
;
Hp(s
( (S- 'S)o) commutes, i.e., g. is an isomorphism. Thus, we have an isomorphism of integral homology groups
Hp(BGL(R)) "~ H p ( s
Hp((S-18)0).
Since (,S-18)0 is an H-group, 7r1((8-~8)0) is Abelian, and is thus isomorphic to H i ( ( 8 - 1 8 ) 0 ) ~ HI(BGL(R)) ~ G L ( R ) / E ( R ) (see (A.47), (A.52)). Hence from the universal property of the plus construction, we have a diagram
BGL(R) ~- Bf..
h ;
BGL(R) +
B ( S - 18)0 - Hp(B(8-1S)0) is an isomorphism, since where Bg + 9 Hp(BGL(R) +) h, Bg are homology isomorphisms. But BGL(R) +, B ( 8 - 1 S ) o are H-spaces with the homotopy types of CW-complexes, and so Bg + is a homotopy equivalence (A.54). Thus we have proved: T h e o r e m (7.4). B ( S - 1 S ) is homotopy equivalent to Ko(R) x B G L ( R ) +, where 8 - Iso(7~(R)). (7.5) T h e Loop S p a c e of BQ79(R). Our final goal in this chapter is to prove Theorem (5.1), i.e., that there is a natural homotopy equivalence
BGL(R) +
~ (fl BQP(R)) ~
In fact we prove below (Theorem (7.7)) that there is a natural homotopy equivalence fl B Q P ( R ) - - ~ B ( 8 - 1 , S ) , which, combined with (7.4), yields the result.
7. Comparison of the Plus and Q-Constructions
137
Let 8 be a monoidal category as in (7.1), and let f 9X - - ~ y be a functor between categories with S-action, such that f preserves the action. If ,5 acts trivially on Y, we say S acts fiber-wise with respect to f; S does indeed act on the fibers f - l ( G ) , G E Y. If in addition, f makes X fibered over Y, we say t h a t the S-action is Cartesian. In this case, one can show (details left to the reader) that S - i f 9 8 - 1 X ~ Y is also fibered, with fibers 8 - 1 f - l ( G ) , G E 3), and base change maps induced by localizing those of f. Similarly, if f makes X cofibered over Y, we say t h a t the Saction is co-Cartesian; then S - i f . S - 1X ." 3) is still cofibered, with fibers S - l f - l ( G ) , G e Y. Now suppose S is as in (7.1), and S acts on X. Assume (i) every arrow in X is monic; (ii) V F E X, the functor S on Hom-sets).
, X, B ~-~ B + F, is faithful (i.e., injective
Then as in the proof of (7.1), one checks that the projection S x X .- X induces a cofibered functor q 9 S - 1X .~ (8, X), with fibers q - ~( F ) ~ S (S , q - l ( F ) is given by A ~ ( A , F ) ) , such that cobase-change m a p s become translations on S. Define a new S-action on S - 1 X , through the action of S on S • X on the first factor, A + (B, F ) = (A + B, F). Then this action is clearly co-Cartesian with respect to q. Localizing, we obtain a cofibered functor S - tq . S - 1S - 1X , (S, X), each fiber of which is isomorphic to S - 1S, with base change maps given by translations, which are invertible on S - 1 S . Thus Theorem B yields a homotopy Cartesian square (for each F E (S, X)) s-lc.~
1
{F}
;
S-1S-1X
.-
(S,X)
1
The functor 8 - 1 3 : ~ S - 1 S - 1 X is (A,B) ~-, ( A , ( B , F ) ) . If (S,X') is contractible, then ~q- l~q ~. S - 18-1 X, is a homotopy equivalence.
Let S be as in (7.1), and let X satisfy (i), (ii) above. Suppose ( S , X ) is contractible. Then for each F E X, S - 1 3 ~ S-1X given by (A, B) ~-~ ( A , B + F) is a homotopy equivalence.
Lemma
(7.6).
P r o o f . As seen above, 3 - 1 3 - - ~ S - 1 S - 1 X , (A,B) ~-, ( A , ( B , F ) ) is a homotopy equivalence. Clearly S - 1 3 ---4 3 - 1 S , (A,B) ~ ( B , A ) is a homotopy equivalence (which is its own inverse). Hence 3 - 1 3 9 3-13-1 X, (A, B) ~-~ (B, (A, F ) ) , is a homotopy equivalence.
138
Algebraic K-Theory
On the other hand, we have a functor S - 1 X " ---. S - 1 8 - 1 X ', (A,F) (0, (n, F)), which preserves the S-action given by B + (A, F ) = (B + A, F) on ,5-1X, and B + (C,(A,F)) = (C,(B + A,F)) on S - 1 8 - 1 X . But B + (A, F) = (A, B + F) gives a functor 8 - 1 X ---, ,5-1X which is a homotopy inverse for the above translation by B on 8 - 1 X , since there is a natural transformation from the identity functor to either composite (B, B + (A,F) ----. (B + A , B + F)) (where we use the translation B + (A,F) = (B + A, B + F), used in defining ,~-Ix'). Thus, the new S-action B + (A, F) = (B + A, F) on ,q-1X is invertible. Hence S - 1 X 3 - 1 S - 1 X , (A,F) ~ (0, (A,F)) is a homotopy equivalence, by (7.1). Finally, consider the triangle 8 -18
~ 8-18 -1X
\
/
8-1X corresponding to S - 1 $ ,- , S - I S - I , ~ ', (A,B) ~. ( B , ( A , F ) ) , S - 1 8 8 - 1 2 , (A,B) ~ (A,B + f ) , 8 - 1 2 ----, , 5 - 1 S - I X , (A,F) ~-~ (0, (A,F)). The composite 8 - 1 S ~S - 1 X : S - l ~ q - l X is ( A , B ) ~-, (A, (A, B+F)). The arrows in 8 - 1 8 - 1 X
( B , B + (0, (A, B + F))-Z-.(B, (B + A, B + F))) (0,0 + (B, (A,F))--~(B, (B + A , B + F))) (where u is induced by the arrow (B, B + (A, F ) - - ~ ( B + A, B + F)) in $-1X') give a chain of natural transformations connecting the two functors ~q- 18 ---. 3 - 1 S - 1X'. Hence the triangle commutes up to homotopy. Since two sides axe homotopy equivalences, so is the third. This proves (7.6). Now we can compute ~ BQP(R). More generally, let P be any exact category where all exact sequences split; let S = Iso P. T h e o r e m (7.7). There is a natural homotopy equivalence
~2B Q P
~B S - 1 S .
P r o o L Let s be the following category. An object of E is a short exact sequence in 7~. An arrow in g
(o--, A---, B
C - - . O)
, (0
A'-.
o)
7. Comparison of the Plus and Q-Constructions
139
is defined to be a n equivalence class of d i a g r a m s 0
~ :
A
I
0
.~ A'
0
;
~
B
--~
B
sl
A'
--,
sl
;
C
:
~.
0
;
0
;
0
T
C1
I o I
BI
,
C'
where t h e s q u a r e m a r k e d [3 is a pullback; a d i a g r a m isomorphic to t h e above one w i t h a n i s o m o r p h i s m inducing t h e i d e n t i t y m a p s on A, B , (7, A ~, B ' , C ' defines t h e s a m e arrow in E. T h e r e is a f u n c t o r f " E - - * Q7 ~, (0 --, A --~ B --, C --, 0) ~-4 C. T h e fiber f - l ( C ) = E c is t h e c a t e g o r y of s h o r t e x a c t s e q u e n c e s (0 ~ A --, B - , C --, 0), with m o r p h i s m s given by isomorphisms 0
.~
A
0
-~ A'
~
B
---,
C
,
0
---,
B'
;
C
,
O.
We claim f m a k e s E fibered over Q P . defined as follows: if q" C ' --* C, t h e n ( q! ) , " E c ,
T h e base change m a p s are
; Ec
is given b y (0 if i" C ~
, A
, B ~ :C'
(0
, u - 1(ker q)
C ' , t h e n (i~)* 9 E c , (O--~A ~
,0 ; B qo=, C
, 0);
~. E c is
- , B----,C'----~0)
(0 ~ : A ----, B x c , C
, C----,
O).
O n e checks t h a t t h e s e definitions do make f a fibered functor. Let S act on s by (A') + (0 =(0
, A
,B
; A' ~ A
;C
, O)
, A ' $ B ~=- , C
~ 0).
O n e verifies t h a t this S - a c t i o n is Cartesian with r e s p e c t to f . Also, S C0 given by A ~
(0
is an equivalence of categories.
, A ~A,A ---~ 0
, O)
140
Algebraic K-Theory
Lemma
( 7 . 8 ) . F o r a n y C e 7~, (~q,s
is contractible.
P r o o f . We define a product Ec x Ec
* Ec by
(0 --. A --, B --, C ~ O) + (0 --4 A ' ~ = (O --, A ~ A ' ~ B x c B ' ~
B ' --. C ~ O)
C --, O).
One checks easily t h a t this makes (S, Ec) into an H-space, which is homotopy commutative and associate, with identity element 0 --,
0
~, C I - ~ C
- - - , O.
Next, since every short exact sequence in 7v is split, there exists an isomorphism A + (0 ~ 0 - - , C ~ C
~ O) ~ (0 ~
A ~
S -~ C ~ 0),
giving an arrow (0 --, 0 --, C ! % C
~ O) ~
(0 ~ A ~
B ~
C ~ O)
in (S, Ec). Hence (S, Co) is a connected, h o m o t o p y associative H-space, and is hence an H-group (see (A.47)). Finally, for any (0 --. A --, B --, C -~ 0) E s the diagonal map B , B x c B provides a natural isomorphism A +(0--, A ~ B--.C-~0)
~ (0--, A ~ A
~
B Xc B ~C--,O).
This can be viewed as giving a natural transformation from the identity functor on ( S , s to the functor corresponding to multiplication by 2 in the H - g r o u p structure. Thus, in the group (under pointwise multiplication) of h o m o t o p y classes of self maps of B ( S , E c ) , we have the identity x = x 2. Thus, this group is trivial, i.e., B ( $ , E c ) is contractible. This proves (7.8). Lemma
(7.9). T h e square 8-18
,
S-1E
{o)
,
qp
is h o m o t o p y C a r t e s i a n .
P r o o f . By Theorem B, it suffices to prove t h a t the base change functors for the fibered functor 3 - 1 1 are homotopy equivalences. Since every arrow in Q7 ) can be factored as q!o i! where i is admissible mono, q is admissible epi, and for i" C ~-. C', q" C ~ -* C we have iv,! = i,. o iv!, q~, = q! o q~ (where i c " 0 H C , q c " C --~ O, etc.), it suffices to prove this for arrows ic! and qb'
7. Comparison of the Plus and Q-Constructions The base change (q~)"* ( S - 1 s (A', (0 --o A ~
~
B --o C --o 0)) ~
8-1s
141
is given by
(A', (0 -4 B l S ; B --o 0 ~ 0)).
Identifying 80 with ~, this is just 8 - 1 8 c
) 8-13,
(A', (0 --* A --. B - . C --. 0) ~ (A', B). Since (8, s
is contractible, (7.6) implies that 8 - 1 S
, 3 - 1 oCc given
by (A', A) ~ (A', 0
,A
, A~ C
, C ---, O)
is a homotopy equivalence. The composite with (q~)* is the functor 8 - t 8 ~ 8 - 1 8 given by (A',A) , ( A ' , A (9C). This is just t r a n s l a t i o n by C on ~q-16", which is a homotopy equivalence, since S acts invertibly on 8 - 1 8 . Hence (q~) is a homotopy equivalence. Similarly (iv!)* : ~ - 1s .~3 - 1 t;0 is given by (A',0
:A
".S
(A', 0
,C
~0)
~ A I-~AA
". 0
which is identified with the functor 8 - 1 s (A', 0 ~
A
,B
,C
". 0),
~8 - 1 8 , ,0)~
Now the composite functor S - 13 ---, ,~-1s identity. This proves (7.9).
(A', A). ----* ~ - IS is in fact the
L e m m a (7.10). S - 1 E is contractible. P r o o f . For any category X, let Sub X denote the category whose objects are the arrows of ~', such t h a t an arrow f - , g in Sub ~' is a pair of arrows h, k of k' satisfying k o f o h = g. !
g
Then the functor ( f : M ; N) ~-. N, Sub X' ; k' is an equivalence of categories, with inverse given by N ~-, (1U : N ." N). Let X' be the subcategory of QP with the same objects, whose arrows are all the maps i.,, for admissible monos i in :P. Then 0 E X" is an initial object, so t h a t X', and hence Sub X', are contractible.
142
Algebraic K-Theory
T h e r e is a functor Sub A" ) s given by (i) : M ) N ) ~-, (0 M--L-,N ; cokeri ) 0). An arrow il) ) is! in S u b X , for i i 9M i ~-* N j , j = 1, 2 consists of a square of admissible monomorphisms M1 ;
"Nl
sI
i~
Ms > i, " N s yielding the diagram 0
)M1
0
) M2
0
)M2 ~
Thus, Sub A'
i~ ) N:
) cokeril
I,o,, ii ii : o ) N1
N2
~ 0
T :
) coker (il o i'x)
)0
" cokeris
" 0.
; 8 is an equivalence of categories, with inverse
(0 ~
M----,N ~
P
i
, 0 ) ~ - , (i!: M -
, N).
Hence ~" is contractible. Thus S acts invertibly on E; hence by (7.1), ) S - 1~: is a homotopy equivalence. Hence S - 1C is contractible. T h i s proves (7.10). F r o m (7.9) and (7.10), we have a h o m o t o p y equivalence
E
B8-18 ~
F ( B S - l f, {0})
to the homotopy fiber of B S - l f : B , S - I ~ ) BQ79. But by (7.10), t h e n a t u r a l inclusion f~ BQ79 c F ( B S -1 f, {0}) (where f~ BQ79 is the space of loops in BQ79 based at {0}) is a homotopy equivalence, where this inclusion is given by w ~ (xo,w) where x0 is a base point of B S - I E , w : I , BQ'P a loop based at 0. Indeed, one sees easily t h a t a deformation retraction of BG-1F_. into {x0} induces one of F ( B S - l f , {0}) into f~ BQ79. T h u s we have the chain of homotopy equivalences
B S - I , s . , F ( B S - l f, {0}) ~ which proves Theorem (7.7). equivalence
f~ BQ79
As noted earlier, this gives the h o m o t o p y
B G L ( R ) + ~ f~ BQ79(R)o of T h e o r e m (5.1).
7. Comparison of the Plus and Q-Constructions
143
We end this chapter with a brief discussion of the relation between the transfer (or norm) maps on K-groups defined using the plus and Qconstructions. The situation we consider is as follows: let R, S be rings, with a homomorphism R --+ S making S into a finitely generated projective R-module. On the one hand, there is a 'direct image' functor f . : ~ ( S ) --~ 7~(R) between the categories of finitely generated projective modules over S and R respectively, given by f . ( P ) = P, regarded as an R-module via the homomorphism R --, S. This is exact, and so yields a functor QT~(S) --, Q:P(R), and a map on K-groups f . : K~(S) --+ K~(R). On the other hand, we may choose a finitely generated projective R-module P0 such that there is an isomorphism f . ( S ) ~ 9 Po ~- R Sm for some m > 0, which we fix. This yields a compatible family of isomorphisms f . ( S •") ~ Por ~- R Cmn, giving a compatible family of (injective) homomorphisms GLn(S) --+ GLmn(R), and hence a homomorphism GL(S) --+ GL(R). This clearly maps E(S) into E(R), hence yields a map of spaces f.' : B G L ( S ) + -+ BLG(R) +. One checks by the methods of Chapter 2 that the homotopy class of this map is independent of the choices made, so that there is a well defined map on K-groups f.~ : Ki(S) --. K i ( R ) for each i > 0. For i = 1, 2 these are the transfer maps defined in Chapter 14 of Milnor's book Introduction to Algebraic K-Theory. We will sketch a proof that f . = f.~ on Ki(S) for all i > 0. Let 3 s = IsoCP(S)), SR = Iso(:P(R)) be the monoidal categories associated to 7~(S), :P(R) respectively; then the exact functor f . : P ( S ) --, P ( R ) induces a monoidal functor f. : 3 s - , 3R. Next, let s s be the categories obtained by the extension constructions over P(S), 7~(R) respectively. Again, there is a functor f . : s --, E(R). These functors f . yield a map between the homotopy Cartesian squares of lemma (7.9) associated to the rings S and R, respectively. Hence there is a homotopy commutative diagram
flBQP(S)
~- ,
f.l f~BQP(R)
B,eslSS If,
~- ,
BS~I,gn
showing that f . 9 BSsX.~s --+ B S ~ I S R induces the transfer map f . on Ki for all i. One now checks that the diagram
BGL(S) + fl,~ BGL(R) +
,
BSslSs
,
If. BS~I,qR
is homotopy commutative. This implies the claimed equality of the two
144
Algebraic K-Theory
transfer maps. We will use f . to denote either of them. It is tempting to try to directly define products on K-groups using the S~ISR construction, for which various functorial properties (like the projection formula) are obvious, and thus to avoid some of the work done in Chapter 2. There is an 'obvious' approach to doing this: one tries to use the tensor product of projective modules to define a functor S~ 18a x S~ 1Sa - . S~*3R, which is given on objects by
(A, B) x (C, D) H (A |
B (9 B
D (9 B |
C),
with brackets inserted at suitable places, and using the natural isomorphisms inherent in the structure of a monoidal category with tensor products to define what must happen on morphisms. However, it turns out that this fails for a somewhat subtle reason: it is impossible to make the assignment on morphisms compatible with composition! For details, see: R. Thomason, Beware the phony multiplication on Quillen's A-I.A, Proc. Amer. Math. Soc. 80 (1980) 569-573. The method of monoidal categories can be used to prove that ,~-1S is an infinite loop space; see, for example, J.P. May (with contributions by F. Quinn, N. Ray, J. Tornehave): Eoo-Ring Spaces and Eoo-Ring Spectra, Lect. Notes in Math., No. 533, Springer-Verlag (1976).
8. T h e M e r k u r j e v - S u s l i n
Theorem
The main references in this chapter are: [MS] [CS] [M] [Mi] [AS] [AS2] [S] IT]
A.S. Merkurjev, A. A. Suslin, K:-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Math. USSR lzv. 21 (1983) 307-340 (English translation) C. Sould, /(2 et le groupe de Brauer (d'apr~s A. S. Merkurjev et A. A. Suslin), Sdminaire Bourbaki 601, Astdrique 105-106, Soc. Math. France (1983). A.S. Merkurjev,/(2 of fields and the Brauer group, Contemporary Math. Vol. 55, Part II, Amer. Math. Soc. (1986). J . S . Milne, I~tale Cohomology, Princeton Math. Set. 33, Princeton (1980). A . A . Suslin, Algebraic K-theory and the norm-residue homomorphism, J. Soviet Math. 30 (1985) 2556-2611. A . A . Suslin: Torsion in K2 of fields, K-Theory 1 (1987) 5-29. J.-P. Serre, Local Fields, Grad. Texts in Math. No. 67, SpringerVerlag (1979). J. Tare, Relations between /(2 and Galois cohomology, Invent. Math. 36 (1976) 257-274.
Notation: If F is a field, we will use multiplicative notation for the group operation on F*, and on Steinberg symbols. Otherwise, we will use additive notation for the group operation in an Abelian group. If A is an Abelian group, n an integer, and nA : A --, A is multiplication by n, then we let nA = ker nA and n . A = im hA. (8.1) T h e G a l o i s S y m b o l Let F be a field, F a separable closure of F , G = Gal (-F/F). Let n > 0 be an integer relatively prime to char. F, and let #n denote the group of n t h roots of unity in F*. We have the Kummer sequence of G-modules (with -
for
e
)
0--* ~,~ -4
~
-*0
giving an exact sequence of GMois cohomology groups
F* ~ F* --4 Hi(F, ]z,) --4
HI(F,-F*)
Jl
0
146
Algebraic K-Theory
where the last group is 0 by Hilbert's Theorem 90. Thus we have an isomorphism Xn = Xn,f " F* | Z / n Z --* H I ( F , # n ) . (We will abuse notation and also use Xn to denote the surjection F* --~ H i ( F , p , ) . ) The cup product gives a m a p F* |
--* (F* |
--, H 1(F, ~ u , ) |
Pn) ~ H2(F, lZn@2).
Lemma
8.1 If a e F*, a ~t 1, then xn(a) U Xn(1 - a) - O.
Proof. Let T" - a = --[Ik i=1Pi(T) TM be the decomposition into prime factors over F[T]. Let ~, e F be a root of pi(T) = 0, and F~ = F(cq). Then k
k
a = Hp,(1)
1 -
TM
i=l
= H gF,/F(1 -- C~,)n'. i=1
Hence
Xn,F(1 -- a) = ~
n, N F , / f o Xn,F, (1 -- ai),
where NF,/F" H l ( F i , # n ) --* g l ( F , lzn) is the norm (i.e., the corestriction). By the projection formula (in Galois cohomology),
x,,,~(~) u x,,,~(x - ~) = ~
~_, n, NF,/~
n,x,,,~(~) u ( N ~ , / F
o x . , F , (I - ~ , ) ) =
(X.,F, ( - ) U X . , F , (I - ~ , ) ) =
i
niNF,/F (Xn,F, (a'~) U Xn,F, (1 -- C~i)) = 0. i
C o r o l l a r y 8.2 The map F* @z F* --, H2(F, # ~ )
given by
a | b ~ x.,F(~) u X.,F(b) is a Steinberg symbol (see (1.14)). Definition:
Let
Rn,F " K2(F) |
Z / n Z --, H2(F,/Zn@2)
be the homomorphism given by (8.2); it is called the norm residue homomorphism or the Galois symbol (see (1.16)). Suppose the group of n th roots of unity # , C F; let r E /Zn be a primitive n th root. Given a, b E F*, let Ar b) be the central simple F algebra with generators X, Y subject to the relations X n = a, y n = b,
8. The Merkurjev-Suslin Theorem
147
~ X Y = Y X . One verifies easily that Ar is indeed a central simple algebra over F of degree n (i.e., dimF Ar = n2). Ar is called a cyclic algebra since it is split (i.e., isomorphic to the algebra of n x n matrices) over the cyclic extension field F(~fb). More generally, let E / F be a cyclic Galois extension of degree n, and a E Gal (ELF) a generator. Given b E F*, consider the F - a l g e b r a A(a, b) which is an E-vector space of dimension n with basis 1, Y , . . . , y n - 1 and algebra structure given by the rules y n = b, Yc = a(c)Y for c E E, and ( Y~+J
yiyj =
bYi+j-'~
if I < i,j and i + j < n, if I _< i, j _< n - 1 and i + j _> n.
If E / F is a K u m m e r extension, so that E = F ( ~ r j ) , this reduces to the earlier construction. We call A(a, b) the cyclic algebra over F associated to the cyclic extension E l F , the generator a E Gal (ELF) and b E F*. For example, if char. F = p > O, and S - F[X]/(X p - X a) is an Artin-Schreier extension, a E Gal (ELF) given by a ( X ) = X + 1, then A(a, b) has generators X, Y subject to the relations X p - X = a, YP = b,
Y X = X Y + Y. If E l F is a cyclic extension of degree m, a E Gal (ELF) a generator, b E F*, then there is another way of describing [A(a, b)] E Br (F), the Brauer class of A(a, b). There is a unique character X" Gal (ELF) -~ g / m g given by X(a) = 1 (mod m). Let ~ : Gal (-F/F) ~ Z / m g be the induced character. From the Bockstein exact sequence (associated to the sequence of trivial Gal ( F / F ) - m o d u l e s 0 --, Z :-~ g --, g / m Z --~ O)
9..--. H'(F, Z) --. Ht(F, Z / m Z ) -~ H2(F, Z) ---, .-. we have a class ~(~) E H2(F, Z). The cup product
H2(F, Z) |
g~
~ H2(F,-F*) = Br (F)
yields a class ~ ( ~ ) O (b) e Br (F). It can be shown (see A. Weil, Basic Number Theory, XII, w t h a t ~f(~)U (b) = [A(a, b)] e S r (F). Now suppose/zn C F ; fix a primitive n th root ~ E Dr,. Given a Kummer extension E = F ( ~ f a ) and a generator a E Gal (E/F), there is a unique i(a)(mod n) such t h a t a ( ~ ) = r ~f~. This defines a character g a l (E/F) --, Z/nZ, a H i(a)(mod n). Let ~ be the induced character of Gal (F/F). Now ~(~)U (b) e nBr (F). But from Hilbert's Theorem 90 and the K ummer sequence, we have an exact sequence
0 --~ H2(F, #n) --* e r (F) "~, Br (F) --, 0, so that h e r (F) ~ H2(F, lzn). Thus 6(~) U (b) U (~) E H2(F, ~ 2 ) .
148
Algebraic K-Theory
L e m m a 8.3 Rn,F({a, b}) = ~(~) U (b) U ((~). See Serre's Local Fields, XIV, Prop. 5.
Proof.
We denote the induced map K2(F) |
by R t n , F
Z / n Z --, , B r (F),
{a, b} H 6(~) U (b),
.
L e m m a 8.4 (a) Let E l F be a cyclic extension of fields of degree n, a E Gal (ELF) a generator, X : Gal (ELF) --, Z / n Z the character given by a ~-. 1. Then the map F* --, Br (F), b ~-, 5(~) U (b) gives an isomorphism
F*/NE/FE* ~- Br (ELF) - ker(Br (F) --, Br (E)). (b) Suppose #n C F. Then P~,FC(a, b}) = 0 r r r
(a, b} e n . K2(F)
b = NE/F(C) for some c e E*, where E - F(
b - NE,/F(C) for some c e E'*, where E' = F[T]/(T n - a).
Proof. (a) One knows (see Serre, Local Fields, X, w Cor. to Prop. 6) that S r (ELF) = ker(Br ( f ) --, Br (E)) -~ H 2 ( E / F , E*). Thus we must show t h a t if 5(~) E H2(E/F, Z) is obtained using the boundary map 6 in the exact sequence
... _._, H I ( E / F , Z ) - - ~ H I ( E / F , Z / n Z )
~ H2(E/F,Z)--, ...,
then b ~-, 6(~)U (b) gives an isomorphism F*/NE/FE* ~- Br ( E / F ) . But b H 6(~) (9 (b) is precisely the periodicity isomorphism for the Tate cohomology groups H ~ E*) ~ ~I2(E/F, E*), for the cohomology groups of the cyclic group Gal ( E / F ) (see Serre, Local Fields, VIII, w (b) Clearly b = NE/F(C) for some c E E* r b = NE,/F(CJ) for some c E E ~*. Hence it suffices to prove: {a, b} e n - K 2 ( F ) ~ P~,F({a, b}) = 0 r
b = NE/F(C) for some c e E*.
From (8.3) and (8.4)(a) applied to E l F , we see that
{a, b} e n . K2(F)=~R~,F({a, b}) = 0 r
b = NE/F(C) for some c e E*.
Hence it suffices to observe that if NE/F : K z ( E ) --* Kz(F) is the transfer map (the map f . obtained from (5.11) for the morphism f : S p e c E --. Spec F ) , then from the projection formula, we have b =
b} = NE/F{
,
= NEll{
8. The Merkurjev-Suslin Theorem
149
which lies in n - K z ( F ) . Here, the projection formula we need is verified in Milnor's book Introduction to Algebraic K-Theory, Ch. 14, with a different definition of the transfer map on K2: one regards K2(F) as H 2 ( E ( F ) , Z), where E ( F ) C GL (F) is the group of elementary matrices; now choose a basis for E as an F-vector space, to get an embedding GL (E) ~-~ GL (F), and an induced embedding of commutator subgroups E ( E ) ~ E ( F ) . The transfer map is the induced mapping on H2. The inclusion GL (E) r GL (F) induces B G L ( E ) + --, B G L ( F ) +, giving maps f . : K i ( E ) ~ K~(F); this yields the above map for i - 2. As discussed at the end of Chapter 7, this agrees with the transfer map defined using (5.11). With this background, we now state the theorem of Merkurjev and Suslin, which is the main goal of this chapter. T h e o r e m 8.5 (Merkurjev-Suslin) Let F be a field, n > 0 an integer not divisible by char. F. Then the Galois symbol R~,F" K 2 ( F ) @ z Z / n Z --, H2(F, #~2)
is an isomorphism. (8.2) P r o o f of t h e M e r k u r j e v - S u s l i n T h e o r e m We first make some preliminary reductions. We begin with a useful lemma due to Bass and Tate. L e m m a 8.6 Let p be a prime number, and let F be a field which has no non-trivial finite extensions of degree < p. Let E / F be an extension of degree p. Then K2(E) /s generated by symbols {a, b} with a e F*, b E E*. Proof. Let F = E(t), so that t satisfies a monic polynomial over F of degree p; thus every element of E is a polynomial (over F ) in t of degree < p. Since F has no non-trivial finite extensions of degree < p, any polynomial over F of degree < p factors into a product of linear factors. Hence every element of E* is a product of linear polynomials in t. Thus K 2 ( E ) is generated by symbols {at + b, ct + d}, a, b, c, d e F, (at + b), (ct + d) e E*. If ac = 0 or ad = bc, then {at+b, ct+d} lies in the image of F * | --~ K2(E) (use the relation { u , - u } = 1 in K2, if ad = bc). If a c ( a d - bc) ~ O, write {at + b, ct + d} =
c(at + b) a(ct + d) c a(ct + d) a { b c - a d ' a d - bc }{ b c - ad' ad - bc }-1{at § b, a d - b--'----c}-1" But
+ b)
+ d)
+ =1; bc - ad a d - bc using {u, 1 - u} = 1, we deduce that {at + b, ct + d} e image(F* |
E*).
150
Algebraic K-Theory
L e m m a 8.7 Let E / F be a finite separable extension of fields, n a positive integer not divisible by char. F. Then we have a commutative diagram K2(E)
R24E H2(E,/~n@2)
NE/F ~
~ NE/F
g2(F)
R.,__,F H2(F ' #n@2)
(In fact the lemma is true without the separability assumption, but we do not need this.) P r o o f . It suffices to consider the case when n = pk for some prime number p. If F ' / F is a finite extension ofdegree relatively prime to p, then since the composite
H2(F ' fln@2)Resff/F g 2 ( F , , pn@2) N ~ F H2(F ' #~2) is multiplication by [F' 9 F], it is an isomorphism. Let F be a maximal algebraic extension of F such that any finite sub-extension of F contained in F- has degree relatively prime to p. Then E |
F- ~
YI
i--I
Ei for certain
extension fields Ei of F (this is true because E / F is separable). Then we have a diagram
/ K2(E)
1-I K2(E,)
[
;
// H2(E, #~ 2)
H2( ,
K2(F)
N2(F)
H H2(Ei, ~n@2)
)
H2(F,
where the vertical arrows are norm maps, the horizontal maps are Galois symbols, and the maps joining the front and back faces are induced by the inclusions of fields. Since the map !h is injective, the lemma for E / F follows from the lemma for each of the extensions Ei/F.
8. The Merkurjev-Suslin Theorem
151
Thus, to prove the lemma, we may assume that n = pk, and F has no finite extensions of degree prime to p. Hence any finite separable extension of F is a tower of cyclic Galois extensions of degree p. If F c E C E ' are finite extensions, then NE,/F = NE/F o NE,/E (where N denotes the norm on e i t h e r / ( 2 or H2). So it suffices to consider the case when E l F is cyclic of degree p. Now by lemma (8.6), K2(E) is generated by symbols {a, b) with a E F * , b E E*. But
Rp, FNE/F{a, b} - Rp, F{a, NE/Fb} = Xp,F(a) U (Xp,F o NE/F(b)) -Xp,FCa) U (NE/F o Xp, s(b)) = NE/F(Xp,E(a) U Xp,E(b)) - NE/FRp,E{a, b}. To prove (8.5), t h a t P~,F is an isomorphism, it clearly suffices to do this when n = pk is a power of a prime number p. From the above lemma, we may also assume t h a t F contains a primitive pta root of unity. From now on, we will fix such a prime number p. 8.8 Let F be a field containing a primitive pth root of unity, such that Rp,f is an isomorphism. Then Rpk,f is an isomorphism for all k > O.
Lemma
Proof. By induction on k, we may assume k > 1, and Rp~,F is an isomorphism for 1 < k. We have a commutative diagram
F* |
pp ~ K2(F) @z Z/pk-1Z ~ K2(F) |
H1(F, I~2) ---, H2(F, pph-,) |
Z/pkZ ---* K2(F) |
--* H 2(F,p~)
Z/pZ ---, 0
---, H2(F,I~ 2)
in which the b o t t o m row is exact, the top row is a complex which is exact except possibly at K2(F) | Z / P k - I Z , and the vertical arrows other t h a n Rp~,F are given to be isomorphisms. A diagram chase now implies t h a t Rpk,f is also an isomorphism. Hence we are reduced to proving t h a t Rp,f is an isomorphism, where p is a prime number, and F contains a primitive pth root of unity ~. T h e main idea of the proof in this case is to compare the Galois symbol for F with t h a t for an extension field F ( X ) , where X is a Severi-Brauer variety over F associated to a division algebra of degree p over F , and F ( X ) is its function field. Along the way, we also prove an analogue f o r / ( 2 of Hilbert's T h e o r e m 90 (see Theorem (S.15)), which is of independent interest. In C h a p t e r 5, we gave Quillen's computation of the K - g r o u p s of a Severi-Brauer variety (see (5.29), (5.40)). Let p be a prime and let D be a central simple algebra over F of degree p (i.e., D is a simple F - a l g e b r a
152
Algebraic K-Theory
with dimF D = p2, and D has centre F). Then either D is a division algebra, or D "~ Mp(F), the algebra of p x p matrices over F . Let X be the Severi-Brauer variety over F associated to D (see Serre, Local Fields, Ch. X). We have an isomorphism, by (5.40),
K~(X) ~- K~(F) (9 g~(D) (9... (9 K i ( D |
( 9 . - . (9 Ki(D|
V i > O.
If D = Mp(F), so that D | = Mp~(F), we have canonical isomorphisms K~(D| ~ K,(F). Indeed, for any m > 0, if we fix an isomorphism Mm(F) ~ End F(W), where W is an F-vector space of dimension m, then W can be regarded as a left Mm(F)-module. Then W is an indecomposable projective module, and we have a (Morita) equivalence of categories 7~(F) --. :P(Mm(F)), Y ~-, W| where V e :P(F) is a finite dimensional F-vector space, and 7)(Mm(F)) is the category of finitely generated projective M m ( f ) modules. Hence QP(F) ---. Q~(Mm(F)) is an equivalence of categories, giving the desired isomorphisms K~(F) ~ K~(Mm(F)). Thus in the situation when D = Mp(F), (5.40) gives an isomorphism
g i ( x ) ~ K~(F) (9 gi(D) (9... (9 Ki(D |
(9... (9 gi(D |
= Ki(F) ~p.
On the other hand, X = IP~-1, so that (5.29) gives an isomorphism
Ki(F)r
~
(aj)o<j
K,(iPPF-l), r = [Opn-,(--1)] e g 0 ( ~ - - 1 ) .
J We claim that the above two isomorphisms K~ (F) r - . K~(IP~-1) coincide. To see this, note that if 7r 9 IP~- 1 - , Spec F is the structure map, then in the notation of (5.40), fl = V | Opp-~(--1), where V = F(IP~- 1 0 ~ - t ( 1 ) ) Hence ff| = V| | Op.-~ (--n), and f.End (ff| ~ End F(V| ~ End F(V *| (we identify coherent sheaves on Spec F with finite dimensional F-vector spaces). Since a simple left module for End F(V *| is V *| the map (given by (5.40)) -
K , ( F ) U K,(End F(V*|
---, K,(PPF-1)
is that given by the functor "P(F) --. P(IP~-1), Ul ~-4 ~'|
|
.
End (V*|
V*|
n
|
Ul.
For any F-vector space W, regarded as a right End F(W)~ have a canonical isomorphism W @End F(W)~
= F,
we
8. The Merkurjev-Suslin Theorem given by the trace mapping W |
7 (F)
W*
just
---~
V1 ~ Opn-i ( - n ) |
153
Hence the above functor
F.
V1,
which corresponds to the map K~(F) --. K~(P~-I), a ~ a . ~ , where = [Op,-~ (-1)]. This proves the claim. If D is a division algebra, E C D a maximal subfield, then DE -E| ~ Mp(E). For any such E, i f X E = X • then X E is the Severi-Brauer variety associated to DE, i.e., XE ~ IP~-1. If f : X E --~ X , then we have a diagram
g,(x) p--1
(gj=oKi(D|
---, ~
K i ( X E ) ~ K~(PPE- 1) r
p-1
X
K i ( D ~ j) "~= K i ( E ) $n
where the vertical arrows are the isomorphisms of (5.40). In particular, for i = 0, we see that K o ( D | '~ Z, Ko(E) ~ Z, and the natural map Ko(n | ~ K i ( E ) is either (i) an isomorphism, if j = 0, or (ii) is identified with multiplication by p, if 1 <_ j < p. To prove (ii), it suffices to show that D | = Mr,,-~ (D j) for some division algebra Dj of degree p over F. Now [D | E Br (F) is an element of order p if 1 <_ j < p - !, so that D| ~ Mrs (D j) for some division algebra Dj over F of degree > 1, such that Dj is split by E. So it suffices to show that any divison algebra Dj over F, which is split by E, has degree dividing p. This follows from the next lemma, shown to me by R. Sridharan. L e m m a 8.9 If E / F is a finite extension of fields, D a division algebra over F which is split by E, then Mm(D) contains a maximal commutative subring isomorphic to E, for some m > 0 (thus m deg D = [E" F]). P r o o f . We note that E also splits the opposite algebra D ~ so that there is an isomorphism of E-algebras E | D ~ --* Mn(E), n = deg D; we can use this isomorphism to regard E Sn as a left-E right-D bimodule. If m is the dimension of E ~n as a right D-vector space, then E ~ End ~ ( E r ~ v
Mm(D); by counting dimensions, E is a maximal commutative subring its dimension over the centre F is the square root of the dimension of the algebra. The following corollary, needed later, is immediate from the lemma. C o r o l l a r y 8 . 1 0 If D is a central division algebra over F of degree n, the maximal subfields of D are, up to F-isomorphism, precisely the extensions of F of degree n which split D.
154
Algebraic K-Theory
Next, consider the above map f* : K1 (X) --* K1 (XE). T h e corresponding map g l ( D | - K I ( D j ) ~ g~ (E) is given as follows. By the corollary to Example (1.6), K I ( D j ) ~ Dj/[Dj, Dj]. Further, K I ( E ) ~ E*, and the map g l (Dj) --, K1 (E) is given by the composite Dj)* ~ G L p ( E ) d~t E*
Dj --* ( E |
where det denotes the determinant. By the definition of the reduced norm for a division algebra, which we denote Nrd, this factors as Dj
Nrd
F* r
E*.
We have the following result of Sh. Wang, On the c o m m u t a t o r group of a simple algebra, Amer. J. Math. 72 (1950) 323-334. L e m m a 8.11 Let D be a central division algebra over F of prime degree p. Then [D*, D*] = ker(Nrd" D* ~ F*). P r o o f . We prove the result in 3 steps. We recall t h a t if c~ E D* and K is a maximal subfield of D containing F(c~), then N r d ( a ) - NK/F(~). Since Br ( F ) ~= 0, F is infinite. (i)
Suppose c~ E D* with Nrd(c~) - 1, and c~ E K* C D* where K is a maximal subfield of D containing F(c~), and K is a cyclic Galois extension of F. If a E Gal ( K / F ) is a generator, then by Hilbert's Theorem 90, NK/F((~) = 1 =~ ~ -- a(/3)//3 for some /3 E g * . By the Skolem-Noether theorem, a " K --~ K is given by conjugation by some element r E D*. Then = a(f~)f~- 1 = Tf~r- 1f~- l E [D*, D* 1.
(ii)
Next, suppose (~ E D* with Nrd((~) - 1, and for some extension field E of degree n of F, we have a E [D~, D~]. Regarding E as an F-subalgebra of the matrix algebra M n ( F ) , so t h a t D~ C M n ( D ) , we see that the n x n diagonal matrix
=
I ~ ~1 ~176176
0
e[ D
~
c
[GLn(D) , GL, (D)I 9
a
Hence the Dieudonn~ determinant of a,~ is trivial, i.e., c~n E [D*, D*]. In particular, since D has prime degree p, D has a splitting field E of degree p over F , so t h a t a p E [D*, D*] (if E splits D, then [D*E,D*E] = [ G L p ( E ) , G L p ( E ) ] = S L p ( E ) , since E is an infinite field).
8. The Merkurjev-Suslin Theorem (iii)
155
By (ii), we may replace F by a maximal algebraic extension F ' such that all finite extensions of F in F ~ have degree coprime to p. Indeed, if we write F ' = lim F~ where F i / F is finite of degree prime to p, t h e n i
ker Nrd(D*F, ) ---, F'*
[D*F,,D*f, ]
= lira -~"
ker Nrd(D*F ) ---, F~ [D~,, D~,]
and for each i, ker Nrd(D*F) ~ F"
[D*F,D*F]
r
ker Nrd(D*F, ) ~ F~
F,I
[D~,, D*
So we m a y assume t h a t any finite extension of F has degree equal to a power of p; in particular, any separable extension of F of degree p is a cyclic Galois extension. Now let c~ E D* with Nrd(a) = 1, and let K be a maximal subfield of D containing F(a). If K / F is inseparable, then 1 = Nrd(a) = NK/F(C~ ) -----C~p, SO that a = 1. If K / F is separable, then it must be a cyclic Galois extension. But then a e [D*,D*] by (i). Thus, the m a p f* 9 K I ( X ) --. K I ( X E ) is identified with
F" (9 Nrd(D~) ( 9 . . . (9 Nrd(D~) (9... (9 N r d ( D ; _ 1) ~-~ E r where D | ~ Mp~-, (Dj). Now for any division algebra D over F of prime degree p, N r d ( D * ) = (3L/F(L*), where L runs over the maximal subfields of D. Equivalently (by (8.10)), we may let L run over all extensions of F of degree p which split D. Since [Dj] = j[D] in Br (F), Dj and D have the same splitting fields. Hence Nrd(D*) = Nrd(D~). We will use the above computations of K-groups of Severi-Brauer varieties to s t u d y certain K-cohomology groups. Let p be a prime number, D a division algebra over F of degree p. Let E be a maximal subfield of D, so that E / F is a splitting field of D. Let X be the Severi-Brauer variety over F associated to D, f : XE ---* X the natural map. Proposition
8.12
With the above notation, we have
(a) the Chow groups C H i ( X ) ~ C H i ( X E ) "~ Z for 0 ~_ i <~ p - 1, and f* 9 e l i ' ( X ) ---, C H ' ( X E ) is identified with multiplication by p for i>O; (b) H I ( X , IC~,x) ~- Nrd(D*), HI(XE,]C.2,E) ~ E*, and the map f* 9H 1 (X, ]C2,x ) ---* H 1(XE, IC2,E) is identified with the inclusion Nrd( D* ) ~ E* .
156
Algebraic K-Theory
(a) We have seen that K o ( X ) "~= Z ~p, Ko(XE) "= Z Sp and f * ' K o ( X ) --~ Ko(XE) is identified with the map Z Sp --, Z $p,
Proof.
(n0, nl,... ,rip-l) ~ (n0,Pnl,pn2,... ,PnV-1 ). By the projection formula, f, o f* equals multiplication by the class [f, Ox~] = p[Ox] e K0(X); hence the map f , 9 Ko(XE) ~ Ko(X) must be given by Z r --, Z Sp, (no, n 1,... . ,np_ 1) ~ (pno, n 1,... . , rip_ 1)" If 7 = [OxE(-1)] e Ko(XE), then the isomorphism Z r --~ Ko(XE) is given by (no,... ,np_l) ~-* ~ ni'y i. We claim that the filtration by codimension of support on K j ( X E ) is given by ( 7 - 1)iKj(XE) 9 This is the content of the next lemma. L e m m a 8.13 For any field F, and integers i, j, m ~ O, we have
F i K j ( P ~ ) = (Tin - 1)iKj(IP~), where "I'm = [Op,- ( - 1)] e Ko (1P~). Proof. If f f*Tm = 7m-i, tion, using the in IP~, so that the projection
9 IP~ -i ~-~ IP~ is the inclusion of a linear subspace, then and f,([Oe~-,]) = (Tin - 1)' e K0(IP~) by direct computafact that f(IP~ -i) is a complete intersection of i hyperplanes its ideal sheaf is resolved by a Koszul complex. Hence by formula, m-i
rn-i
f, Kj(IP~ - i ) = f , ( ~ ( T m - i -
1)SKi(F)) = f * ( Y ~ f*('Ym - 1 ) ' K j ( F ) )
a=0
a=0
m
= E(Tm-
1 ) ~ K j ( F ) = (~/m- 1)'Kj(IP~).
S--i
Clearly f . K j ( P ~ -~) C FiKj(P~). Hence (Tin - 1)'Kj(IP~) C F i K j ( P ~ ) . We claim that, on the other hand, the natural composite map
i--1
E(7,,-
1)'Kj(F)~
Kj(P'~) --, Kj(P'~)/FiKj(P'~)
s--O
is injective. To prove this, observe that
FiKj(P~) =ker/Kj(P~)-o
li__m K'j(P~-Z)). codlinZ ~_i
8. The Merkurjev-Suslin Theorem
157
Hence it suffices to show that if Z c IP~ is any subscheme of codimension > i, then the composite map ~-1
)-~(~m - 1)SKi(F) ~ Kj(IP~) --* Kj(IP~ - Z) s--0
is injective. But for any such Z, we can find a linear subspace g" IP~-1 IP~ with Z n i m g -- q}. Hence it suffices to note that the composite i-1
E(Tm
- 1)SKi(F) ~-~ Kj(IP~)--* K j ( I P ~ - Z) g-~ Kj(IP~-1)
s----0
is an isomorphism. This proves the lemma. Now Ko(XE) - Ko(E)[7]/((7- 1) p) is a free Abelian group with basis (7-1) ~,0-
image(Figo(XE) ~ Ko(XE) |
Z / p Z ) = ( 7 - 1)'(Z/pZ)[7]/(('),- 1)P).
On the other hand,
image(Ko(X)|
~ Ko(XE)|
= Z/pZ C (Z/pZ)["/]/((7-1)P).
Hence for i > O,
I * K o ( X ) f3 FiKo(XE) C pFiKo(XE). Now f * F ' K o ( X ) c FiKo(XE), and f.FiKo(XE) c FiKo(X); hence
pFiKo(XE) = f* o f.F~Ko(XE) C f*FiKo(X) C f*Ko(X)r F~Ko(XE). This implies that
pFiKo(XE) - f*FiKo(X) - f*Ko(X) n FiKo(XE) for i > 0. Thus for 0 < i < p - 1, we have gr ~FKo(X) ~- gr iFKo(XE ) ~- Z, and the induced map f* - d r ~FKo(X) ~ gr~FKo(XE) is either an isomorphism (i = 0) or is identified with multiplication by p on Z; in any case the map is injective. We may regard F i as the filtration associated to the BGQ-spectral sequence, so that gr iFgo(X ) ~ E~o-i(X), griFgo(XE)"~ E ~ - ' ( X E ) . Thus f* 9 E~o-~(X) ~ E~o-'(XE) is injective. diagram
CHi(XE) ~ E 2i,-i (XE) f*T
~'~,
T f*
C H , ( X ) .~ ~.'2 ~ , - i (X)
We have a commutative
9 '(XE) E~o T f*
~r
E~o-i(X )
158
Algebraic K-Theory
where ~he horizontal maps are the (surjective) cycle maps (associating to an irreducible subvariety of codimension i the class in K o / F i + I K o of its structure sheaf). From the Riemann-Roch theorem 'without denominators' (see W. Fulton, Intersection theory, Ergeb. Math. Band 3 Folge 2, Springer-Verlag (1984), Ex. 15.1.5 and Ex. 15.3.6), we know that kerr is annihilated by ( i - 1)!. The map r is an isomorphism (being a surjection Z --~ Z). Since f . : CH~(XE) --~ C H ~ ( X ) has the property that f . o f* : CH~(X) ~ CH~(X) is multiplication by p - d e g f , we have that k e r r * - ker(r o f * ) = ker(f* or - kerr is annihilated by p. Since ( i - 1 ) ! and p are relatively prime, ~ is an isomorphism. Thus C H ~( X ) ~ Z, and f* : CH~(X) ~ CH~(XE) is identified with multiplication by p for O
K I ( X E ) ~ (E*) r
and f* is the map induced by the inclusions F* c E*, Nrd(D*) C E*. The isomorphism E *r --~ K I ( X E ) is given by ( a 0 , . . . , a v - 1 ) H ~ i 9a~, and for i < p,
F i K I ( X E ) - ( 7 - 1)iK,(XE) "~ (7 - 1)'E" + . - - +
(7 - 1) p-IE*-
We claim that for 0 < i < p,
F ' K I ( X E ) N f * K l ( X ) = ( 7 - 1)'Nrd(D*) + ... + ( 7 - 1 ) v - l N r d ( D * ) . Indeed, if X --" ao -}- ~ a l q - . . .
q- 7 P - l a p - 1
= ('T -- 1)ibi + " "
+ (7 - 1)p-lbp-1
where ao E F*, aj E Nrd(D*) for j > 0, and bj E E*, then comparing coefficients of 7 j, we see by descending induction that bj E Nrd(D*), so that x e (7 - 1)'Nrd(D*) + . . . + (~/- 1)p-INrd(D*). Next, if L C D is any maximal subfield, then X L ~_ p~-l, and we have a fiber product diagram with finite flat maps XLx E
k~ XL
h
----~
X E
g~.
~f X
so that by (5.13), we have
f* o g. = h. o k* . K I ( X L ) --+ K , ( X E ) ,
8. The Merkurjev-Suslin Theorem
159
and both compositions m a p F i K I ( X L ) into FiKI(XE). IfTL = [OXL( - 1 ) ] 6 = h*7 in Ko(XL• where 7L• = [OX~• Hence
Ko(XL), then k*TL = 7L•
h. o k*(FiKI(XL)) =
h. o k*(TL- 1)iKI(XL)
h . ( h * ( 7 - 1) i" k*KI(XL)) ( 7 - 1) i" h. o k * g l ( X L ) (7 - l)if" o g . g l ( X L ) .
= = =
Now g. 9 KI(XL) --~ K I ( X ) is a map L *ev ~ F* (9 Nrd(D*) ep-1.
8.14 In the above situation, g. 9K1 (XL) ~ K1 (X) is induced by the norm NL/F " L* ---, F*, where we note that NL/F(L*) C Nrd(D*).
Lemma Proof.
First note t h a t from (5.13) applied to the diagram 9
X L
i
Sped L
'
X
l
--.
Spec F
the composite Ki(L) --, Ki(XL) ~ Ki(X) factors through the norm m a p K,(L)--. K , ( F ) . Next, for each j > 0, the composite
Ki(L) ~ Ki(XL)--, K i ( X ) ,
is induced by the exact functor Aj 9 :P(L) --, 7)(X), where for a finite dimensional L-vector space V1, we take
Ai(VI) = g . ( O x r ( - j ) ~L Yl). If J is the locally free O x - m o d u l e of rank p described in (5.40), then g*.7 = V | L OX L (-- 1 ), where V = H ~(XL, OX L (1))" Hence
OXg ( - - j ) -~ g* (J| (g'J|
v*|
~)End (V'|
|
V*|
where we have identified DL = D| with End L(V*) (so t h a t M f : , (Dj) DL~j ~ End L(V *| for 1 < j < p). This implies that
Aj(V1) = J|
where Bj" P(L) --~ P(D |
|174
Bj( V*|
is given by Bj(V~)
= V "~ |
V,,
|
V1),
160
Algebraic K-Theory
regarded as a D| using D | r DL@j ~ End L(V*| However, we see easily that V *| is a simple DL@J-module which is also a simple D | ~- Mpj-1 (Dj)-module. Hence, Bj is isomorphic to the composite of the change of rings functor :P(L) --~ 79(Dj) with the Morita equivalence functor 79(Dj)~ 79(0| In particular, the map g l (L) --, g l (X),
equals the composite
KI(L) ~ KI(Dj) ~ g l ( D |
--~ K I ( X ) ,
where the third map is that given in (5.40). Identifying KI(Dj) with Nrd(D~) = Nrd(D*), we see that K1 (L) --, K1 (D3)is the map
L" NL__~/ r Nrd( D" ), since the reduced norm on D~ restricts to the field norm on L* C D~. This proves the claim. Thus for i > 0, we have a sequence of inclusions
f* o g.FiKI(XL) = (7 - 1) / NL/F(L* ) + . . . + (.y- 1)P-INL/F(L *) C f * f i g l ( x ) C f * g l ( x ) fq F'Kx(XE) c
(7 -
1)'Nrd(D*) + . . . +
(7 -
1)P-~Nrd(D*).
But Nrd(D*) = (JLNL/F(L*), where L ranges over the maximal subfields (up to F-isomorphism) of D. Thus all but the first of the inclusions are equalities. We deduce that g r ~ g l ( X ) ~ Nrd(D*) (i > 0), g r ~
-~ F*,
gr ~ K t (XE) ~ E* V i, and f* 9 g r ~ K l ( X ) --, g r ~ K I ( X E ) is identified with either the natural inclusion F* ~-, E* (for i = 0), or the inclusion Nrd(D*) ~-, E*. In particular, f*" gr ~K1 (X) --, gr ~K1 (XE) is injective for all i. We use this fact for i = 1 to prove (8.12)(b). We have H i ( X , K:2,x) E~'- ~(X). The relevant differentials for Er (X) are Erl-r,r-3 __~ E1,-2,
Erl,-2 __. E r + , , - 1 - r .
Here E 1-r'r-3 = 0 as 1 - r < 0, while E~ +1'- 1-r(X) -~ E ~ I , - ' - ~ ( X ) CHr+'(X) from (8.12)(a) proved above. Hence E~'-2(X) ~- E ~ - 2 ( X ) ~Nrd(D*); by a similar argument, EI'-2(XE) ~ E~-2(XE) ~- E*, and f * " HI(x,]C2,x) ---. HI(XE,]C2,XE) is identified with the inclusion Nrd(D*) E*.
8. The Merkurjev-Suslin Theorem
161
We use the above computations of K-cohomology to prove the /(2analogue of Hilbert's Theorem 90. T h e o r e m 8.15 (Hilbert's Theorem 90 for K2) Let F be a field, E / F a cyclic Galois extension, g E Gal ( E / F ) a generator. Then we have an
exact sequence K 2 ( E ) 1-~; K2(E) N~(~ K2(F) (i.e., x E K 2 ( E ) satisfies NE/F(X) -- 0 r K2(E)).
X = y - a.(y) for some y E
We will need only the special case when E l F is of prime degree, and give the proof in this case. Proof. Let p = [E 9 F]. For any extension field L of F , L | E is a cyclic Galois algebra over L. If a E Gal ( E / F ) is a generator, a determines a generator (which we also denote by a) for Gal (L | E / L ) . T h e action of a . on K~(L| is obtained from the functor a. 9P ( L | --* 7)(L| on the category of finitely generated projective modules, M ~-, a . M , where a . M is the Abelian group M with new module structure given by (~-m a -1 ( a ) m (the expression on the right is computed using the original module structure). Next, the norm m a p s NL| " K~(L | E) --. Ki(L) are induced by the forgetful functor :P(L| E) ~ 7~(L) which associates to an L| E-module the underlying L-vector space. Since M and a . M have the same L-vector space structure, NL| oa. NL| on K~(L | E). Thus NL@~E o (1 - a . ) = 0 on K~(L | E). Let NL = NL| " Ki(L | E) "-* g i ( L ) , and let =
Y(n)
ker NL (1 - a . ) K 2 ( n |
=
E)
Thus, Hilbert's Theorem 90 f o r / ( 2 for E / F is the assertion t h a t V ( F ) 0. If L / F is a finite extension field of degree d, we have a norm m a p
NL/F " V(L) --4 V ( F ) such t h a t the composite V(F) --, V ( L ) NL/[ V ( F ) is multiplication by d. Since E | E ~ E cp, such that for any i >_ O, NE " Ki(E | E) --. K i ( E ) is given by the addition map K i ( E ) $p --. K i ( E ) , and a . 9 K~(E | E) --, K~(E | E) cyclically permutes the factors in K i ( E @F E) ~- K , ( E ) cp, one sees easily that V(E) = O. Hence p V ( F ) -- O. Thus if L is a finite extension of F of degree relatively prime to p, then V(F) r V(L), so it suffices to prove the result for L | E / L . Note also t h a t if L - limLi for suitable extensions Li of F , then i
V(L) -li__m V(Li) (this follows from (5.9), for example). Let L be a maximal algebraic extension of F such that any finite sub-extension has degree
162
Algebraic K-Theory
relatively prime to p. Then V(F) ~ V(L), and it suffices to prove the theorem for L (~f E / L . Thus to prove V (F) = 0, we may assume without loss of generality t h a t every finite extension of F has degree a power of p. L e m m a 8.16 For b E F*, let X be the Severi-Brauer variety associated to
U(F(X)) Proof.
Since E trivializes A(a, b), XE
~'~: ~ - 1 ,
and we claim t h a t
HO(XE,]C2,E) ~-- K2(E) ~ K 2 ( E ( X ) ) , where E ( X ) is the function field of XE. Indeed, H~ K:2) C H~ K:2) C K2(E(IPn)) for any n, from (5.27); hence it suffices to prove that H~ ~ K2(E) ~ K2(E(A'~)) = K2(E(IPr')). This is easy for n - 1, from K j ( A ~ ) ~- K j ( E ) and the BGQspectral sequence, which degenerates at E2 since dim A~ = 1. In general, writing A~ = A~- 1 x E A~, HO(A~,]C2) r
H o(AE(A,),n-1]C2) -- K2(E(AI)) C K2(E(A'~))
by the induction hypothesis for A n - 1; now from the commutative diagram with exact rows 0--~ 0 --~
H~
--4 K2(E(An))---*
T
K2(E)
T
--* K2(E(A1))--*
we see that H~ K:2) = K2(E) Now suppose u E K2(E) such for some v ~. K2(E(X)) (here uL K2(E) --~ K2(L ~:~F E)).We have spectral sequence
~ze(A~)~E(x)*
T
(gze(A~),E(x)*
~-~ K2(E(An)), as desired. that NE/FU = 0, and UE(X) -- v - a . v e K2(L | E) is the image of u under the complex of E1 terms of the BGQ-
0 ~ K2(E(X)) ~ (9~,ex~E(x)* ~ (9~,ex~Z ~ O. Since dl(UE(x)) ----- 0, dl(v) -- dl(a.v) = a.dl(v). Thus dl(v) = f*w for some w ~. (gyex~F(y)*, under the natural map f* 9 EII'-2(X) --~ E~'-2(XE). Now dl o d l ( V ) = 0, and f*" EI~'-2(X) = (9~ex2Z --~ @xex~Z = E 2 ' - 2 ( X E ) is injective. Hence dl(w) = 0, and w determines a class in E21'-2(X) -~ H I (X, K:2,x), which by construction, lies in
ker(f* . H l ( X , lC2,x) ~ H I ( X E , ~ 2 , x E ) ) .
8. The Merkurjev-Suslin Theorem
163
But by (8.12)(b), this map f* is injective. Hence w = d l ( v ' ) for some v' e K 2 ( F ( X ) ) . Hence u E ( x ) = v - a . v = ( v - f * v ' ) - - a . ( v -- f * v ' ) , and d l ( v - f * v ' ) = dl (v) - f * w = 0, so t h a t v - f * v ' E H ~ K:2) -~ K 2 ( E ) . Hence u E (1 - a . ) K 2 ( E ) , as desired, proving (8.16). We now inductively define a sequence of fields Fro, m >_ O, as follows. Let Fo = F ; let F2n+ 1 be the compositum of all the function fields F z n ( X ) , where X ranges over the Severi-Brauer varieties associated to the cyclic algebras A ( a , b ) , b E F:~n; let Fsn+2 be a maximal algebraic extension of Fsn+ 1 of degree relatively prime to p (i.e., such that any finite subextension of F2n+l has degree relatively prime to p). Let Foo = lim Fn. T h e n V ( F ) ---. --..., n
V(Foo) is injective, every finite extension of Foo has degree a power of p, and [A(a, b)] = 0 in Br (Foo) for all b e F~o, i.e., by (8.4)(a), if Eoo = EFoo, then gE,,~/F~ " E~o ~ F~o is onto. Hence we may assume without loss of generality t h a t every finite extension of F has degree a power of p, and N E / F " E* ~ F* is onto. We claim t h a t under these conditions, N E / F " K2(E) --* K s ( F ) induces an isomorphism
K2(E) im(1 - a . )
--. K s ( F ) .
To see this, let ~ " F * | --. K 2 ( E ) / ( i m ( l - a . ) ) be defined by ~o(a| ( a , b} where t~ e E* is a solution of N E / F a = a. Since any other solution cd satisfies N E / F O / a -1 = 1, the usual version of Hilbert's Theorem 90 implies that {t~%-i,b} e ( 1 - a.)g2(E). Thus ~0 is well defined and bilinear; further the composite N E / F o ~ 9 F* | F* ---* K 2 ( F ) is just the Steinberg symbol. By lemma (8.6) of Bass and Tate, K2(E) is generated by the symbols (t~, b} with a E E*, b E F*, so t h a t ~o is surjective. If we prove t h a t for any a e F*, a ~ 1, we have ~o(a | (1 - a ) ) = 0, then we would have (by (1.14)) an induced map ~ " K s ( F ) --~ g 2 ( E ) / i m (1 - a . ) which is surjective, such t h a t N E / F o ~ is the identity on K 2 ( F ) , i.e., ~, W E / F are inverse isomorphisms. So it suffices to prove t h a t if a E F*, a ~ 1 and c~ E E* such t h a t then { c ~ , l - a } e ( 1 - a . ) K 2 ( E ) . Let T p - a = l-I, p , ( T ) " ' be the factorisation into irreducible factors over FIT], and let a i be a root of p~(T) = O, Fi - F(o~), Ei = E | F~. T h e n NEIFOt--a,
1 - a = 17IP,(1)"' = H i
{~, 1 -
a}
N E , / E ( 1 -- a , ) " ' ,
i
= H NE, IE{a, 1 - cei}"'= H NE, IE{a(~', 1 - cei}n' i
i
164
Algebraic K-Theory
(using {c~i,1 - a i } = 1). Now NE,/F~COzo~ 1) = 1, SO by Hilbert's Theorem 90, ~c~ -1 = ilia(f/i)-1 for some ~i E E~. Then
{t~, 1 - a} = H NE,/E(1 -- a.){13i, 1 -- c~i}TM e (1 - a . ) K 2 ( E ) , i
from the commutativity of K2(Ei) NE, I E I K2(E)
5
K2(Ei) I NEilE K2(E)
~-;
(which follows by (5.13) from a corresponding commutative diagram of rings Ei-% Ei
T
T
E-%
E)
This completes the proof of (8.15). We now proceed with the proof of (8.5), following [M]. L e m m a 8.17 Let E = F(~r~) be a cyclic Galois extension of F of degree p, and let a e Gal ( E l f ) be the generator such that a(~r~) = r ~r~. Suppose Rp, F is injective. Then:
(i) ker (K2(F) | Z / p Z ~ K2(E) | ements {a, b} with b e F*
Z/pZ) consists of precisely the el-
(ii) the square K2(F) |
Z/pZ
R;,.__F nB r (F)
K2(E) |
Z/pZ
R',,,___~ pBr (E)
is a pullback (iii) Rp, E i8 injective. Proof. (i) Let u e ker (K2(F) | Z / p Z ~ K2(E) | Z/pZ). Then R'p,F(U ) e ker (vBr ( f ) --. pBr (E)). Hence by (8.4)(a), there exists b e F* ! ! I such that Rp, F(U ) = Rp, F({a, b}). Since Rp,F (and hence Rp,F) is injective, this means u = { a , b } ( m o d p . K2(F)). (ii) Let u E K 2 ( E ) | Z/pZ, v E vBr(F) have the same image in pBr(E). We must show that there is a unique y e K 2 ( F ) | Z/pZ such that YE = u, and R~,F(y ) = v (here YE denotes the image of y in K2(E) | Z/pZ). Since Rip,f is injective, such an element y is unique, if it
8. The Merkurjev-Suslin Theorem
165
exists. Also, by (8.4)(b) and (i) of this lemma, we see t h a t the two vertical arrows in the square have the same kernel. So it suffices to prove t h a t u E image ( K 2 ( F ) | Z/pZ). ! Now R n , F ( N E / F ( U ) ) = NE/F( Rp,E(U)) ' = NE/F(VE) = p v = 0. Since R~, F is injective, NE/F(U) = 0 in K 2 ( F ) | Z/pZ. We claim t h a t *)
9. .
ker ( N E / F " K2(E) |
Z / p Z -* K 2 ( F ) |
(1 - a . ) K 2 ( E ) |
Z/pZ) =
Z / p Z + image K 2 ( F ) |
Z/pZ.
If z E K 2 ( E ) | Z / p Z with NE/F(Z) = O, and ~ E K 2 ( E ) is a preimage of z, then NE/F(Z~ -- p ffJ for some ~ E K 2 ( F ) , so that N E / F ( Z - - WE) -- O. Hence by (8.15), ~ = ~ E + (1 -- a.)(t--) for some t e /(2 (E). If ~ ,-, w e K2(F) | Z/BZ, t~--, t e K 2 ( E ) | Z / p Z , then z = wE + ( 1 - a . ) ( t ) in K2 (E) | Z / p Z . In particular, for our given u, we can write u = WE + (1 - - a . ) ( t ) with w E K2(F) | Z / p Z and t e K2(E) | Z/BZ. First suppose p = 2. If NE/F(t) = S, then SE = (1 + a . ) ( t ) = (1 --a.)(t). Hence u -- (w + S)E is in the image of K 2 ( F ) | Z / p Z as desired. Now suppose p is an odd prime. We claim by induction on j t h a t (1 - a . ) ( t ) ~_ image K 2 ( F ) |
Z / p Z + (1 - a . ) J K 2 ( E ) |
Z/pZ
for each j = 1, 2 , . . . , p - 1. This is clear for j = 1. Suppose 1 < j < p - 1, and (1 - a . ) ( t ) - (1 - a.)J(s) is in the image of K 2 ( F ) | Z/pZ. Then (1 - a . ) J + l ( s ) = (1 - a.)2(t). Hence t
(1 - ~ . ) J + ~ R ~ , ~ ( ~ )
= (1 - ~ . ) !
= (1 - ~ . ) n . . ~ ( ~
- ~)
2
t
R~.~(t)
t
= (1 - ~ . ) R ~ , E ( ( 1
= (1 - ~ . ) ( . E
- (R'~.F(~))E)
- ~.)(t))
= O.
Since j + 1 < p - 1, we get (1 - a . ) P - I R p,E(S) ' = O. In the polynomial ring Z/pZ[X], we have 1 + X + . . . + X n-1 = (1 - X ) n-1. Hence _
!
(NEzF o Rn,E(s))F. = ( l + a . + - . . + a P . - ' ) R ' p , E ( S ) = (1--a.)n-lR'p,E(S) -- O. Hence by (8.4)(a), NE/F o R'p,E(S ) = R'p,F((a,b)), for some b e F*. We then have I ! Rp, F o NE/F(S) = N ~ / F o R'p,E(S ) = R p' . ~ ( { ~ , b } ) R =p , F O N E / F ( { C ~ , b } ) ,
where the last equality is because p is odd. Since Rip,F is injective, by assumption, we have NE/F(S -- { ~fa, b}) = 0. Hence by (.), we must have s - { ~fa, b} e (1 - a . ) K 2 ( E ) |
Z / p Z + image K 2 ( F ) |
Z/pZ,
166
Algebraic K-Theory
so t h a t (1 - a . ) J ( s ) - {r
e (1 - a . ) # + l g 2 ( E ) |
This proves the claim for j + 1. Now applying the claim for j - p (1 -
ZIpZ.
1, we get t h a t
a.)(t) - (1 - a . ) P - l ( s ) E image K2(F) |
for some s E K 2 ( E ) |
Z/RZ
Z/pZ. But
(1 - f f , ) P - l ( s )
-- (1 + a , + - - . + a,P-1)(s)
= (NE/F(S)) E e image K 2 ( F ) |
Z/pZ.
Hence ( 1 - a . ) ( t ) e image K 2 ( F ) | and so u e image K 2 ( F ) | (iii) This follows from (ii) and the injectivity of R'p,F. Our next goal is to determine the kernel of K2(F) |
Z / p Z --~ K 2 ( F ( ~ ) )
|
Z/pZ
without first assuming (as we did in lemma (8.17)) t h a t Rp, F is injective. We will accomplish this in Proposition (8.23), by deducing it for an arbit r a r y field F from a suitable 'generic' situation, where it is easier to prove. To do this, we need to first prove the injectivity of Rp,F when F is a pure transcendental extension of a finite field or of a number field. This is done in the next two lemmas. T h e reduction of the a r b i t r a r y field to the 'generic' situation is done using lemma (8.20). Note t h a t if F contains an algebraically closed subfield F0, then (as explained later in more detail) we do not need the next lemma in proving Proposition (8.23), and so can avoid the reference to class field theory made below. 8 . 1 8 (Tate) Let F be a field which is algebraic over its prime subfield, and p a prime such that F contains a primitive p t h r o o t of unity. Then Rp, f is injective.
Lemma
Proof. We may assume without loss of generality t h a t F is a finite algebraic extension of its prime subfield. If F is a finite field, then K 2 ( F ) = 0. Hence the lemma is trivial in this case. If F is a number field, then the lemma follows from lemma 8.4(b), and the following two assertions: (i)
given a,b,c,d e F* such t h a t R'p,F({a,b}) - R'p,F({C,d}) , there exist x, y E F* such t h a t
R'p,F((a,b}) = R'p,F((X,b})= R'p,F({x,y})= R ' p , F ( ( c , y } ) - R~,F({C,d})
8. The Merkurjev-Suslin Theorem
167
(ii) given a, b, c, d E F*, there exist x, y, z E F* such that
R~,F({a, b)) = R~,F((X, Z}) and R~,F((C, d}) = R~,F({y, z}). Indeed, granting these assertions, we first see, from (i) and (8.4)(b) above, that if R~,F({a, b}) = R~,F((C, d}), then {a, b} = {c, d} in K2(F) | Z/pZ. Then (ii) implies that any element of K2(F) | Z / p Z is the image of a symbol (x, y} for some x, y e F*. Now (i) implies that R~, F is injective. We first prove (ii). We claim that since F is a number field, any finite number of elements { a l , . 9 a t } of pBr (F) have a common cyclic splitting field E with [E : F] = p. We will deduce this from the exact sequence describing the Brauer group of a number field (#)
0-+ Br(F)--,
(~
Br(Fv)
~'~vinv" O / g ,
--, 0
vEMp
which is obtained from class field theory (see for example Algebraic Number Theory, J. W. S. Cassels and A. Frhhlich, eds., Academic Press Reprint (1990), Ch. VII, 9.6). Here ME denotes the set of places of F, and invu : Br (Fv) - . Q / Z is the natural inclusion given by local class field theory (invv is an isomorphism for non-Archimedean v, and is the inclusion Z/2Z -~ Q / Z if Fv ~- R, the real field). Let S be the finite set of places v of F such that some ai has a non-zero image in pBr (Fv). For each v E S, there is a cyclic Galois extension Ev of Fv of degree p which splits the generator of pBr ( F v) ~ Z/pZ. By Kummer theory, we have E~ = Fv(~rd~) for some dv E F~. We can find d E F* which gives a sufficiently good Fv-approximation to dv, for all v E S. Then du/d E (F~,) p for all v E S, so E , = F~(~rd) for all v E S; hence E = F(~fd) splits all the a~. This proves the claim. Since by lemma 8.4, ker(pBr (F) --~ pBr (E)) consists precisely of the elements R~,F((a,d}) for a E F*, (ii) follows at once. To prove (i), we again use the exact sequence (#). First observe that since pBr (F~) is cyclic, (i) holds if we replace F by Fv. Indeed, identifying pBr (Fv) with Z / p Z (assuming pBr (Fu) ~ 0), we may view R~,F, ' as a bilin, ! ! ear form on F v | Z/pZ. We may assume Rp, F,,({a,b}) = Rp,F. ((c, d}) a ~ 0. Then the linear forms x ~-. R p,F,,((x,b}) ~ t F,,((x,d}) and x ~ Rp, are both non-zero. If they are equal, or are linearly independent, then take y = d, and we can solve (for x) the system of linear equations Rp, F,,({x,b)) = R~,F,,((x,d}) = a. If the linear functionals are not equal but are dependent, then p > 2, and the bilinear form is alternating (any t E I F~. is either a pth power or is a norm from F~( ~7~); hence Rp,F, ' ({t, t}) = 0). Now consider the linear forms x ~ R~,F, ' ({x, c}) and x ~ R~,F,"((x, d}). These are linearly independent since R~,F,,((c,d)) - ~ ~ O. Now take
168
Algebraic K-Theory !
!
T h e n Rp,F, ' ((c, y}) -- Rp, F, ' ((c, d}) -- ~, and the linear forms ! I x ~-~ Rp,F~({x,b}) and x H Rp,F,,({x,y}) = Rp,F,,({x, cd}) are linearly t t independent, so t h a t we can solve Rp,F,,({x,b}) = Rp, F~ ({x,y}) = c~ for y = cd.
I
x E F~ |
Z/pZ. !
I
Now let a = Rp,F({a,b}) = Rp, F(iC, d}). Let S be the finite set of places of F such t h a t ~ H c~v r 0 E pBr (Fv). For each v E S, we can t h e n find xv, Yv E F~ such t h a t =
'
=
=
? p , F , ( { c , y , } ) - Rp,F,,((c,d})
R ?
.
From the last equation, y v / d is a norm from Fv(~f~) = Ev, say y v / d = NE,,/F,, (tv). Choose t E F(~fc) = E such t h a t t / t v E (E~) p for each v E S. Take y = d N E / F ( t ) . Then ?
?
=
Y})
=
and =
for each v E S. We now need to find x E F* such t h a t !
b})
!
Y})
=
=
By construction, we know t h a t we can find a local solution xu E F~ for each v E S, and there is the trivial solution xv - 1 for all v ~ S. Now we are reduced to (the case r = 2 of) the following assertion: if ~ 1 , . . . ,c~r E p B r ( F ) , and a l , . . . , a t E F* such t h a t for each place v ! of F , there exists xv e F~ satisfying { R p , F , ( { a , , x v } ) = (a~). in pBr (Fv), then there exists x E F* satisfying R'p,F({a~, x}) -- c~ for each i. This last assertion follows from class field theory; it is a reformulation of Cassels and Frhhlich, eds., loc. cir., Ex.2.16.
Let L be a finitely generated pure transcendental extension of F. If P~p,f is injective, then so is Rp, L.
Lemma
8 . 1 9 (Bloch)
Proof. By induction, we may assume L - F ( t ) where t is an indeterminate. From the BGQ-spectral sequence for A~ (which degenerates at E2) and the isomorphisms K i ( F ) ~ K i ( A ~ ) for all i, we have H~ = g i ( F ) , and H J ( A ~ , K:i) = 0 for j > 0, for all i. In particular there is an exact sequence
0 ~ K2(F) ~ K2(F(t))
~
T~
(~xE(A~.)' F(x)*
-~ O,
8. The Merkurjev-Suslin Theorem
169
where T~ : K2(F(t)) --* F(x)* is the tame symbol associated to the discrete valuation v~ determined by x (see (9.12)). The corresponding sequence of p-torsion subgroups is 0 --. pK2(F) --, pK2(F(t))
--.
r
Up ~ 0,
where the surjeetivity of the map on the right is clear, because Tx({a, ~)} ~v~(a) for any pth root of unity ~ and any a e F(t)*. Hence 0 --, K2(F) |
Z/pZ -, K2(F(t))|
Z / p Z --, @~e(nxF),F(x)'/F(x) *p --, 0
is exact. There is a corresponding exact sequence of Galois cohomology groups
0 --4 H2(F, 1~2) --, HZ(F(t) , #~2) E.. 0~. , (~xecn~)~F(x)*/F(x) *p --4 O, where H 2 ( F , # ~ 2) is identified with the ~tale cohomology group H~2t(A~,#~ 2) (see Milne, Etale Cohomology, Ch. VI, Cor.4.20, for example). This exact sequence of Galois cohomology groups may be deduced from the Gysin exact sequence in ~tale cohomology, for the smooth Fvariety A~ (see Milne's book, Ch. VI, 5.4(b)), as follows. For any open subset U C A~., and Z = A~- - U, we have a Gysin exact sequence i i i--1 --- ~ He,i - - 2 (Z, Up) --+ Het(A~', # ~ 2 ) .__, Her(U, # ~ 2 ) ~ He t (Z, Up) H~+~(A~.,#~ 2) ---, 999
2) Since H'(F, #~2) ~ H~t(A~ ' ~ 2 ) , the map H ' ,(AF, #~2 , is injective for each i (if F is infinite, then U has F-rational points; if F is finite, U has two closed points of relatively prime degree over F). Hence the long exact sequence above splits into short exact sequences for each i
0--, H ' ( F , U~ 2) --, H~t(U, #~ 2) ~ H ~ [ ' ( Z , # p ) - , O. Since Z consists of a finite set of closed points,
H~89
= (9~ez H 1 (F(x),#p) = O x e z F ( x ) * |
Z/pZ;
hence the above sequence for i = 2 reads
0 -~ H2(F, D~2) __, He2(U, #~2) ~ ~ x e z F ( x ) * |
Z / p Z ~ O.
Taking the direct limit over all finite sets Z of closed points, we obtain the desired sequence of Galois cohomology groups.
170
Algebraic K-Theory We claim that for each closed point x E A~, the square
K2(F(t))
--,
F(x)* |
Z/pZ
H2(F(t),tt@p 2)
--,
F(x)* |
Z/pZ
commutes up to a (universal) sign. Both composites clearly vanish on symbols {a, b} with a,b E O*a~,=. So it suffices to compare the values of both composites on elements {a, b} with a E
,=, for a local parameter
b E Oa]~,z. Since Rp,f(t) is defined via cup products using Xp,F(t) " F(t)* |
Z / p Z ~-; H I ( F ( t ) , p p ) , which is a boundary map in an exact cohomology sequence, the claim in this special case follows from the formula for the cup product of a boundary (see Milne's book, Ch. V, w and in particular, Prop. 1.16). The claim yields a commutative diagram with exact rows 0 --. K 2 ( F ) |
Rp,F
Z / p Z --4 K2(F(t)) | Z / p Z Rp, F(t) ~ H 2 ( f ( t ) , lt@p2)
o
~ (9=e(A~)lF(x)*/F(x) *p --, 0 ~
~,,. ' a~. (~=e(A~), F ( x ) * / F ( x ) *p -o 0
A diagram chase immediately implies the lemma. 8.20 Let R = k[X1,... , Xn] be a polynomial ring over a field k, T an integral domain which is a finite R-algebra, and L the quotient field of T. Let Iz : T --. F be a homomorphism of k-algebras to an extension field F of k, and r : R --~ F the induced homomorphism. Let P = ker r P - ker p. Suppose T~ is a fiat and unramified (i.e., dtale) Rp-algebra. Then there are (non-canonical) homomorphisms Lemma
81 : L* | 92: K2(L) |
Z / p Z ---, F* |
Z/pZ,
Z / p Z --4 K 2 ( F ) |
Z/pZ,
such that (i) (ii)
02{a, b} = {Ol(a), 01 (b)} for all a, b E L* o,(y) = , ( y )
all y e
Proof. Let/z|174174174 extensions of # and r If P ' = ker r | 1 and Q = ker # | 1, then (T | F)Q is an 6tale algebra over (R | F)p,. In particular, (T | F)Q is a regular local ring, and hence an integral domain. Hence there is a unique minimal
8. The Merkurjev-Suslin Theorem
171
prime Q' of T | F contained in Q. We may now replace R by R | F , and T by T | F / Q ' . Hence we are reduced to the case when k = F . Next, if r = ai, we may change variables in R to Yi = X i - ai. Now under the given hypotheses, we have an isomorphism of completions k[[Y1,...,Yn]] ~- R p ~ T~. If M is the quotient field of k [ [ Y 1 , . . . , Yn]], then we have a natural inclusion L ~-~ M giving a commutative diagram --,
1
L
---,
1
M
Hence it suffices to construct homomorphisms 01 "
M* |
02" K 2 ( M ) |
Z / p Z - - , k* |
Z/pZ,
Z / p Z ~ K2(k) |
Z/pZ,
such t h a t (i) (ii)
02{a, b} = {01 (a), 01 (b)} 0t(y) = y(0, 0 , . . . , 0) (mod k *p) for all y e k[[Y1,..., Yn]]*-
We may well-order the monomials in Y l , . . . , Yn using the lexicographic order: Y ~ Y2n2..- Y~" < Y1q~Y2q2... Ynq" if for the first i such t h a t p, ~ q~, we have pi < qi. T h e n any non-zero element z E k[[Y1,.. 9, Yn]] is uniquely expressible as z = a . in(z) + zl, where a E k*, and in(z) is the initial monomial of z, i.e., the smallest monomial (with respect to our chosen ordering) appearing, in the power series z with a non-zero coefficient. Here a is the coefficient of in(z) in z, so that every monomial appearing in zl is strictly larger than in(z) with respect to the chosen ordering. Then we have: (a) if z is a unit, we have in(z) = 1, and the unique expression is z =
z ( 0 , . . . , 0). 1 + (b) for any non-zero z', Z ?! E k[[Y1,..., Yn]], we have in(z'z") = i n ( z ' ) i n ( z " ) . Now an a r b i t r a r y element y E M* can be (non-uniquely) expressed as y = y~/y", where y ' , y " are relatively prime (non-zero) elements in k[[Y1, . . . , Yn]]. We can uniquely write y' = a.in(y')+y'l, y" = b . i n ( y " ) + y ' 1' where a, b E k*. Define 01 (y) = image of a/b in k* @z Z/pZ.
172
Algebraic K-Theory
We easily verify using (b) above t h a t ~1 is well defined, and is a homomorphism. Now (a) implies t h a t for y e k[[Y~,... ,Y.]]*, we have 01 (y) = y ( 0 , . . . , 0) (mod k'P). We now define 02({y,z}) = {Ol(y),01(z)}. It only remains to show t h a t this is well defined. This follows provided 82({y, 1 - y}) - 0 for all y E M - { 0 , 1 } , which we now prove. I f y = y r / y , as above, t h e n 1 - y -- ( y " - y ' ) / y " . If in(y") < in(y'), then i n ( y " - y ' ) = in(y"), so 01 (1 - y) = 1. On the other hand, if in(y") > in(y'), then we similarly conclude t h a t ~ 1 ( 1 - y) = -~1 (y). Finally, suppose in(y') -- in(y"). In this case ~1(1 - y) = 1 - Ol(y). In all three cases, clearly {01(y),01(1 - y)} is trivial in K2 (k) @z Z/pZ. R e m a r k . T h e subring of M, whose non-zero elements are the fractions yl/y2 with in(y1) <_ in(y2), is a valuation ring with quotient field M , and the maps 0i are particular cases of specialization maps for K groups associated to valuation rings. 8.21 Let F be a field, p a prime ~ char. F. Let f : F * -~ F* | Z / p Z . Then K 2 ( F ) | Z/BZ = (F* | Z/pZ)|
Lemma
where A is the subgroup generated by f (x) | f (y), where x, y e F* range over all pairs such that y is a norm from the field F(~fx). Proof. As in the proof of (8.1), we see t h a t f ( a ) | f ( 1 - a) e A for all a E F* - {1 }. Hence (F* | Z / p Z ) | is a quotient of K 2 ( F ) | Z / p Z . On the other hand, if y = NF(~f~)/F(Z ), then in K 2 ( F ) , we have
{x, y} = NF( ~f~)/f {X, z} = NF( ~/~)/F{ ~-X, Z}p e p. g 2 ( F ) . Hence A is in the kernel of the (surjective) symbol m a p (F* |
|
~ K2(F)|
In the next lemma, we require the following notation. For any positive integer n, let . 4 , = ( ~ : { 1 , 2 , . . . ,n} -~ (0, 1 , . . . , p -
1} I (~(i) r 0 for some i}.
For any a 1 , . . . , an E F*, and a E . 4 , , let n aa
=
H 7(`) a
,
i~-I
Fa = F(C/-da), N a -
NF~/F : F~ ~ F*.
L e m m a 8.22 Let a 1,... , am E F* have Z/pZ-linearly independent images in F* | Z/pZ, and let bl, .... , bm E F*. Then the following are equivalent.
8. The Merkurjev-Suslin Theorem (i)
Him=l{ai, bi} is trivial in K a ( F ) |
(ii) There exist am+ l , . . . ) an cl , . . . , c~ E F* such that
173
Z/pg.
E F* , xa E F~* f o r each a ~ A n , and
8, = b, l-i N.(~.)"(')
,
1 <_i
i--1
where bm+ l = . . . .
b,., = 1.
P r o o f . (ii) =~(i) is easy, and does not use the Z/pZ-linear independence of the ai. Assuming (ii), we compute (in K 2 ( F ) | Z/pZ) that m
gl
H{...b,)= H H
=
i=I
n
i=1
i=1 aEA,,
n
= H l-I(-? ('), No(~o)}-' : 1] (~., No(~o))-', ~ E.A,, i = 1
ckE,A,~
which is trivial. To prove (i) =~(ii), we use the equivalent definition (8.21) of K 2 ( F ) | Z/pZ. Since Hi~1{ai, b[ 1} is trivial in K2(F) | Z / p Z , we have EiL1 f ( a i ) | f ( b ( 1 ) e A, i.e., m
T"
~=I
f(ai) ~ f(b; 1) - ~ f(ej) | f(gj), j=l
where e j , g j E F*, the ej are all distinct, and gj is a norm from F(~fej). Let
f(a~),...,
f(~m),
f(~m+~),..
be a Z/pZ-basis for the subgroup of F* | f(al),.
.. ,
f(am),
9 f(a,,)
Z / p Z generated by
f(el),...
, f(er).
Then we can find c~j E .An, for 1 _< j _< r, such that f ( e j ) -- f(ac~j), and c~j ~ C~k for j ~ k. Clearly also F(~fej) - F(~faa~) -- Fa~, so gj - Na~ (xj) for some xj E F ~ . Define f
Za
t
1
if a = c~j for some (necessarily unique) j, otherwise.
Then we have m
f ( a i ) @ f ( b ? 1) = ~ i=l
f(ao,) ~ f(go,(xo~))
c~E.An n
- ~ : Z: :(~,)| :(No(~) ~ (~E.A,, J = l
n
= 5-'. :(a,)| :( H No(~:('))). i=l
aE.A~,
174
Algebraic K-Theory
Since the f(a~) are Z/pZ-linearly independent in F* | have f(bi H N~ 1
Z / p Z , we must
aEA,~
for 1 < i ~ n, where b~ - 1 for i > m, i.e., bi H
Na(xaa(i))=~
aEA,~
for some ci E F*, for 1 < i < n. This is w h a t was to be proved. P r o p o s i t i o n 8.23 Let F be a field, p a prime # char. F, such that F contains a primitive pth root of unity ~. Let a E F * - F *p, and E = F ( f f ~ ) . Then ker ( K 2 ( F ) |
Z/pE ~ K2(E) |
Z / p Z ) = {{a, b} [b e F* }
P r o o f . This was proved in lemma (8.17) under the additional assumption t h a t Rp,F is injective. We now prove it in general. Let x = 1-Iim__l{a,,bi} be an element of the kernel. To prove the result, we m a y change x by multiplying it by {a, b} for any b E F*, since such elements are certainly in the kernel. We may thus assume w i t h o u t loss of generality t h a t a x , . . . , am yield Z/pZ-linearly independent elements of E* | Z / p Z . This is because ker (F* | Z / p Z --, E* | Z / p Z ) is just the subgroup generated by the image of a, from K u m m e r theory. Now by lemma (8.22), we can find
(i)
am+l,
(ii) cl, (iii)
-
-
-
o
*
-
,an E E*
,cneE*
x~ e E a = E(~fS~) for each a e An,
such t h a t for each 1 <_ i < n, we have
=b, 1-I Here, recall that .An is the set of non-zero functions {1 . . . , n} --, {0, 1 , . . . , p - 1}, and for a ~ An, we have a s = 1-I,~l a7 ('). Also N ~ - E~ --, E* is the norm, and we take bm+l = "." = bn -- 1. Let w - ~/r~. Then 1 , w , . . . ,w p - I is an F-basis for E, so t h a t we can uniquely write p-1
ai =
E
j=O
aijw 3, m + 1 < i < n, aij E F,
8. The Merkurjev-Suslin Theorem
175
p-1
C4
Z c~jwJ' l < i < n, cq 6 F. j=0
Further, (~/an)iw
for each a 6 .An, we have [Ea " F] = p2 with j, 0 _< i,j _< p - I. Hence we can uniquely write
a basis
given
by
p-I Xn
= E xqn(~raa)'wJ, V a 6 .An, xijn 6 F. i,j=O
Let F0 be the subfield of F generated by the algebraic closure in F of the prime subfield, and the elements a and r (the given primitive pth root of unity). Consider the following independent variables over F0"
Ai
Aq Bi
c,j
Xqn
(1 < i < m ) (m+l
(O
ot6An).
We may then define the following rings and fields: Eo = f o ( C Z ) = ro(
)
R = Fo[Ai, Bi, Aq, Xqn] F1 = Fo(A~, Aq, Bi, Xqo,) (the quotient field of R) S = R[ C/r~ = R | E , = FI(~/~) = Fx |
E0 (the quotient field of S).
Also define p-1 A~ = ~_0= o Aqw-i for m + 1 _< i < n
An = I'Ii"--I a ~ (i)" Then Ai, An E S for all i, a. Let El,n = E, (~rAn), for each a 6 An. Finally, we may define elements p-1
.
Xn = ~ i , j = 0 X q a ( ~fAn)iw j for all a 6 .An (thus Xn 6 El,n) Di = Bi I]neA. N*,o,(Xo,)n(~), for all 1 <_ i <_ n
D = l-Ii~l NEI/FI (Di). Here NI,~" E*x,n ~ E~ is the norm map. Note that D~ 6 S, and D 6 R. There is a homomorphism of F0-algebras r R --, F given by Ai ~ ai, 1 < i < m, Aq ~-~ aq, m + 1 < i < n, 0 < j < p - 1 , Bi ~-* b~, 1 <_ i<_m, Xqn ~ x q o , , o t E . A n , O < i , j < p - 1 . There is an induced
176
Algebraic K-Theory
E0-homomorphism ~" S = R | E0 --* F | E0 = E, which maps Ai to as for all 1 _< i < n, and maps N I , a ( X a ) to N ~ ( x a ) . Let P C R b e the kernel of r Then R p is a regular local ring, in which As, 1 _~ i < m, Bs, 1 < i < m, are units. Also, Q = P S is a p r i m e ideal, which is the kernel of ~/; the ring SQ is a regular local ring, in which A~, 1 < i <: n, Bs, 1 ~ i ~ m, NI,~(X~), ~ E An, 0 ~ i , j < p - 1 are units. Let T = R[Csj] be the polynomial ring over R in the variables Cij, 1 < i ~ n, 0 _ < j < p - 1 . Thus T is the polynomial ring over F0 in all of the variables As, 1 ~ i < m, Aij, m + l < i < n, 0 < j ~ p - 1 , Bi, 1 ~_ i ~ m, Csj, 1 ~_ i <_ n, 0 <_j ~_ p - 1, X~j~,, c~ E .An, 0 ~_ i , j ~ _ p - 1. T h e homomorphism r R -~ F extends to a homomorphism p " T ~ F , by letting # In= r and #(Csj) = csj. Let T | E0 = U, and let p-1 Ci = ~ Csjw j f o r l < i _ < n . j=O
T h e n p and 7/have a common extension to a h o m o m o r p h i s m v : U --, E, with v(Cs)--c~. Clearly V = S[Csj] is a polynomial algebra over S. If a E GaI(E1/F1) is the generator such t h a t a(w) = Cw (where r is the given primitive pth root of unity), then S is a Galois R-algebra, U is a Galois T-algebra, and a also generates Gal ( S / R ) and Gal ( U / T ) . Clearly 7? and v are a-equivariant, where we also identify a with the corresponding generator of Gal (ELF) ~- Gal (Eo/Fo). If we set p-1 p-1 z,k =
c,
=
j =0
c,jr
1 < i <
0 < k < p-
1,
j =o
then Zsk are another set of independent variables generating the polynomial algebra U over S, since det[ak(wJ)]o<j,k
fi =
Csjw j
- Di, l < i < n.
In terms of the variables Zsk, we may rewrite this as
fs=ZPo-Di,
1
If we uniquely write p-1
fs= ~ j=O
gijw ~ , l < i ~ n ,
8. The Merkurjev-Suslin Theorem
177
let J C T be the ideal generated by the gij. Then J U is the ideal generated by Z~k --ak(Di), l <_ i <_ n,O <_ k < p - 1 . Again, this is because p-1
Z ~ - ak(D,)
= ak(fi)
=Z j=O
so that the ~rk(fi) are E0-1inear combinations of the gij obtained by applying the invertible p x p matrix [ak(wJ)]. Also note that J U fq T = J (this is true of any ideal J of T). We claim the elements ak(Di), 1 < i < n, 0 <_ k < p - 1, yield Z / p Z linearly independent elements of E~ | Z / p Z (recall E1 is the quotient field of S). To see this, since S is a polynomial algebra over the field E0, it suffices to find an E0-homomorphism S --. M to some field M such that the images of the ak(Di) are multiplicatively independent elements of M* modulo pth powers. Let M = Eo(Yi), where Y~, 1 < i < n are independent variables over E0. Let at e An be the mapping given by at(l) = 1, a t ( j ) = 0 for j # I. Consider the homomorphism S --, M given by (i) Ai ~--~Y~ for l < i < m, (ii) A i 0 ~ - - ~ Y i f o r m + l < i < n ,
Aij~--~0form+l
(iii)
Bi~-~lforl
(iv)
for a e ~ 4 n - { a l , . . . , a n } , i , j < p - 1, (i, j) ~ (0, 0),
(v)
for 1 < l < n, X0oa~ ~-* 1, Xloa, ~ - 1 , l
we haveX00a ~-* 1, Xija H 0
for 0 <
Xolao H 1, Xij,:,,, ~ 0 if
Then Ai ~-~ Yi for all 1 < i < n, and Xa H 1 for a r at for all l, while p-1
N1,,~,(Xon) ~ H ( 1 4 - w - ~'-Yl~k) -- (I + w ) p - l~. k=O
Hence ak(Di) ~ ( l + r 1 < i < n, 0 < k < p - 1 . Since I / 1 , . . . , Y, are independent variables over E0, and for 0 < k < p - 1, the elements (1 + ~kw)P in E~ are distinct, the images of the ak(Di) are multiplicatively independent, as desired. By K ummer theory, this implies that J U C U is a prime ideal, and the quotient field E2 of U / J U is a (solvable) Galois extension of F1 (the
178
Algebraic K-Theory
quotient field of R) of degree pnp+l, generated by w and the pth roots o f a a ( D ~ ) , 1 < i ~ n, 0 < k < p - 1. Hence J = J U N T is a prime ideal in T, and the quotient field F2 of T / J is an extension of FI (in E2) of degree id'n. If G = Gal (E2/F1), H : Gal (E2/F2), then G has order ppn+l, and H c G has order p. Hence there is a chain of subgroups H = Ho C HI C ... C Hn = G where Hi-1 is a normal subgroup of Hi of index p. There is a corresponding chain of subfields F1 = Lo C L~ c . - .
c Lp, = F2
where each F~ is a Galois extension of Li-1 of degree p. Further, (T/J)[D -1] is a fiat, unramified RID -1] algebra (i.e., an 6tale algebra). Indeed, S[D -1] is faithfully fiat over R[D -~] (since S is faithfully fiat over R), so it suffices to prove that (U/JU)[D-1] is an 6tale algebra over 8[n-~] (see Milne, E,tale Cohomology, I, 2.24). But this is true, since (U/JU)[D -1] is obtained by adjoining to S[D -1] the pth roots of a set of units of S[D-1]. Since u(C~) = c~, we have u ( n , ) = ~ = u(Z~). Hence u(f~) = 0 for 1 <_ i <_ u. Hence also ~,(ak(f~)) = 0 for 1 < i < u, 0 <_ k <_ p - 1 . Thus # factors through T / J , and y factors through U/JU. The kernel of # 9 T / J --~ F is a prime ideal P contracting to P c R, such t h a t Q - P(U/JU) = keru " U / J U --, E, and Q contracts to Q c S. Since y(Di) - c~i ~ 0, so that u(aJ(D~)) - aJ(ci) for 0 < j < p 1, we have aJ(Di) ~ Q for all i, j, and hence n ~ Q N R - P. Thus ( T / J ) ~ is localisation of (T/J)[D-I], and is hence an ~tale extension of Rp. Hence by lemma (8.20), there are homomorphisms
01" F~ --~ F*, Z/pZ ---, K 2 ( F ) |
02" K2(F2) |
Z/pZ
such that (i)
01(Y) -- P(Y) for y e ( T / J ) p
(ii)
02{Yl,Y2}= {OI(Yl),OI(y2)} for all Yl,y2 e F~.
In particular, m
e2(H(A,,B,}) i=1
m
= yI(~,,b,} = x e K2(F) | i=1
also, for any Y ~ F~,
02{a, Y} = {a, 01 (Y)}.
Z/pZ;
8. The Merkurjev-Suslin Theorem
179
In E2 = F2(~fa), we have that p-1
(~CowJ)P=
Nl,"(X'~)'~(i)' l<_i<_n,
B, H
j=O
aEA,,
Bi=l,
m+l
Here we abuse notation, and also use the symbol Cij to denote the image of Cij in F2. Hence by lemma (8.22), m
X = H { A i , B,} e ker (K2(F2)|
ZIpZ---+ K 2 ( E 2 ) |
Z/pZ).
i--1
Now F1 is a pure transcendental extension of F0, which (we recall) is generated over an algebraic extension of the prime field by ~ and a; hence F1 is a pure transcendental extension of a field which is algebraic over its prime subfield. By lemmas (8.18) and (8.19), the Galois symbol Rn,F1 is injective. Now by l e m m a (8.17) and induction, we see that the Galois symbol is injective for each of the fields Li, and hence in particular for F2. Hence again by lemma (8.17), we see that
ker(g (F2) |
Z / p Z --, K=(E2)
|
z/pz)
consist of symbols {a, Y} with Y E F~. In particular, there exists Y E F~ such t h a t X -- {a, Y}. Then x = o = ( x ) = o={a, Y } = {a, o , ( Y ) } ,
i.e., x is a symbol (a, y} for some y E F*, which proves the proposition. Note t h a t the reference to lemma (8.18), i.e., to class field theory, is not needed if F contains an algebraically closed field, since lemma (8.18) is trivial for the algebraic closure of a prime field. This helps to make our discussion more self-contained, for t h a t case. We can now finish the proof of the main theorem (8.5). We first prove that for any field F containing a primitive pth root of unity ~, the m a p in Rp, F is injective. Suppose x -- 1-I,=t{ai, bi}, and Rn, F(X ) = 0. We prove by induction on m t h a t x = 0 in K2(F) | Z/pZ. For m = 1, this follows at once from l e m m a (S.4)(b). We may assume am is not a pth power in F, else x is a product of m - 1 symbols in K2(F) | Z / p Z , and we are done by induction. Now let E = F({/-am). Then XE = YIi~=l1{hi, b,} in K2(E) | Z/pZ, and Rn, F_,(XE) = O, SO by induction, we have xE = O. Hence by Proposition (8.23), x = (am, y} in K 2 ( F ) | Z/pZ, for some y E F*; hence we are done again by the case m = 1.
180
Algebraic K-Theory
Now we show Rp, F is also surjective. By the usual norm (i.e., transfer) argument, we may assume that any finite extension of F has degree a power of p. Let u E pBr (F), and let E be an extension of minimal degree splitting u. We work by induction on [E : F]. Let L be the normal closure of E over F. The Galois group Gal ( L / F ) is a p-group, since every finite extension of F has degree a power of p. Hence we can find an intermediate field ! F C F1 c E such that [F1 9 F] = p. Then UFl E image Rp, F~ by induction, since UF1 is split by E, and [ E : F1] < [ E : F]. By lemma (8.17)(ii), the square R~,.F , B r (F) K2(F) | Z / p Z
1 g2(F~) ~z Z/pZ
R e p , F1 ---k
1 pBr (F1) !
is a pullback. Hence UF1 E image Rp, F~ ~ u E image Rp, F. This concludes the proof of Theorem (8.5).
(8.3) Torsion in K2 We now prove a theorem of Suslin, which describes the torsion subgroup of K2(F) in many situations. T h e o r e m 8.24 (a) Let F be a field, p a prime such that F contains a primitive pth root of unity ~. Then the p-torsion subgroup of K2(F) is precisely the subgroup {{r
I a e F*} C K2(F).
(b) In (a), if Fo C F is the algebraic closure in F of the prime subfield, then for any a E F* with {r a} - 0 in K2(F), there exists b E F*, c e F~ such that a = b pc, and {r c} = 0 in K2 (Fo). (c) If F has characteristic p > O, then K2(F) has no p-torsion. C o r o l l a r y 8.25 Let F1 C F2 be an inclusion of fields such that F1 is algebraically closed in F2. Then K2(F1) --* K2(F2) is injective. Proof. We may assume without loss of generality that F1 and F2 are finitely generated extensions of the prime subfield; let Fo C F1 be the algebraic closure of the prime subfield in F2. Then we can find a purely transcendental extension F3 of F1 in F2 such that F2 is finite algebraic over F3. Now K2(F1) --~ K2(F3) is injective, since K2(F3) -l i m K 2 ( F l [ t l , . . . ,td, f-1]), where d is the relative transcendence degree I of F3 over F1, and f ranges over the monic polynomials, and K2(F1) --*
8. The Merkurjev-Suslin Theorem
181
K2(Fl[tl,... ,td, f-1]) is a split inclusion for any field F1 (if F1 is finite, K2(F1) = 0; if F1 is infinite, F1 ---* Flit1,... ,td, f -1] is a split inclusion of rings). On the other hand, by the usual transfer argument, ker(K2(F3) --* K2(F2)) is an n-torsion group, where n = [F2 : Fa]. Hence it suffices to prove t h a t K2(F1) --+ K2(F2) is injective on p-torsion subgroups, for each prime p. By (c) of the Theorem (8.24), we may assume F1 is not of characteristic p; hence by the usual tranfer argument, we may also assume F1 contains a primitive pth root of unity. Now by (a) and (b) of Theorem (8.24), we are done. P r o o f of T h e o r e m (8.24) (a): Let K = F(U), L - F ( U ) [ T ] / ( T P - U ) = F(T), so t h a t K, L are each isomorphic to the function field of A~, and the inclusion of fields K ~ L corresponds to a morphism f : X --~ Y over F, where X = SpecF[T] ~ A~, and Y = SpecF[U] ~ A~. Then L / K is a cyclic Galois extension of degree p, which we may think of as a "universal K u m m e r extension over F". Let a E Gal ( L / K ) be the generator determined by or(T) = ~-1T. There is a commutative diagram K2(L)
T
NL,/__..K K2(K) T
K2(F)
'P;
K2(F)
where the (injective) vertical maps are induced by the inclusions of F in L, K respectively, and .p denotes multiplication by p. This follows immediately from the diagram of categories and functors T~(L)
N
~
aT T~(F)
P(K)
TZ
-p
V(F)
where c~(V) - L | V, and (.p)(V) = V an , ~(V) - K | V for any V E P ( F ) , and N ( W ) is the underlying K-vector space of an L-vector space W. The choice of a basis for N(L) gives a natural isomorphism of functors N o c~ -~ f~ o (.p). Thus from Theorem (8.15), p K 2 ( F ) = K 2 ( F ) gl ker(NL/K" K2(L) --. K 2 ( K ) )
= K2(F)
N (1 -
a.)(K2(L)).
So it suffices to prove that K 2 ( F ) n (1 - a.)(K2(L)) = ( ( r
[a E F*} C K2(F).
182
Algebraic K-Theory
Let u E K2(F), v e K2(L) such that u -- v the exact sequence 0-~ K2(F)~
K2(L) ~
a . ( v ) in K2(L). Consider
@xex,F(x)*
~ O,
obtained from the degeneration of the BGQ-spectral sequence for A~, and H~ = K2(F), HI(A~,K:2) = 0 (see the proof of (8.19)). We have O(u) = O, so t h a t cO(v) is invariant under a. If we write =
then the subgroup of a-invariant elements is ~yey1F(y)*.
The map f* 9 E~'-2(Y) --, E~'-2(X) of E1 terms of the BGq-spectral sequences is given as follows: let Y0 E Y and x0 E X be the respective origins, so t h a t X - {x0} --* Y - {y0} is unramified, and f - l ( y o ) -- Xo. Then for y ~ y0, f* induces the natural inclusion F(y)* ~ ~f(x)=~F(x)*
as the subgroup of a-invariants. If y -- y0, then F ( y o ) = F ( x o ) -- F , and a acts trivially; however, F ( y o ) * ~ F ( x o ) * is multiplication by p, since f - 1 ( y o ) is a fiber of multiplicity p in X. Indeed, the terms F ( y o ) * , F ( x o ) * are obtained by ddvissage from K1 groups of suitable categories of Artinian modules, and for any Oy, yo-module M of finite length, f * M has length equal to p(length(M)). Thus we can find w E K2(K) such t h a t O ( W L ) - - i ) ( v ) lies in the s u m m a n d F ( x o ) * . Now we can find a e F* such t h a t c 9 ( ( T , a } ) = O ( W L ) - O(v), since (up to sign) cO is the sum of tame symbols associated to the discrete valuations given by points of X 1 (see (9.12)). Hence, after altering w by adding an element of K 2 ( F ) , we can write v - WL + { T , a } . Hence u = (1 - a . ) (v) = {r a}, as desired. P r o o f o f (8.24)(c): This is very similar to (a), using the "universal Artin-Schreier extension" g = F ( U ) ~ F ( U ) [ T ] / ( T p - T - U) = F ( T ) = L. Since the corresponding morphism A~ --. A~ is unramified everwhere, the argument above shows that if u E K 2 ( F ) , u = v - a . ( v ) , then in fact cO(v) = O(WL) for some w E K 2 ( K ) . Again, after altering w by adding an element of K2(F), we may assume O(v) = c9(wL); hence u = v - a ( v ) = O. P r o o f of ( 8 . 2 4 ) ( b ) : The proof uses continuous (e-adic) Galois cohomology, which we now discuss, as well as results of Tate [T].
8. The Merkurjev-Suslin Theorem
183
Given a topological group G, and a topological G-module M, let
I)i(G,M) denote the group of continuous maps G i+l --, M; then the Di(G, M) can be made into a complex with differentials given by ~+1 dr(go,... , g i + l ) - " Z ( - 1 ) J f ( g o , .
-. , ~ , - . .
,gi+l)
j=O
(here ~ means t h a t gj is omitted). G acts on the complex D* (G, M) by
( g f ) ( g 0 , . . . ,g,) = g . and the subcomplex of G-invariants D*(G, M) v = C*(G, M) is defined to be the standard complex of (continuous) cochains on G with values in M. We may define Hi(G,M) = Hi(C*(G,M)). If F is a field, F a separable closure, G = Gal (F/F), M a continuous G-module with the discrete topology, then H~(G,M) is just Hi(F, M), the Galois cohomology group considered earlier. If M is a finitely generated Zt-module with its g-adic topology, which is a continuous G-module, we can consider the continuous (~-adic) cohomology groups Hi(G, M). A notable example is given by M : Zt(1) = lim ]zt,,, where ~ is a prime ~ char. F. T h e assertions below n
are proved in Tate's article [T]. (i) If G is any topological group, and
0--, M' --%M ~ M"---,O is an exact sequence of continuous G-modules such that M ~ has the subspace topology, and 13 has a continuous section (which may not be either G-equivariant or a homomorphism), then there is a long exact sequence
9..--~ H'(G, U') 24 Hi(G, M) ~ Hi(G, M") ---~... (ii)
Next, if M is a finite Ze-module with its g-adic topology, and G a topological group acting continuously on M through Zt-linear automorphisms, then we have an exact sequence 0 - ~ lim 1Hi-1(G,M/P"M) ~ Hi(G,M) --, limHi(G,M/enM)---* O, ~
n
where lim 1 is the first derived functor of lim. (Recall that if n
n
9- - - ~ G . ~ G . _ I
]n-
1
-~'"~Gl
f2
184
Algebraic K-Theory is an inverse system of Abelian groups, and we let A = 1-In=~176 1 Gn, and define T " A ~ A by T ( g l , g 2 , . . . ) - (Hi - f 2 ( g 2 ) , g 2 - f 3 ( g 3 ) , . . . ) , then we have ker T = lim G?% and coker T - lim 1Gn. Further, if each ?%
n
Gn is finite, then lim 1Gn = 0; more generally, this holds if the inverse n
system {Gn} satisfies the Mittag-Leffler condition, that for each n, im (G,~+m ~ G?%) is independent of m for all large enough m.) 8 . 2 6 Let G be a compact topological group, H C G a closed normal subgroup, such that G ~ G / H has a continuous section. For any topological G-module M, we have an exact sequence
Lemma
0---, H I ( G / H , M H) ~ H I ( G , M ) - - * - M ~ H 2 ( G / H , M H) --~ H 2 ( G , M ) where M C H I ( H , M ) G/H. Proof. This follows from an analogue of the Hochschild-Serre spectral sequence. A continuous section s 9 G / H ~ G gives a map of complexes ~ 9 T)*(H,M) ~ I)*(G,M) which is H-equivariant, such t h a t and the restriction map I)*(G,M) ~ T)*(H,M) are chain h o m o t o p y inverses. If g* - 7)* (G, M) H, then g* computes the continuous cohomology H*(H, M); further ( g . ) c / g = t).(G, M)C _ C*(G, M) computes the continuous cohomology H*(G, M). Now g~ c T)~(G, M ) is the subset of H equivariant continuous maps G n+ 1 ~ M. This can be topologised with the compact-open topology (see Appendix (A.1)), so that gn becomes a continuous G/H-module (since G is compact, the evaluation map is continuous). One checks that g n is an induced G / H - m o d u l e (i.e., g n = H o m (G/H, N ) for some topological group N, where the group of continuous homomorphisms is given the compact-open topology); using this one shows t h a t H ' ( G / H , E n) = 0 for all i > 0, and H ~ ") = ( ~ n ) G / g .~ C?%(G,M), for all n. Consider the double complex
A'J =Cd(G/H, Ei). Taking cohomology first in the /-direction, the above computation of H * ( G / H , g n) shows that the first spectral sequence degenerates at E2, and the cohomology groups of the total complex are H*(G, M). If ZJ ker(gj __~ gi+1), then the second spectral sequence has E1 terms
E~'j = Ci(G/H, ZJ)/(imageCi(G/H,$J-1)). In particular
E ~ = H j (g*) - H j (H, M), E~,~ - Ci(G/H, Z ~ - C~(G/H, MH).
8. The Merkurjev-Suslin Theorem
185
The differential dl " E ~ --* E~ 'j is a map
Z j I B j = H j ( g , M) ~ C 1(G/H, ZJ)/(imageC 1(G/H, g j - 1)), where BJ = image(~:J-1 --o EJ). It is induced by the map
ZJ ---~CI(G/H, ZJ), z~--, .fz, where fz " ( G / H ) 2 ~ Z j is given by
.fz(goH, gtH) = go(z) - gl (z). If fz ~ 0 in E~ 'j , i.e., fz lifts to an element of C 1 (G/H, e j - l ) , then g o ( z ) g~(z) ~. B :~, for any go, g~ ~ G, so the class [z] e H J ( H , M ) is G / H invariant. Thus
E ~ C H j (H, M) c / " . Let M = E ~ C H i ( H , M) G/H. Then the five-term exact sequence of low degree terms (for the above spectral sequence of the double complex) is
0--~ H I ( G / H , M H) ~ H I ( G , M ) ~ - ' M
~ H 2 ( G / H , M H) --~ H 2 ( G , M ) ,
which is the desired exact sequence. This proves (8.26). Let F be a field, finitely generated over the prime field, and s a prime char. F. One sees easily t h a t lim 1H~ #t-) = O, since H~ #t-) is n
finite for each n. Thus we have an isomorphism (Kummer theory)
HI(F, Ze(1)) ~ limF* | .._._
Z/e nZ,
n
giving a character
XF,Zt " F* --~ H i ( F , Zt(1)). Tate shows (in [T]) t h a t if a E F*, a ~ 1, then the cup product
XF,Zt(a) U XF, Zr (1 -- a) = 0 C H2(F, Zt(2)), where Zt(2) = Z t ( 1 ) | Zt(1) (as a topological Galois module). Thus we obtain a homomorphism (Tate's g-adic Galois symbol) K 2 ( F ) --. H2(F, Zt(2)). In IT], Tate proves the following important result. 8.27 (Tate) If F is a number field, then for any prime ~, the s adic Galois symbol K 2 ( F ) --~ H2(F, Z t ( 2 ) ) / s an isomorphism on g-primary components.
Theorem
186
Algebraic K-Theory
We will now use this to prove (8.24)(b). We first prove: 8 . 2 8 Let F be a field which is finitely generated over the prime subfield, and Fo C F the algebraic closure of the prime subfield in F. Then the natural map U ' ( F 0 , Zt(2)) --* H'(F, Zt(2))
Lemma
is an isomorphism for i = 1 and an injection for i = 2. P r o o f . (Sketch) Let F be a separable closure of F , and F0 the separable ( = algebraic) closure of F0 in F. Since F is finitely generated over F0, and F0 is algebraically closed in F , we can find a normal projective variety X defined over F0 whose function field (over F0) is F. Let X = X~oo, so t h a t the compositum FoF is the function field of X over F0. T h e n FoF is a Galois extension of F with Galois group isomorphic to Gal (Fo/Fo). We can apply (8.26) with G = Gal (-if/F), H = Gal (-ff/-~oF), and G / H ~Gal (Fo/Fo), for the module M - Zt(2). We then obtain an exact sequence of continuous Galois cohomology groups
0--* HI(Fo, Z t ( 2 ) ) ~ H I ( F , Zt(2)) --~ M - - * H2(Fo, Zt(2))---* H2(F, Ze(2)), where -M C H~
Hl(-~oF, Zt(2))). So it suffices to prove t h a t H~
Hl(-~oF, Zt(2))) = 0.
m
i
Let Div (X) denote the group of Cartier divisors on X; being a subgroup of the (free Abelian) group of Weil divisors, Div (X) is a free Abelian group, each of whose elements has a finite orbit under G / H - Gal (-~o/Fo). F r o m the family of exact sequences of G/H-modules (Pic X is the Picard group
of x ) 0--~ t . P i c X --, (FoF)*|
~ (Div X ) | 1 6 3
(PicX)|
we obtain an inverse limit exact sequence
0 --, Tt(Pic X) -~ li__m(~oF)* |
Z/enZ --* (Div X) |
Ze.
n
Since
HI(~oF, Z~(2))~ HI(-FoF, Ze(1))|
Ze(1) ~ lim (FoF)* | n
we obtain an exact sequence of G/H-modules 0 --* T t ( P i c X ) |
Zt(1) --* H I ( T o F , Z~(2)) --* ( D i v X ) |
Z~(1).
8. The Merkurjev-Suslin Theorem
187
Now (Div X ) | Zt(1) has no G/H-invariants, since every element of Div is fixed by a subgroup of finite index, and the cyclotomic character G/H --. Z~ has infinite image. Next, Te(PicX) = Tt(Pic 0~), where Pic 0(~), the Picard variety of X, is an Abelian variety (since X is a normal projective variety) by a theorem of Chevalley. By Weil's theorem on the eigenvalues of Frobenius acting on the Tate module of an Abelian variety, we see immediately that Tt(Pic X) | Zt(1) has no G/H-invariants, since the Frobenius associated to a (suitably chosen) finite field ]Fq has eigenvalues of absolute value ql/2 on Tt(Pic X ) | Q, while the cyclotomic character has value q-1 on this Frobenius. This proves HI(-FooF, Zt(2)) c/H = 0, proving the lemma. To prove (S.24)(b), we may assume without loss of generality that F is finitely generated over its prime subfield. Consider the following diagram with exact rows
HI(Fo, Zp(2))
--, HI(Fo,Qp(2))-~
gl(Fo, Qp/Zp(2) ) o H2(Fo,Zp(2))
H I ( F , Zv(2) )
~ H I ( F , Qv(2) ) --+
HI(F, Qv/Zp(2))
_~a H2(F, Zp(2) )
Here a is injective, and f~ is an isomorphism, from the previous lemma, with g = p; also Hi(F0, Qp (2)) = Hi(F0, Zp (2)) | Q, and similarly for F. Suppose c e F* such that {r c} = 0 in K2(F). If
c9' " lim#p.(F) = H~
Qp/Z.(1))-4 HI(F,Z.(1))
n
is the boundary map in the exact cohomology sequence associated to 0 ~ Zp(1) then one checks that 0'(r
--, Qp(1)
-~
Qp/Zp(1) --, 0,
= XF,Z,,((). Hence the cup product
7 = (() U XF,z,,(c) e HI(F, Qp/Zp(1)) satisfies o ( 7 ) = 0'(r
u xv, z,,(~) = z~,z,,(r
o •
= RF, z . { r
= 0.
Hence 7 lies in the image of H~(F, Qp(2)). By a diagram chase, this implies that "t lies in the image of
pHi(F| Qp/Zp(2)) ~ rill(F, Qp/Zp(2)) (note that ~ has order dividing p). We have a commutative diagram with exact rows
g~
Qp/Z~(2))
go(F, Qp/Zp(2))
-~ g l ( F 0 , # ~ 2) --. pgl(Fo, Qp/Z.(2))
--, 0
--. g l ( F , # ~ 2) -4
--, 0
pHI(F, Qp/Z.(2))
188
Algebraic K-Theory
Since 3, is the image of (r Xp,F(C) e H l (F, p~2), we see that (r is in the image of Hl(Fo,p~v 2) ~ H l ( F , ; t ~ 2 ) . But the cup product with (r gives an isomorphism H*(Fo,#v) ~- H*(Fo,#~2), and similarly for F; hence Xp,F(C) lies in the image of Hl(Fo,#p) ~ H I ( F , # p ) , i.e., we can write c = dPe with d q F*, e ~ F~, from the Kummer isomorphisms
HI(Fo, #r,) "~ F~) |
Z/pZ, H i ( F , I-%) ~ F* |
Z/pZ.
Hence {r = {~,e} lies in the image of vK2(F0) --+ vK2(F). If F has positive characteristic, then K2(F0) = 0. If F0 is a number field, Tate's theorem (8.27) and the injectivity of H2(F0, Zp(2)) --. H2(F, Zp(2)) imply that {r e} = 0 in K2(F0). This proves (S.24)(b).
(8.4) Torsion in
CH
2
We now discuss applications of the results of Merkurjev and Suslin to the study of torsion in the Chow group of codimension 2 cycles. We restrict ourselves to a typical situation, though greater generality is possible. Let X be an irreducible smooth quasi-projective variety over an algebraically closed field k, and n an integer prime to the characteristic. We will show that r, C H 2 ( X ) , the n-torsion subgroup of the Chow group of cycles of codimension 2 modulo rational equivalence, is a finite group. Our starting point is the Gersten resolution (5.27) (1)
..- 0 --+ K:2,x --~ i . g 2 ( k ( X ) ) T @xex,(i:r.).k(x)* o ~ u e x 2 ( i y ) . Z --~ 0
which gives H2(X, IC2,x) - C H 2 ( X ) ; here 0, T are the divisor map on rational functions, and the sum of tame symbols, respectively (since the Gersten resolution has these maps, up to sign). We have the complex of n-torsion subsheaves (2)
...
0 ---+ .lC2,x ~ i . . g 2 ( k ( X ) )
--% (gxex, (iz).l-t. ~ O,
where tr is induced by T; the exactness of (1) implies that this complex is exact, except possibly that a may not be surjective. But by considering the commutative diagram i.k(X)*
t,,,
i. ,.,K2(k(X) (where 8(f | - (f,r deduce the exactness of (3)
o@1
,.
'
o,
@zex, (ix).tt.
we see that c~ is surjective. From (1) and (2) we
Z / n Z ---+ i . g 2 ( k ( X ) ) |
--- 0 --, K:2,x | 9
Z/nZ T
z/nz o
- - , O.
8. The Merkurjev-Suslin Theorem
189
Consider the exact sheaf sequences (which define 5~) (4) (5)
--. 0 - . , g 2 , x ~ K:2,x --* Y --* 0,
Z/nZ --, O.
--- 0 - . ~ --. g2,x --' K:2,x |
From (2), we have H'(X,,]C2,x) = 0 for i >_ 2, so that from (4),
H I ( x , IC2,x) .~;HI(X,~-),
H 2 ( X , ~ ) ~- H2(X, IC2,x).
From (5) we thus have a diagram with an exact row (.p denotes multiplication by n) H 1(X, ~ )
.~ H I (X,
T /
HI(X, IC2,x)
H 2 ( X , ~ -)
~ H2(X, M2,x)
I-I2 ( X, g2,x ) We obtain an exact sequence
0 --~ HI(X, IC2,x) |
Z/nZ ---, HI(X, IC2,x |
Z/nZ) ~ ,~CH2(X) ~ O.
Let T/~(/z~ j) be the sheaf on X for the Zariski topology associated to the presheaf V ~-, H~t(V,p~J ). By results of S. Bloch and A. Ogus, Gersten's conjecture and the homology of schemes, Ann. Sci. Ecole Norm. Sup. 7 (1974) 181-201, there is a flasque resolution
o --, n k ( ~ , ~ j) ~ i . H ' ~ , ( k ( X ) , i,~ j) ---, ~ x ,
(i.).H'j'(k(.),
f,~.J-') ~ . . .
Further, there is a commutative diagram
i, g 2 ( k ( X ) ) |
~.H~(k(X), ~ )
-, --,
(which may be checked as in the proof of the commutativity of a similar diagram in the proof of lemma (8.19)). Combining this with the result of Bloch and Ogus for 7 ~ ( p ~ 2 ) , we obtain an isomorphism of sheaves
JC~,x ~z z / n z ~ ~ : (~ .~ ).
190
Algebraic K-Theory
In fact, the only part of the results of Bloch and Ogus which we need for this argument is the exactness of 0 --. "H~ ( / ~ 2) --~ i . H e 2 ( k ( X ) , # ~ 2) --. @ . e x , H~t(k(x), # . ) , which boils down to the exactness for each local ring R of X of
0 ~ He2t(n, Un@2) --~
Hfft(k(X),U~n 2) -"+ (]~htT~=lHelt(k(P),~n).
Since #n C k, this is equivalent to 0 --, nBr (R) ~ nBr ( k ( X ) ) --, (~ht 7~=lglt(k(~D), Z / n Z ) . We give an ad hoc proof of this. In fact, we have: 8.29 Let X be an irreducible smooth variety over an algebraically closed field k. Then we have an exact sequence
Lemma
0 ~ . B r X --~ nBr k ( X ) - . ~ e x a H ~ t ( k ( x ) , Z / n Z ) ,
where Br X = H2(X, Gin) is the cohomological Brauer group. Proof. Br X - , Br k ( X ) is injective (see Milne, Etale Cohomology, III, (2.22)). Hence it suffices to prove t h a t there is an exact sequence
He2t(X, ~n) ---, He2(k(X), l.tn) ---. ~x~xa H l ( k ( x ) , Z / n Z ) . For any effective reduced divisor Z c X, consider the exact sequence
9.. ~ H2t(X, Iz.) ---* H 2 t ( X - Z , U . ) ---* H ~ ( X , u . ) ---* H3t(X,t-t.) ---* ... We have limH~t(U,#n ) = Hiet(k(X),I.tn). So it suffices to prove t h a t we n
have a natural inclusion
H3 (X, u.) --"
**~z"Hl,(k(~), U-)-
For any W C Z of codimension 1 (so t h a t W has codimension 2 in X ) , we have an exact sequence
H~v(X, #n) ~ H ~ ( X , #,,) ~ H ~ _ w ( X - W, #n) Taking W to contain the singular locus of Z, we have ( T h o m isomorphism)
H3_w(X- W,U.) ~ H'~,(Z- W,Z/nZ) =~ HI~(Z- W,u.)
8. The Merkurjev-Suslin Theorem (fix a choice of a primitive n th root of unity). sequence
191
Hence we have an exact
lim H~v(X, #n) '-* H~(X, #n) --* @xez'Hlet(k(x), #n) WCZ
So the lemma follows from the formula H ~ ( X , #n) = 0 for codim W > 2. We prove this formula by induction of the dimension of W; for dim W = 0, W is smooth, so H 3 ( X , # n ) ~ H e3-2dimX(w,#~n1-dimx) = 0, since dim X = codim W > 2. In general, let W1 be the singular locus of W, and consider the exact sequence
H3, (X, #n)
H3(X, #n)
4.4
HW 3 - W a (X - WI~#,.,)"
The first t e r m is 0 by induction, and the third term by the Thorn isomorphism theorem, since W - W1 is smooth of codimension > 2 in X - W. From the isomorphism K:2,x | sequence (6) . . . 0 --~ H I ( X , IC2,x) |
Z / n Z ~ 7 ~ ( # ~ 2 ) , we obtain an exact
Z / n Z --. H ' ( X , TI2x(#~n2)) ~ n c g 2 ( x ) ~ O.
There is a Leray spectral sequence associated to the morphism of sites
f " Xet ---* Xzar,
E~ 'q = H p ( X , R qA , .
@2 )
ge,p+q (X, p@n2).
Since Rqf.#~n2 is the sheaf associated to the presheaf U ~-. H'~q(u, #~2), have R q f . l t @ n 2 "~ 7"/~(/~n@2), and s o E 1'2 = g l ( z , 7"/~f(/~n@2)). T h e differentials for ~-:,I,2 a r e
we
~Ir-r,l+r
__+ ]~Tr1,2 --+ / ~ r r + l , 3 - r
where E r1 - r ' t + r = 0 as r >_ 2. We claim E 3'' = E 4'~ = 0. Indeed, R ~ 2 is the constant sheaf #~2 for the Zariski topology on X , which is a flasque sheaf; hence E~ '~ = 0 for p > 0. Next, from the K u m m e r sequence on Xet we get an exact sequence
O.X .n ' Ox, -" R1 f . m , --' Rl f.Gm where R l f . G m = 0 by Hilbert's Theorem 90 (see Milne, Ethic Cohomology, III, (4.9)). T h u s (since #p ~ Z / p Z ) we have R l f . p p ~ O*x | pp. From the flasque resolution
o - . o*x |
--. i . ( k ( x ) *
e=
x. ( i . ) . . , ,
o,
192
Algebraic K-Theory
we see t h a t Rlf.lZp has cohomological dimension < 1. Hence E~ '1 = 0 for p >_ 2. Thus E~ '2 ~ E~2; further E 3'~ = 0. Since E ~ 3 C E ~ we obtain an exact sequence (7) ..-
0 --,
H'(X, ~2x(p~)) ~ He3(X,~2) ~ HO(x,~/3(p~2)).
Since X is a smooth variety over an algebraically closed field, H3e(X,D~2) He~(X,Z/nZ) is finite. Hence nCH2(X)is finite. If X is a surface, then ~/3 (#~2) = O, since for any aIIine variety of dimension d over k, the 6tale cohomological dimension of an n-torsion sheaf is < d, by the Weak Lefschetz Theorem (Milne, JEtale Cohomology,VI, (7.1)). Thus Hi(X, 7~c(#~2)) ~ He3(X,#~2). In particular, if X is a smooth affine surface, then nCH2(X) = 0, since H3t(X, iz~2) = 0 by the Weak Lefschetz Theorem. If X is a smooth projective surface, Poincar6 duality gives isomorphisms (Milne, l~tale Cohomology,VI, (11.2))
H3t(X, U~ 2) ~
Horn
(g~t(X, Z / n Z ) , Z / n Z )
Horn (He89(X, #.), I~n) Horn (n Pie X, ~u,~) -.~.~ Horn (,,Pie ~ #n) -~ n Alb X. The last isomorphism is via the en-pairing between the n-torsion subgroups of P i c ~ and its dual Abelian variety Alb X. Since the N~ron-Severi group NS(X) = PicX/Pic~ is finitely generated, its torsion subgroup is finite, and so ker H3t(X,#n)-~nAlb X has order bounded by a number independent of n. On the other hand, the torsion subgroups of CH2(X) and Alb X are divisible (this is true for CH2(X) because the subgroup generated by cycles of degree 0 is a quotient of a direct sum of J acobians of smooth projective curves). Further, we claim that for any m, the natural map mCH2(X) --~ mAlb (X) is onto. In fact, we claim t h a t if C c X is a smooth hyperplane section, the map J(C) --~ Alb X is surjective on m-torsion; this is equivalent to the claim that for any connected ~tale Galois Z/mZ-covering Y --. X, the pullback Y • x C is connected. This is true because Y • x C is the pullback to Y of an ample divisor on X; hence it is an ample divisor on the non-singular projective variety Y (see Hartshorne, Algebraic Geometry, III, Ex. 5.7). In particular, Y x x C is connected (Hartshorne, Algebraic Geometry, III, Cor. 7.9). We deduce that ker(CH2(X) --, Alb X ) is a divisible group. Since
ker(nCH2(X) ~ ~Alb X) = n(ker(CH2(X) --~ h l b X))
8. The Merkurjev-Suslin Theorem
193
has order bounded independently of n, we see that n C H 2 ( X ) --, nAlb X is in fact an isomorphism for each n (relatively prime to char. k). This was first proved by A. A. Rojtman (his proof is given in "The torsion in the group of 0-cycles modulo rational equivalence," Ann. Math. 111 (1980) 553569), for smooth projective varieties of any dimension. A proof along the lines given here was first obtained by S. Bloch (see his book Lectures on Algebraic Cycles, Duke Univ. Math. Set. IV, Durham (1976)); he also shows that the result for smooth projective surfaces implies it for smooth projective varieties of any dimension. Milne has used an analogous argument with flat cohomology, combined with results of Kato (which describe K 2 ( F ) / p K 2 ( F ) for function fields f of characteristic p) to show that n C H 2 ( X ) ~ nAlb X is an isomorphism even when n is a power of the characteristic; see J. S. Milne, Zero cycles on algebraic varieties in non-zero characteristic: Rojtman's theorem, Compos. Math. 47 (1982).
9. Localization for Singular Varieties
In this chapter we prove a localization theorem of Quillen for singular varieties, and a generalization of it due to Levine. These are then used to prove the so-called "Fundamental Theorem" (9.8), which computes K~(A[t, t - i ] ) , and to relate the study of 0-cycles on normal surfaces to modules of finite length and finite projective dimension over the local rings at singular points. We begin with Quillen's localization theorem, proved in Higher Algebraic
K - Theory II. Suppose X is a quasi-compact scheme which supports an ample invertible sheaf, j : U , X the inclusion of an affine open subscheme, such t h a t U = X - D for some effective Cartier divisor D in X. Let Z = O x ( - D ) be the ideal sheaf of D. Let T / b e the category of quasi-coherent O x - m o d u l e s Y such t h a t j * Y = 0, and Y has a resolution of length 1 0
-'El
;~:o
; ~- ---~ 0
with Co, C1 E :P(X), the category of locally free Ox-modules of finite rank. Since every coherent sheaf on X is a quotient of a locally free sheaf, ~ is closed under extensions, and is thus an exact category. Theorem
(9.1). There is a natural long exact sequence
Ko(X)-, No(u). P r o o f . Let 1) be the category of vector bundles on U which extend to vector bundles on X, and let 7)1 denote the category of quasi-coherent O x - m o d u l e s ~ which have a resolution of length 1 0 . 's
'8o-
.'.F
,0
with ~:i E P(X). Thus 7~ C Pl; further P C P l is a homotopy equivalence, by the resolution theorem. Since U is affine, every exact sequence in l) splits. Let ~: ~. Q~) be the extension construction over Q]) (~: is the category constructed in the proof of Theorem (7.7), associated to the exact category V). Thus by (7.7), if T - Iso(1)), then T - I ~ : , Q1) is fibered with fibers homotopy
9. Localization for Singular Varieties
195
equivalent to T - 1 T , and T - 1 E is contractible. Let 7) = 7~(X), 8 = Iso7 ~. Then the restriction functor j* 9 ,~ , T is a monoidal functor, making S a subcategory of T. L e m m a (9.2). Let T act on X, and let S act on X by the induced action.
Then S - 1,~ ~
~/-- 1,~ i8 a homotopy equivalence.
P r o o f . For any object A E T, there exists B E 7" such that A ~ B "~ j * C for some C E S. Hence if T acts on A', and S acts by the induced action, translation by C is a homotopy equivalence if and only if the translations by A, B are homotopy equivalences. Hence T acts invertibly on A' ~ S acts invertibly on X. Now for any category X with a T-action, we have homotopy equivalences by (7.1)
S-IX
~T - 1 S - 1 X ,
T-1X
~,8-17"-1X.
But,~xTxX - T x 3 x X , ( A , B , F ) ~ ( B , A , F ) induces an isomorphism 8 - 1 T - 1 X ~ T - 1 S - 1 X . Now the composite T - I X ,~ , S - 1 T - 1 X ~T - 1 , S - 1 X is just the functor obtained by localizing X , S - 1 X with respect to T. We claim the triangle 8-1A,
T-1S-1X
/
7 - - 1 r~'
commutes up to homotopy, where S - 1 X .~ T - 1 X is induced by the inclusion S C T; this will prove S - 1,~, , T - 1X is a homotopy equivalence. Now 3 - 1 X - - . T - 1 S - 1 X is given by ( A , F ) ~ (0, ( A , F ) ) , while the composite ~ - 1A" .~ T - 1A' .~ T - l S - ~A" is given by (A, F) H (A, (0, F)). There is a chain of natural transformations, giving a homotopy (0, (A, F)) ,.L (A, ( A , A ~ F ) ) ~ ( A , (0, F)), where u is the arrow in T - 1( ~ - 1X) given by (A, A + (0, (A, F)) _E. (A, (A, A @ F))), and v is induced by the arrow in ~-1,~, given by
( A , A + (O,F) ~ This proves the lemma.
( A , A (~ F)).
196
Algebraic KoTheory
We construct below a diagram of categories with S-action (where 3 acts trivially on QP, Q V, Q ~ )
c
s
Q~
z)
P--~ Qp
c
,
QV
and show that f, h are homotopy equivalences. Localization then yields a diagram
S-1C
~
S-1D
__~
I
Qp
I
where S - 1 s ~ T - 1 s is contractible. This yields a long exact sequence
K,+~(V)
,K,(X)
,K,(P)
,K,(V)
,...
where K i ( P ) = K i ( X ) , and V c P ( U ) induces maps K i ( V ) L e m m a (9.3). Ki(V) -~ K i ( U ) i s monomorphism for i = O.
, Ki(U).
an isomorphism for i > O, and a
P r o o f . Let S ~ - Iso/)(U), where U - Spec R is given to be affine. T h e inclusion V c )o(U) induces T c 3 ~. If 8n c 3 t is the connected component of the free module R ~n, then since R ~n E ~), Sn C T . Thus the functor s - (St-18')0, used to give the h o m o t o p y equivalence B G L ( R ) + B ( S ' - i S t ) 0 of (7.3), factors through ( T - 1 T ) o , ( 8 t - l S ' ) o . Hence the proof of (7.3) yields homotopy equivalences
B G L ( R ) + '~ B ( T - ' T ) o
'~ B ( S ' - I s ' ) 0 .
Thus Ki(V) ~ Ki(U) is an isomorphism for i > 1. Since all exact sequences in 7~(U) are split, and R Sn E ]) for all n, we see that for A, B E V, [A] = [B] in Ko(V) ~
A (9 R e'' "~ B (9 R $'~ for some n
< : - [ A ] - [B] in Ko(V). Hence Ko(V) : ; Ko(U) is injective. R e m a r k . Gersten has shown that if ~4 C 7~ are exact categories, such t h a t every exact sequence in P splits, and A4 is cofinal in 7) (i.e., V A E P , there exists B E P with A (9 B e jL4), then BQ.A4 - ~ BQT' is h o m o t o p y equivalent to a covering space. Hence Ki(~&4) ; Ki(7 ~) is an isomorphism for i > 0 and a monomorphism for i - 0, from the corresponding properties of ~ + 1-
9. Localization for Singular Varieties
197
Returning to the proof of (9.1), we note that s is equivalent to the category (again denoted E, by abuse of notation) whose objects are admissible epimorphisms B ~.;C with B, C E ~', where a morphism (B ~; C) (B' ;; C') is defined to be an equivalence class of diagrams (in 1)) B
~
C
B
~
C1
I o I B'
~
6"
where the bottom square is a pullback. Define the category T) to be the pullback
D
P__~ Qp
!
s
1
,
QV
Thus, an object o f / ) is a pair ( B , Z ---o j ' B ) with B E ~P, Z E ~) (where j : U ~ X , P = P ( X ) ) . An arrow in 7) is represented by an equivalence class of pairs of diagrams B
Z
T
II
B1
Z
I
I
B
Z'
,
j*B
T
~
j'B1
~
j*B
I
(where the right hand column is obtained by applying j* to the left hand column). Since s ~ QV is fibered, so is p : 7) , Q7~. Next, let C be the category whose objects are admissible epimorphisms (in P l ) L -o M (9 B, with L , B E P , M E ~f, where an arrow (L -~ M (9 B ) ~ ( L ' -~ M ' (9 B ' ) is an equivalence class of diagrams L
II
L
I
LI
~
M(gB
~
Ml(gBI
T
~
I
M' ~ B I
198
Algebraic K-Theory
where the right hand column is the direct sum of columns representing arrows M , M ' , B ~ B' in QT/, Q p respectively. There is a functor C ~ Q ( ~ / • p ) given by
(L
, M (g B) ~-. ( M , B ) ;
this makes C fibered over Q ( H • P). Let g : C , Qp, h : C , Q~/be obtained by projection; then g, h are also fibered functors. Let f 9C .- :D be the functor
(L
~';M (9 B) ~-~ ( B , j * L ~ j ' B )
(where we note that j * M = 0). 3 - Iso P acts on C by A + (L r M (9 B) = (A (9 L (0,r.-- M (9 B), and on C (through the usual action of T ) by A + (B
r ,, C) = (A (9 B (0,r,, C).
S acts on T) by the induced action, where we take the trivial S-action on
Qp, Qv, Q~t. Lemma
(9.4). h : C
~ QTl is a homotopy equivalence.
P r o o f . Since h is fibered, as remarked above, it suffices to prove the fibers of h are contractible. For M E ~ , let A M be the category whose objects are admissible epimorphisms (in :Pl) L ;~M with L E T~, where an arrow is a diagram L
L'
J
where L ~ L' is admissible mono in ~ . Then the category Sub ~ M (see the proof of (7.10)) is equivalent to h - l ( M ) , by associating to the above diagram (an object of Sub 7~M) the object L' ,, M @ L ' / L in h - ~ ( M ) . Since Sub 7~M is equivalent to R M , it suffices to prove ~ M is contractible V M E 7-/. But if L --~ M is a fixed object of ~ M , we have natural transformations of functors ~ M -~AM
(L ~-, M ) ~-~ (L (9 L'
~-M)~-~(L'
which connects the constant functor T~M functor.
, {L
,, M ) - M } to the identity
9. Localization for Singular Varieties Lemma
(9.5).
199
If E E 79 is a vector bundle on X , then E C j . j * E , and j,j*Z = U Z-"E = U E | n>O
Ox(nD).
n>O
P r o o f . Clear. Lemma
- - ~ D is a homotopy equivalence.
(9.6). f-C
P r o o f . We have the triangle of categories and functors (g = p o f ) C
-
f
/)
;
Q~ where g, p are fibered functors. C l a i m . For each B E Q79, the functor (induced by f ) g - l ( B ) ~ p - l ( B ) is a h o m o t o p y equivalence. To prove this claim, let TCB be the category whose objects are admissible epimorphisms (in 79) L ;; B with L E 79, and whose arrows are diagrams L* ~
III
L
--~
B
B
where L ~ --, L is an admissible mono in 791 whose cokernel lies in ~/, i.e., j*L I , j * L is an isomorphism. Then the functor Sub TCB ---, g - 1(B) given by LI
L
.
B
B
is clearly a h o m o t o p y equivalence. Let W = j ' B , so t h a t p - l ( B ) = s must show t h a t the functor Sub TCB ; s
L~
.-. B
,III L
To prove this claim for B , we given by
~ B
~-~ (j*L
~;j*B = W )
200
Algebraic K-Theory
is a homotopy equivalence. This functor factors through the range functor Sub T~B -~R B , L'
.
B
~-.(L--~B) L
~
B
which is a homotopy equivalence (see (7.10)). Hence it suffices to prove that w 9 T~B ~s (L "-" B ) ~-. ( j * L ~.; j * B - W ) is a homotopy equivalence. To do this, by Theorem A, it suffices to prove that w / ( Z W) is contractible for any Z --~ W in tYw. An object of w / ( Z ~ W ) is a pair, consisting of an object L --~ B of T~B, together with an arrow (i.e., an isomorphism) (j'L
, W)
, (Z ~
W),
given by an isomorphism j * L ~ Z making the triangle j*L
",,, /
"-
Z
W
commute. Let/;at. be the partially ordered set of vector bundles L E 7:' such t h a t i) L c j , Z , ii) j * L = Z, iii) the image of the composite map L ----, j , Z ,, j , W = j . j * B is B C j , j * B . The partial ordering is given by inclusion. From iii), every L E Eat has a given admissible epimorphism (in 7)1) L , B , such that j * L , ; j * B = W is just Z , , W . Thus /:a_At is identified with a full subcategory of w / ( Z , W ) , such that every object of w / ( Z ---~ W ) is isomorphic to an object of Eat, i.e., the inclusion Ea__At ~ w / ( Z ;'. W ) is an equivalence. So it suffices to prove that Eat is filtering, hence contractible. To prove that Eat is filtering, since Eat is clearly nonempty (since Z E QI), Z extends to a vector bundle on X), it suffices to check that if L1, L2 E Eat, there exists L 3 E /:a__Atwith L1 C L3, L2 C L3. We have an exact sequence in ~) 0
~Y
;Z
~.W
~.0
where since Y E ~), Y = j * C for some C E 7~. Then Z ~ j * ( C ( g B ) . Consider the corresponding lattice C (9 B C j . Z . If L ; j . Z is any lattice, then since L ~j . W = j . j * B has image B, by definition of Eat, we must have
9. Localization for Singular Varieties
201
Since L is locally finitely generated and X is quasi-compact, L C I - n C @ B for some n > 1. Take L3 = ( Z - n C (9 B) C j , Z , for some sufficiently large n; then clearly L1 C L3, L2 C L3. This completes the proof of the claim, that g - l ( B ) ; p - l ( B ) is a homotopy equivalence for each B E 7). The lemma now follows from: L e m m a (9.7). Let A
\
f
*
B
,x
C
be a commutative triangle of categories and ~nctors, such that g, h are fibered. Suppose that for each B e C, g - l ( B ) , h - l ( B ) (induced by f ) is a homotopy equivalence. Then f is a homotopy equivalence. We postpone the proof of this lemma, and first finish the proof of Theorem (9.1). We now have the diagram (~- denotes a homotopy equivalence) on which ,q = Iso 7~ acts. r
C
-;
D
,
I
Q~
Q7 ~
I
c
~ QV
Since S acts trivially on Q~/, and Q t / ~ C ~ D, S acts invertibly on C, D so that we have a localized diagram
S-1C
~,
S-1D
,
l
Q~
S-~s
::;
Q~O
l
QV
where the arrows marked ~ are homotopy equivalences, from (7.1). Further, 3 - 1 s is contractible, by (7.10). The base change arrows for the fibered functor S - 1 s ; QV are homotopy equivalences, from the proof of (7.9), so that the same is true for the pullback functor 8-I:D ; Q p . Thus, in the square 8-1~ , Q7:'
l
,.q- l ,f,
,
l
QV
the horizontal maps are quasi-fibrations, by Theorem B, and the maps from the homotopy fibers of the top row to those of the bottom row are homotopy
202
Algebraic K-Theory
equivalences (since the maps on the actual fibers are isomorphisms). Thus the square is homotopy Cartesian; since S - 1 s is contractible, B S - I:D
~BQP
~ BQ1)
has the homotopy type of a fibration. Hence, BQTI
, BQP
~BQV
has the homotopy of a fibration, for a certain homotopy class of maps BQTl ~ F, where F is the homotopy fiber of B Q P ~ BQ]) over a 0-object 0 E ~). This suffices to give a long exact sequence ...K~+:(U)
, K,(7"l)
, K,(X)
, K , ( U ) ~ ...---, K o ( X )
, go(U).
However, in applications, we often need a stronger statement. Let 7)1 be the exact category of coherent Ox-modules of homological dimension _< 1, and let 7"/(U) denote the category of coherent O r - m o d u l e s of homological dimension < 1. Then Q P c Q:Pl is a homotopy equivalence, by the resolution theorem, while Q Y c QT"l(U) is homotopy equivalent to a covering space of QT"I(U), from (9.3) and the resolution theorem. Thus we have a diagram F' , BQPl , BQ~(U)
T
F
,
T
BQP
,
T
BQ])
where F, F ' are the respective homotopy fibers over a given 0-object in V C T/(U), and the arrows marked ~ are homotopy equivalences. We have an inclusion Q ~ / c QPl such that the composite functor QTl ~ QTl(U) maps every object to a 0-object. Since any two 0-objects of T/(U) are canonically isomorphic in QT-I(U), there is a natural nullhomotopy of the composite BQ?-I ~ B Q ~ ( U ) , giving a map r : B Q T l ~ F'. On the other hand, we have already constructed a homotopy equivalence B Q T l F (which is well defined up to homotopy); composing this with F -~, F ~ yield a homotopy equivalence r : B Q T l ~ F t. Using r we want to prove is also a homotopy equivalence; the long exact sequence constructed using r is the one with good naturality properties (see the discussion of naturality, at the end of Chapter 6, for the localization sequence associated to a Serre subcategory of an abelian category). Our proof that r is a homotopy equivalence is based on unpublished notes of Richard Swan. (Swan has now published his notes as an appendix to his paper "K-Theory of Quadratic Hypersurfaces," Ann. Math.
9. Localization for Singular Varieties
203
122 (1985), 113-153.) We first consider the commutative triangles (-~ denotes a homotopy equivalence) S-1E
D
Q1)
'
: ,S-11P
Q:p
This yields the commutative diagram of classifying spaces , BQP
BI)
S
BS-1D
1
, BQP
BE
BS- lg
: BQV
, BQP
which gives a commutative square of homotopy fibers F1
-~ F2
l
1
F3
,
F4
where F1 - - ~ B:D
, B E , F2 ; BQT' ~ B Q V , F3 ; B,S-I:D ; B Q P are the homotopy fibers over corresponding points (i.e., choose any point in BE, and take its image in B Q ) ) and B3-1,f.). Then in fact F2 - F4, and F3 , F4 is a homotopy equivalence B,S-tE., F4
~BQT'
since the front face is homotopy cartesian. Since l:) -~, S - 1 ~ , ~ ~ S--1 are homotopy equivalences, F1 ~ F3 is a homotopy equivalence (for example, compare the long exact homotopy sequences to see t h a t F1 * F3 induces isomorphisms on lh). Hence FI * F2 is a homotopy equivalence. Since the Fi were constructed by starting with an arbitrary point of B E , we see that Z)
---,QP
1ol C
~
QV
is homotopy Cartesian. Now E is contractible by a chain of natural transformations linking the identity functor to the constant functor with value (0 -~ 0 --, 0 -4 0 --,
204
Algebraic K-Theory
0 --* 0) E E; this is represented by the diagram 0
* A
* B
,
0
;
,
B
~ B
T
T
0
II 0
~ 0
,
0
C
~
0
,
0
0
~ 0
(giving arrows in t; from top and b o t t o m sequences to the middle one). Since C is contractible, the homotopy Cartesian square (~3rields a h o m o t o p y equivalence B:D , F, where F is the h o m o t o p y fiber (over 0) of Q p Q1). This homotopy equivalence depends on the choice of a deformation contracting ~ to a point (see the discussion in Chapter 6 of h o m o t o p y Cartesian squares, preceding the s t a t e m e n t of Theorem B). The above chain of n a t u r a l transformations thus gives one choice of a homotopy equivalence u 9 BT) ; F. Regarding a map to F as a pair, consisting of a m a p to B Q 7 ~, and a nullhomotopy of the composite to B Q 1 ) , we see easily t h a t u 9 B:D , F is given by the pair (consisting of a functor, and a chain of natural transformations) (P,O ~ M ~ L --, j * P --~ O) ~
(P,O ~
L -~ j ' P )
ET)
where L denotes the functor (0 --, M --, L --~ j * P ---, O) ~ L, and j * P has a similar meaning (so that (0 ---, M ---, L --, j * P ~ O) ~ j ' P ) is the composite 7:) ----, Q1)); 0 ~ L, L --~ j * P give the chain of n a t u r a l transformations of functors T~ , Q1) 0
, L~
j*P.
Next, let 7~t C Pl be the smallest full subcategory containing :P and ~/, which is closed under extensions ( P ' is easily seen to be the full subcategory of Pl consisting of sheaves M such t h a t there is an admissible m o n o m o r p h i s m N ~-, M with j * N ~- j ' M , and N E :P). Let :D' be the fiber p r o d u c t category /),
1
g
~
Qp'
,
Q1).
l
As above, if F is the homotopy fiber of Q p ' , Q1) over 0, we get a map BT)' , F induced by the above contraction of E (we do not need to know if this is a homotopy equivalence). Since ~ c P ' is a h o m o t o p y
9. Localization for Singular Varieties
205
equivalence, F .~ F is a h o m o t o p y equivalence; since F t h r o u g h / ~ , clearly ~' .~ F ' is a h o m o t o p y equivalence.
~ F ' factors
Let A 9 Q ~ , iD' be given by H ~-~ ( H , 0 --* 0 --, 0 --, j * H ---, T h e n clearly the composite BQT-I ~ BQI)' ~ F ~ F ' is just r BQT"I ~ F ' , since t h e nullhomotopy Q~/ , Q1), as well as t h a t of QT-I ~ Q ~ ( U ) , are given by the canonical isomorphism of j * H with the given 0-object. Hence it suffices to check t h a t the composite BQT"I ~ F, induced by A, is a h o m o t o p y equivalence, in order to prove r is a h o m o t o p y equivalence.
0).
Let i 9 :D , :D' be the inclusion, f 9C ~ T), h 9C functors constructed earlier; we have a diagram of categories C
f
,Z)
i
~ Q?/the
, :D'
h
where f, h are h o m o t o p y equivalences; this diagram does not c o m m u t e , but does so "up to sign", as we explain below. We have a related d i a g r a m of spaces BC
BI
m ,,
, BiD
,~BiD'
0
E
~F
BQTi I
where the composite OoBi 9 BID ~ F is the homotopy equivalence induced by u 9 BiD ; F. Thus Oo B(i o f) 9BC ~ F is a h o m o t o p y equivalence. Now BZ)' is an H - s p a c e under the direct sum ~; since j * 9P : ." )) is surjective on objects, one sees easily t h a t BiD' is connected. Hence BiD' is an H-group. If we show t h a t (A o h) ~ (i o f ) 9C ; / ) ' is nullhomotopic, then if 7- 9 BiD' , BiD' is the inverse map for the H - g r o u p s t r u c t u r e , we see t h a t B ( A o h ) = B ) ~ o B h is homotopic to r o B ( i o f ) . Hence O o - r o B ( i o f ) is homotopic to 0o B A o h . B u t ~" induces multiplication by - 1 on 7rn(BiD') k/n; since 0 o B ( i o f ) induces isomorphisms on h o m o t o p y groups, so does OoT"oB(iof). Hence 0 o T o B ( i o f ) , and thus also 0 o B A o B h , are h o m o t o p y equivalences. Since h is a homotopy equivalence, 8 o B A is a h o m o t o p y equivalence, as desired. So we have reduced t h e desired n a t u r a l i t y s t a t e m e n t to t h e claim t h a t g = (i o f ) $ (A o h ) - t : ~ T~' is nullhomotopic. Now T)' has objects ( P ' , 0 ~ M ---, N ---, j * P ' - . 0) with P ' E 7~', and (0 ---, M ---. N ~ j * P ' ---. 0) E g'. T h e functor s 9C ; iD' is given by
206
Algebraic K-Theory ( 0 ~ M ~ L ---, H (g P - - , O) H ( H (9 P, 0 ~ j * M ---, j * L ---, j * ( H (9 P ) ~
0).
The diagrams below give a chain of natural transformations to a constant functor: H(gP
T
0
,
j*M
,
j*L
II I 0
~ j*(H(gP)
L
0
~
T !! 0
~ j*L
;
0
0
~
0
~
~
I
T
;
O.
j*L
~ O.
0
~ O.
I
(We remark here that though L --~ H (9 P is a natural transformation, L ~.,P is not one, from the definition of morphisms in C; hence the above does not give a nullhomotopy of C , :D). Thus, we have completed the proof of the localization theorem, modulo the topological lemma (9.7). P r o o f o f (9.7). Given the triangle of categories A
f
~
B
C we have a diagram (as in the proofs of Theorems A and B)
A
,I
13
,P'
8(g)
P~,
C~
vl
S(h)
v2
Co p
II
Here S ( g ) is the category of triples (X, Y, v) with X E r Y E C, v" Y , g ( X ) an arrow in C, where a morphism (X, Y, v) ~ ( Z ' , Y', v') is a pair of morphisms u" X
; X', w" Y' V
~ Y such that ~,
g(X)
,
g(X')
~T Y'
I ~(~)
commutes; S ( h ) is defined similarly, and f 9 S ( g ) ---. S ( h ) is given by f ( X , Y, v) -- ( f ( X ) , Y, v). As in the proof of Theorem A, pl " S ( g ) ~ .4,
9. Localization for Singular Varieties
207
pl : S(h)
, 13 are homotopy equivalences. As in the proof of T h e o r e m B, since g, h are fibered, Bp2 : BS(g) ~ BC ~ Bp2 : B S ( h ) , BC ~ are quasi-fibrations, where the fiber of Bp2 : BS(g) , BC ~ over Y E C is B ( Y \ g), and the fiber of Bp2 : BS(h) , BC ~ over Y is B ( Y \ h). By hypothesis, B ( Y \ g) ; B ( Y \ h) (induced by f ) is a h o m o t o p y equivalence, since B g - I ( Y ) , B h - I ( Y ) is one, and g, h are fibered. Since the maps Bp2 are quasi-fibrations, we deduce t h a t in the diagram F(g, Y)
,
1
F(h,Y)
,
BS(g)
;
l
BS(h)
BC ~
!1
,
BC ~
where F(g, Y), F(h, Y) are the respective homotopy fibers over Y, the map F(g, Y) ; F(h, Y) is a homotopy equivalence. Hence, by the 5-1emma, B f 9 BS(g) ~ B S ( h ) induces isomorphisms on homotopy groups, and hence is a homotopy equivalence. This proves (9.7). Our next goal is to prove the so-called Fundamental Theorem. need some notation: if X is a scheme, let
We
N K i ( X ) = coker(Ki(X) ---, K i ( X Xz A~)); if A is a ring, let
N K i ( A ) = N K i ( S p e c A) = coker(K~(A)
, K,(A[t])).
Next, for any scheme X, let N i l ( X ) denote the exact category of pairs (V, a) where V E 7~(X) is a vector bundle on X, and c~ is a nilpotent endomorphism of V; an exact sequence 0
,(v2,c
2)
,0
is an exact sequence 0
t, Vl
i ,V 2
q ,V
3
,0
in 7~(X) such t h a t i o a l = t~2 o i, q o c~2 = t~3 o q. Let Nili(X) = ker K i ( N i l ( X ) )
,
Similarly define Nili(A) for any ring A. Theorem
(9.8) (Fundamental Theorem). There are natural isomorphisms
a) N K i ( A ) ~- Nili_l(A), for i >_ 1 b) Ki(A[t,t-l]) ~- KI(A) (9 N K i ( A ) r
(3 Ki-I(A), for i >_ 1.
208
Algebraic K-Theory
Below, we only prove a), and the following weaker form of b). show t h a t there is an exact sequence
0
} K~(A)
, K~(A[t])@ g,(A[t-1])
}Ki-I(A)
We
~. K~(A[t,t-1])
}0
for i > 1, which can be rewritten as a short exact sequence
0
} K , ( A ) @ N K i ( A ) $2 ~
K,(A[t,
t-l])
} Ki_I(A)
, 0
from a). T h e map Ki(A[t,t-1]) , K~_I(A) can be shown to be a split surjection, where the splitting is given by the composite of the natural m a p K i - I ( A ) --~ K i - l ( A [ t , t - 1 ] ) with the map K i _ l ( A [ t , t - i ] ) } K i ( A [ t , t - 1 ] ) given by the product with t E A[t,t-x] * = K l ( A [ t , t - 1 ] ) (see Higher Algebraic K-Theory H for details). This yields the stronger form
of b).
P r o o f o f (9.8). We will assume A is commutative. Let X = ]P~, considered as the union of the open subschemes SpecA[t] and Spec A i r - l ] ; the inclusion of the open subscheme U - Spec A[t -1] in X satisfies the hypothesis of the localization theorem (9.1), so t h a t there is an exact sequence
, K,(H)
, Ki(X)
, Ki(A[t-l])
, Ki_l('~'~)
'...,
where 7~ is the category of coherent O x - m o d u l e s which have homological dimension _< 1 and vanish on U. Clearly 7t is isomorphic to the category of finite A[t]-modules of projective dimension < 1 whose localization to A[t, t - 1] is zero. Since Spec A[t] } X is flat, we have a diagram of localization sequences
K,(~)
~
K~(X)
II
~
l
K~(~)--,
K~(A[t])~
K~(A[t-1])
--,
1
K~(A[t,t-1])--,
K,_~(U)
II
K~- 1(7-/)
--~ . . .
--* ...
We first observe t h a t 7-/ ~ N i l ( A ) , as follows. If (V,~) E N i l ( A ) , regard V (which is a finitely generated projective A-module) as an A[t]module where multiplication by t is the A-linear endomorphism cz. T h e n A[t, t -1] | V = 0, and we have the characteristic exact sequence
0~
A[t] |
V t - 2 A[t] |
V
,V
, O.
T h u s (V, a) gives an object of T / i n a natural way, giving a functor N i l ( A ) H. Conversely, if M is an A[t]-module of projective dimension < 1 killed by a power of t, then we claim the underlying A-module M is a projective A-module. Indeed, if 0
,Q
,P
}M
,0
9. Localization for Singular Varieties
209
is a projective resolution over A[t], and triM = O, then multiplication by t n on the terms of the above resolution yields (snake lemma) an exact sequence 0--. U , Q/t"Q , PIt"P, and hence an exact sequence
0 --. M
,
Q/t"Q
,
Q/t"P
,0
(t"P C Q as t'~M = 0). Now Q/t"Q is a projective A-module (as Q/tQ is one), and Q / t n p has projective dimension _< 1; hence M is a projective A-module. If c~(t) is the A-linear endomorphism given by multiplication by t, then M ~. (M,a(t)) gives a functor 7/ , Nil(A) inverse to the above functor. By Theorem (5.29), Ki(IP~) -~ K,(A) ~2, where K~(A) ~2 , Ki(IP~) is given by
(x,y) ~-, x + Cy,
r = [Op,(-1)] e K0(IP~).
Thus K~(IP~) ~- K~(A) + r Ki(A). This may be rewritten as Ki(IP~) K~(A) + (1 - ~ ) K ~ ( A ) , where
" g,(A[t]))= g e r ( g , ( F ~ )
Ker(g,(IP~)
= (1
-
~).
, g,(A[t,t-x]))
K,(A),
since Op, (1) restricts to the trivial bundle on Spec A[t] and on Spec Air- 1], while the natural maps
K~(A)
, K~(A[t]),K~(A)
,
K~(A[t-I]),
given by change of rings, are split inclusions. Hence the localization sequence for X = ]P~ breaks up into exact sequences (for i > 1),
0 - - - . K~(A)
~ Ki(A[t-l])
~ K~_x(gil(A))
~ K~_I(A)
~ O,
where we have used 7~ ~ Nil(A). If (P, a) E Nil(A), and we again denote the associated element of 7~ by P, the characteristic exact sequence 0
,~ Om ( - 1 ) |
P t-20p~ |
P
~P
,0
(where t is now regarded as a global section of Oe~ (1)) gives an exact q ence of functor N l(A) , VI(P ), the c t gory of coherent Opl-modules of homological dimension _< 1. Hence the square below commutes:
Ki_1(Nil(A))
-~ x-r
Ki- 1 (~'~)
210
Algebraic K-Theory
where the map f is induced by (P, cz) ~-, P e :P(A), and g is induced by 7"/c P l ( P ~ ) . Hence ker(Ki_l(~/)
~, K i - l ( X ) ) ~
gili_l(A),
and the resulting exact sequence 0
, Ki(A)
, Ki(A[t-1])
, Yil,_l(A)
~0
(which is obviously split) proves (9.8)(a). Next, the diagram comparing the localization sequences of X and Spec Air] gives the Mayer-Victoris sequence ...
, Ki(X)
, Ki(A[t])(9 K i ( A [ t - i ] )
, Ki(A[t,t-1])
....
, Ki-,(X) ~
As above, this breaks up into sequences 0
, K,(A) ,
gi_,(A)
, K~(A[t])(9 K , ( A [ t - ' ] )
, K,(A[t,
t-l])
,0
which is the weak form of b). Finally, the case when A is noncommutative is dealt with using arguments similar to the proof of (5.39). M . L e v i n e ' s L o c a l i z a t i o n T h e o r e m . We now indicate how to modify the proof of (9.1) to yield a generalization due to Levine, and apply it to the study of modules of finite length and finite projective dimension over twodimensional normal local rings (see M. Levine, "Modules of finite length and K-groups of surface singularities", Compos. Math. 59 (1986) 21-40). Let X be a Noetherian scheme which supports an ample line bundle, and let Y c X be a closed subscheme of pure codimension d, such that for each 9 E Y, Ou,y has projective dimension _< d over Oy,x. Fix an affine open subscheme U C Y such that for some effective Cartier divisor Z on Y, Y - Z - U. Thus the ideal sheaf Z z of Z c Y is an invertible Ov-module. Let i 9Y , X, j 9U ~ Y be the inclusions. Let C = Y - U = Zred, We now describe some full additive subcategories of M ( X ) * , the category of quasi-coherent Ox-modules; it is easy to see t h a t they are closed under extensions in A4(X)*, and hence form exact categories" ~ ( Y ) =locally free Ov-modules of finite rank (regarded as a subcategory of M ( X ) * via i.) (for r > d) P ~ ( Y ) = coherent Ov-modules of homological dimension _< r over O x . p r (y, U) =full subcategory of ~" E P r (Y) such t h a t ~" has no associated points in C (i.e., ~ and j * ~ E :P(U).
, j.j*Y= is injective),
9. Localization for Singular Varieties
211
P ~ ( Y ) = full subcategory of Jr E 7~r (Y) such t h a t ~" is supported on C (i.e., j*~" - 0). Theorem ...
( 9 . 9 ) . There is a natural long exact sequence (for i > 0) ,
(u)
, IQ(-pd+~ c (Y)) ' ' Ki(7:)d( Y, U)) ~
Ki(U)
which is compatible with the localization sequence (9.1) for U C Y, i.e., the inclusions "P(Y) C 79d(Y, U), 7/c C p g + l ( y ) (where 7~c = coherent Oy-modules of homological dimension < 1 over Oy and supported on C)
induce a commutative diagram ----} K i + I ( U ) - - " }
II
Ki+I(U)--*
K~(7)~+'(Y))~
l
K,(TY6,)
Ki(7~d(y,u))--+
l
Ki(Y)
~
--,
K~(U)
II
Ki(U)
R e m a r k . Here, the naturality of the sequence means the following--let Y~ C X be a subscheme of pure codimension d with proj.dim.o~.x Ou,y, <_ d for all y E Y', such t h a t Y c Y~. Let C ~ C Y~ be a closed subset supporting an effective Cartier divisor, such that U ~ = Y ~ - C ' is affine, C c C ~, and U A U' is open and closed in U'. Let ii " Y , yt, jl " U N U t } U, h" U N U ~ , U' be the inclusions. Then we have exact functors 9 r'd(Y, U)
,
U')
i,. "/)~+'(Y)
,/)d+,(y,)
h, o j; . 79(U)
,79(U').
The naturality statement is that the induced diagram below commutes: --'}
Ki+l(U)
I
--~ Ki+l(V')
~
Ki(Pd+I(Y))
--,
K,(~~
I
---} Ki(pd(Y, U))
~
I
Ki(pd(y',u'))
~
Ki(U))
--'
Ki(U')
I
P r o o f o f (9.9) ( S k e t c h ) . The proof closely follows that of (9.1). Let V C P(U) be the full subcategory of sheaves of the form j ' M , for some M E 7~d(Y, U). Let T = Iso V, S = Iso 7)d(Y, U). In the following, we write 79d for 7)d(Y, U), and P g + ' for Vg +' (Y). Let E be the extension construction over Q]), so that E has objects given by admissible epimorphisms M "-" N in ]), where a morphism
212
(M
Algebraic K-Theory ;,N) ~
;; N') is an equivalence class of diagrams
(M'
M
;
N
Ii
T
M
,
I
N~
o
M'
i ,
N'
where the square marked [9 is a pullback. ( M ,9 ~ QV; let :D be the pullback D
~
pd
,
Qv
1
g
;; N) ~-~ N gives a functor
1
so that the objects of :D are pairs ( P , M ~ j ' P ) with P 6 pd, ( M --* j* P ) 6 E. Let C be the category whose objects are admissible epimorphisms (in 7)d+l) M ~ ( P (9 B) with M , P 6 7:'d, B 6 p d + l . An arrow ( M ' --~ P ' (9 S ' ) , (M ~ P (9 B) is an arrow ( P ' , B ' ) ~-- (P", B " ) ~ (P, B) in Q..pd • Q p d + l such that there is an induced equivalence class of diagrams L'
-~
L'
P'
(9 B '
P (9 B "
-;
I
I
L
~
P(gB
W e o b t a i n a diagram of categories and functors
C
I
Qc~§ where
D
p,
Q:pd
e
~,
Qv
q(M
f(M
,; N ) = N
;; P (g B) - ( P , j * M ---~j*P)
h(M ~
and let g = p o f :C
Qp(u)
P (9 B) = B,
~. Q7 )d, so t h a t g(M
, P (g B ) - P.
9. Localization for Singular Varieties
213
Let s,p be the natural projections. One verifies easily (as in (9.1)) that g, h, p, q are fibered functors. Clearly 3 c T is cofinal; hence 8 - IS ~ T - 1T, and S - 18 ~ T - 1E. As in (9.1), we see t h a t T - 1 E is contractible, and T-iT
)
1
(point)
)
T-1s
1
Q1)
is homotopy Cartesian, since all exact sequences in Q)) are split. Hence S - 1~ is contractible, and S-1S
1
(point)
)
8-1s
,
QV
1
is homotopy Cartesian. Next, we claim: (i) h" C ---, Q7 ~d+1 is a homotopy equivalence (c.f. Lemma (9.4)) (ii) if P e pd, then P Lemma (9.5))
I.C
) j . j * P is injective, and j . j * P = Un z z n P (c.f.
:D is a homotopy equivalence (c.f. Lemma (9.6)) 3-1l:)
~ Qpe
l
(point) -~ ,~- 1 g
1
,, ,
QV
is homotopy Cartesian (v) B Q P d+l ~ BQP d ) B Q V is homotopy equivalent to a fibration (this follows from (iv) and the homotopy equivalences S-1 :D ~ S - 1 C ~(vi) K , ( V ) ) K~('P(U)) = g , ( v ) is an isomorphism for i > 0 and a monomorphism for i - 0 (c.f. Lemma (9.3)). In fact, the proofs of the above statements closely follow those of the corresponding lemmas cited from the proof of Theorem (9.1). We leave the details to the reader as a (fairly straightforward) exercise. Clearly (v), (vi) give the desired long exact sequence. The compatibility of the above constructions with those in the proof of (9.1), gives the diagram comparing the sequence of (9.1) for U c Y with the above new sequence.
214
Algebraic K-Theory
Finally, naturality (see the remark after the s t a t e m e n t of the theorem) is proved as follows: let E', C ', /),, 1)', S ~ be the analogous categories associated to Y', U t. Then one checks t h a t i l, j l , h define a map from
, S-1I)
t.~-- IC
1
l
Qpd+t(y)
,
QPd(Y,U)
,
QP(U)
1
8-1g
,
QP
: ,5'-12) '
,
QT~d(Y',U')
'
QV'
to the diagram 8,-1C,
1
Qp~+l(y,)
1
s'-lg '
,
QT'(U')
l
This gives the desired map between the two localization sequences. We apply Theorem (9.9) to the following situation. Let X = Spec R, where R is the local ring of a normal point on a surface over an infinite field k, and let Y be a reduced effective Cartier divisor on X , so t h a t Y = Spec R / f R for some non-zero divisor f E R such t h a t R / f R is reduced. If g E R such t h a t (f, g) form a regular sequence, i.e., (f, g)R is p r i m a r y to the maximal ideal ~ of R, then the image 0 E R / f R of g generates the ideal of an effective Cartier divisor Z C Y, such t h a t Y - C - U = Spec ((R/fR)[~-I]) is affine. Then :P~(Y) is the category of R-modules M of finite length and finite projective dimension which are annihilated by f , and 7~1 (y, U) is the category of finite R-modules P of projective dimension 1 and annihilated by f; the condition t h a t P restricts to a vector bundle on U is automatically true, since U ~ Spec k(f), where k(f) is the total quotient ring of R / f R (if f R = 7)1 N . . . N Pn is a p r i m a r y decomposition, then k ( f ) = k(:Pl) x . . . x k(7~n), where k(7 ~) = residue field of P ) . T h u s we obtain an exact sequence
, Ki+l(k(f))
' K,(7)~(Y))
, Ki(7:'I(Y, U))
, Ki(U).
T h e set of reduced Cartier divisors in X form a directed set, and the naturality statement in (9.9) yields a directed family of long exact sequences. Since K, commutes with filtered direct limits of categories, we obtain an exact sequence in the direct limit ; ~ ht
T~=I
g,+l(k(T'))
, K,(CR)
, K,(T"(R))
, ~ ht
g,(k(T'))
T~=I
where :P runs over the primes of height 1 in R, Ca denotes the category of modules of finite length and finite projective dimension, and p l (R) denotes the category of finite R-modules M of projective dimension 1, such t h a t for some f e A n n n M , R / f R is reduced (to see t h a t lim T ~ ( Y ) = CR, we
9. Localization for Singular Varieties
215
need only observe that any M 6 Ca is annihilated by some f 6 R such that R / f R is reduced). In particular, we have an exact sequence
K~Cr'~(R))
, ~)
k(V)* o go(Ca).
ht 7='= 1
We wish to compute the map 0 explicitly. By the compatibility of (9.1) and (9.9), if f 6 R is such that R / f R is reduced, and f R = 791 el--. f] P , is the primary decomposition then we have a diagram
K, (pl(y, U))
n
~ ~ ) k(P~)* i=1
l ,
K1 (Y)
'
g0(v$(Y))
II
~[~ kCP,)*
Of ~
i=1
where Y = Spec R / f R , C = {A4}, and 7/c is the category of (R/fR)modules of finite length and finite projective dimension. Any element of @k(79i) * is of the form ffl/g2, where g,,g2 6 R such that f,g,g2 form a regular sequence, and gi *-" -if'i,6 R / f R . We claim that 0f(gi) = • g,)] e K0(T/c) (for some universal choice of sign). If gi 6 (R/fR)*, both sides are trivial. If gi is a non-unit, there is a local homomorphism
, R/.f n which is flat, since 9i is a nonzero divisor. Hence we have a diagram of localization sequences (by the naturality of the sequence of (9.1))
g l (Tlc)
~
g l ( R / f R)
~
~ Ko(Tlc) J
Kl(k)
,
K , (k[9,](~,))
!
l
,
,
go(k)
II
Z
where k(gi) is the quotient field of the polynomial ring kiwi], and r = 5=1 for some universal choice of sign, by (5.28). The map Ko(k) ; Ko(Tlc) is given by 1 ~-, [R/(f, gi)]This yields the following explicit formula for
O" ~
k(T')*
, go(Ca).
h t T'= 1
Given a finite set T ' l , . . . , 79, of height 1 primes, and ai 6 k(79i) * for all i, choose f e 79, N . . - n T ' r such that R / f R is reduced. Let 791,..., 79r,..., 79,
216
Algebraic K-Theory
be the set of height 1 primes containing f , and set c~ = 1 for r + 1 < i _~ n. Choose gl,g2 E R such that (f, glg2) form a regular sequence, and gig21 H ~ , 1 <_ i <_ n. If a E @k(~P)* is the element given by c~ E k(7)~), 1
)] -
[n/(f
, g2)].
This explicit description of 0 allows us to prove two statements, contained in the following two lemmas. L e m m a (9.10) (Hochster). 0 is surjective, i.e., Ko(Ca) is generated by the classes [R/(f,g)] for regular sequences (f, g) in R such that R / f R is reduced. P r o o f . (This proof is due to Mohan Kumar). If ( f , g ) is any regular sequence, then a "general" k-linear combination f ' = a f + bg, a,b E k will have the property that R / f ' R is reduced, since k is infinite. Since R / ( f , g) = R / ( f ' , g ) , we are reduced to proving t h a t Ko(Cn) is generated by R / ( f , g) for all regular sequences (f, g) in R. Let M E Cn, f , g E AnnRM a regular sequence in R. I f M is not cyclic (generated by one element, as an R-module), let x , y E M be part of a minimal set of generators, and let i 9 f R + g R ~ M be defined by i ( f ) - x, i(g) = y. Since f, g form a regular sequence and gx - f y - O, i is well defined. Consider the diagram obtained by pushout along i, o
0
,
,
(f,g)
,
1
M
n
,
l
( M / i ( f , g))
=
,
1
N
,
n/(f,g)
II
R/(f,g)
,
0
;
0
1
( M / i ( f , g)).
If M is minimally generated by n elements, M / i ( f , g) requires n - 2 generators, so that N requires at most n - 1 generators, also, [MI = [ N I [ R / ( I , g)] in Ko(Cn). Hence, by induction on the number of generators of M , we see that Ko(Cn) is generated by classes of cyclic modules R / I of finite length and finite projective dimension. If R / I ~_ On, choose a regular sequence f , g in I which is part of a minimal set of generators, so that we have an exact sequence
0
,M
, R/(f,g)
, R/I
,0
where M = I / ( f , g) requires n - 2 generators, if I requires n generators. Clearly M E Ca from the above sequence. We have an exact sequence 0
, TorR(M, k)
, TorR2(R/(f, g), k)
, T o r R ( n / I , k)
, ....
9. Localization for Singular Varieties
217
Since proj. d i m R M = 2, Tor2R(M,k) ~ 0; also T o r 2 R ( R / ( f , g ) , k ) "~ k. Hence TorR(M, k) ~ k, and the minimal resolution of M has the form 0
} R
~ R~n-1
: ~ R ~n-2
~M
~0.
The funetor E x t , ( - - , R) 9 (TR , CR is an exact involution which fixes the isomorphism class of any module R / ( f , g); since E x t , ( - - , R) = 0 on CR for i < 1, we see from the above resolution of M that M* = E x t , ( M , R) has a minimal resolution 0
~. R r
; R Cn-1
-;
R
~ M*
~0
i.e., M* ~- R / J E C.R where J requires only n - 1 generators. By induction on the number of generators, we may assume [M*] E Ko(CR) lies in the subgroup generated by all classes [R/(f, g)]. Since the involution is trivial on this subgroup, we deduce that [M] E KO(CR) lies in this subgroup. This proves (9.10). Next, each prime 7) C R of height 1 gives a discrete valuation on K, the quotient field of R, and hence defines a tame symbol T~, : K 2 ( K ) ---* k(7)) * (see Example (1.15)) defined by
T~,(a, b} = r where r : P ~
"(a))
, k(~)*, and v is the valuation. Let
T" K2(K)
, ( ~ k(V)* ht "P= 1
be the sum of the maps T~, (note that for any {a,b} E K 2 ( K ) , a, b E R~, for all but a finite number of primes P, so that T~{a, b} = 0 for all but a finite set of 7~). Lemma
(9.11). 0 o T " K 2 ( K )
Ko(Cn) is zero.
P r o o f . We must show that 0 o T{a, b} - 0 for every a, b E K*. If a, b E R such that R / a b R is reduced, then a, b form a regular sequence. We compute that for any prime 7~ C R of height 1, if Tp is the associated tame symbol,
TT:,(a, b} -
1 E k(P)* ~-1 E k(P)* a E k(P)*
if ab ~_ P if a E P, b r :P if a ~ P , b E P
(where a, b denote the images of a, b in appropriate residue fields). Hence
0 o T{a, b} - [R/(a, b)n] - [R/(a, b)S] - O. We reduce the general case to this case, by showing that K2(K) is generated by symbols {a, b} with a, b E R such that R / a b R is reduced. We use the following terminology: if ], g E K*, we say f, g are "relatively prime" if for any discrete valuation v associated to a height 1 prime 7~, then
218
Algebraic K-Theory
v ( f ) . v(g) = 0, i.e., at least one of f, g is a unit in Rp. If v ( f ) v ( g ) ~ O, we say t h a t 7~ is a common divisor of f, g. Clearly K2 (K) is generated by symbols {a, b} with a, b E R - {0}. Fix {a, b}, and let aR = 7~ rl) r)...N7 ~(r~) be its primary decomposition (7~(n) : R N 7:'nRp). Thus if vi is the discrete valuation associated to Pi, then vi(a) : ri. Let s~ = ~-~j
-- ( a , b } - { b , b } - 1 . ( a~ , l -
~)
-- {a, b} - { - 1 , b}-l - A, 1 -
= {a,b}. { - 1 , b ) - X { A , b - Aa}-{A,b}. Thus it suffices to prove that { ~, b - ,ka} is a product of symbols of the specified type. If aR = 7~ N QI, bR = :P N Q2, then choose c E Q2 such that R / c R is reduced, and a, c as well as b - ,ka, c are relatively prime (this can be done since P is the only common divisor of b, b - ,ka, so t h a t the primes occuring in the primary decomposition of Q2 do not divide b - Aa). If cR = Q2 cI Q3 then d = ~ E R, and acR = 7) N Q1 N Q2 N Q3, so t h a t dR = Q1 N Q3. Thus d, b - ,ka are relatively prime, R / d R is reduced, and
{~~, b - )~a} -- {d, b -
Aa}-{c, b - )~a} -1
is an expression of the desired form. L e m m a (9.12). Let S be a discrete valuation ring with quotient field K and residue field k. Then in the localization sequence K2(S) ~ K2(K) Os g l (k)
II k
~
9. Localization for Singular Varieties
219
the boundary map is given by as(a) = Tff 1 (for universal choices of the signs), where Ts is the tame symbol. (Here the group law in k* is written multiplicatively.) P r o o f . Let H E S be the local parameter. Using bilinearity of the Steinberg symbol and the identity { H , - H } = 1 in K2(K), we reduce to checking that Os{1-[, a} = a ~1 for any a E R*, where a ~-, ~ E k*, and the sign of the exponent is universal. We have a diagram of rings S
.-
T
K
T
Z[u, u-~, t]
, Z[u, u-~, t, t- ~1
where the vertical maps are given by u ~-* a, t ~-, II. Let AJ(S) be the abelian category of finite S-modules, ~vIl(S) the Serre subcategory of torsion modules, and A/I (K) the category of finite dimensional K-vector spaces, so that there is a natural equivalence of categories A/[ (S)/~/f 1(S) --, j~4(K). The map Os is the boundary map in the resulting localization sequence. Let r u - l , t]) be the abelian category of finite Z[u, u - l , t]-mod ules, j~41(Z[u, u-l,t]) the Serre subcategory of modules on which t acts nilpotently; then the quotient is naturally equivalent to A4(g[u,u-l,t, t - l ] ) , the abelian category of finite Z[u, u - l , t , r-1]-modules. Let
.M.(z[u,u-~,d)
c
.M(z[,,,,,-~,d)
be the full subcategory of modules M satisfying Tori(M, S) = 0 for i > 0; similarly define full subcategories
~.(z[~, u-',d) and
c
M'(Zl~,~-~,d)
.M. = (Z[u,,,-'t, t-'l) c .M(z[u, u-', t, t-'I).
Then we have diagrams (where arrows marked -~ are homotopy equivalences, by the resolution theorem) M'(s)
I
.M~.(Z[u,u-~,d )
M'(z[,~,,r',d)-,
---.
M(s)
,
l
-
,
M(K)
l
.M.(Z[u,u-~,tl)
--,
M.(Zlu, u-~,t,t-~])
M(z[,~,,r',d)
-.~
M(z[,~,~-',t,t-'I)
220
Algebraic K-Theory
; K2(S)
l
K2(Z[u,u-l,tl)
".
,
~
K2(K)
~
K2(Z[u,u-l,t,t-i])
,l
KI(s)
,
Kl(k)
,
Ki(Z[u,u-1])
,
---,
,
l
Kl(Z[t,u,u-1])
where r = {H,a}, r = ~. Hence it suffices to prove t h a t j3({t, u}) -- u +1. Now by Theorem (5.2), gu(Z[u, u -1, t]) -~ K2(Z[u, u - l ] )
g2(Z[u, u - l , t , t - 1 ] ) ~- g2(Z[u, u-1]) (9 gl(Z[u, u-1]), and the second isomorphism is obtained from the facts t h a t a is a split inclusion, and/~ a surjection. Now gl(Z[u, u-i]) ~ Z ~ Z/2Z, where a typical element has the form +u :i:n, n E Z. Further, by considering Z[v, v-l,t] ~ Z[u, u - i , t ] , V H U n, w e s e e t h a t fl{t, u} = h(u) e Z[u, u - l ] * satisfies h(u n) = h(u) n, for any n > 1; hence h(u) = u rn for some m E Z. We claim m = 4-1, which will prove the lemma. If not, let p be a prime divisor of m, and let ~ be a prime such t h a t g - 1 (mod p). T h e n the finite field Ft contains a primitive pth-root of unity r E F~; consider the homomorphism of rings Z[u, u-1] -~Fe given by u ~-~ ~. We have a diagram of localization sequences K2(F~[t])
K2(7/.[u,u-l,t])
~:;
-~.
K2 (Ft[t, t-i]) K2(Z[u,u-l,t,t-ll)
'
---.
K1 ( F t )
'
Kx(Z[u,u-1])
..
where # ( ( t , u } ) = (t,~}. Now K2(Ft[t]) ~ K2(Ft) = 0 (see E x a m p l e (1.19)). Also r off{t, u} = r m) = Cm = 1. Hence {t, ~} ~ K2(Ft[t, t - l l ) must vanish. But if T : K2(F~(t)) , F~ is the t a m e symbol associated to the valuation corresponding to the discrete valuation ring ]F~[t](t), then T({t,r = r e IF~ is non-trivial. This contradiction proves the lemma. R e m a r k . One can show t h a t a s = Ts by a more natural, but s o m e w h a t less elementary argument; see Grayson, "Localization for Flat Modules in Algebraic K-Theory," J. Alg. 61 (1979), 463-496.
9. Localization for Singular Varieties
221
T h e o r e m (9.13). There is a presentation K2(K)
T, ~
k(V)*
0 ' K0(CR)
.~O.
ht T'= 1
Equivalently, if U = Spec R - {2r is the punctured spectrum, then 0 gives an isomorphism Hi(U, IC2,u) ~= Ko(Cn). Since U is smooth over k, Gersten's conjecture holds for the local rings of U, and so (from (9.12)). H 1(U, K:2,v) ~ coker T. Hence the two formulations are equivalent. We already have one presentation
g l (7,1(R))
,1
~
k(7~), o_~ Ko(CR) : ; 0
ht "P=I
where 791(R) is the category of (torsion) R-modules of projective dimension 1 with reduced support. We have also verified that 0 o T = 0. So it suffices to prove that image 71 c image T. We have a localization sequence (dim U = 1)
K2(U)
~, K2(K)
,T, ~
k(7~)*
, K,(U)
, KI(K)
h t 7:,= 1
(where the boundary map can be identified with T by (9.12)). Hence it suffices to prove that the composite
Ki(T"(R))
, (~
k(79) *
, K,(U)
ht T'= 1
is 0. Now 791(R) C 7~(R), the category of all finite R-modules of finite projective dimension, and if 79(R) is the category of finite projective Rmodules, then 79(R) ,7~(R) is a homotopy equivalence by the resolution theorem. The map
K~(7='~(R))
, gl(v)
is clearly given by the restriction functor 791(R) factors through 7/(R); thus we have a triangle
K1(791(R))
, 2el(U), and hence
', Kl(U)
/ K,(R)
The composite KI('PI(R)) , KI(U) composite restriction functor 7~1 (R) ,
KI(K) is clearly 0, since the , 79(K) maps every object to a
222
Algebraic K-Theory
0-object. Since
gl (U)
~ g l (K)
",,,
/ K,(R)
commutes, and KI(R) ~- R* ~ K* ~- K I ( K ) , we see t h a t KI('Pl(R)) , K~(R) is 0, and hence KI('Pl(R)) , KI(U) is 0, as desired. This proves
(9.13).
Let f : Z ., Spec R be a resolution of singularities, E the reduced exceptional divisor. Then Z - E ~ U; hence we have a localization sequence
, K, (Z)
el(E)
,, KI (U)
, Go(E)
, Ko(Z).
Since R is the local ring of a point on an algebraic surface X / k , the local rings of Z are regular local rings which are essentially of finite t y p e over k, and hence satisfy Gersten's conjecture. Since dim Z - 2, if we set S K i ( Z ) = F1KI(Z), then
KI(Z) ~- r(Z, O*z)~ SK, (Z) ~- R* @SKI(Z), and we have an exact sequence
H2(Z, IC3,z)
, SKi(Z)
, HI(Z, IC2,z)
') O,
from the B GQ-spectral sequence. Here
~.S K i ( Z ) ) - F 2 g l (Z)
image(H2(Z, K:3,z)
k(x)*
= image( ~
, KI(Z)) C image(Gl(E)
, KI(Z)).
xEZ 2
Since all closed points of Z lie on E, this subgroup lies in the image of G1 (E) ~. KI(Z). Thus, if FoGo(E) C Go(E) is the subgroup generated by the classes of closed points, we have an exact sequence
0
; HI(Z, 1C2,z)/N
, SKi(U)
, FoGo(E)
, O,
where N = i m a g e ( e l ( E ) 9 , S K i ( Z ) -~ H I ( z , IC2,z)). More explicitly, H I ( z , IC2,z) is H I of the Gersten complex 0
,g2(g)
~~
k(x)* 0~ ~
xEZ t
Z
,0;
x6Z 2
then N is the subgroup generated by ker ((gk(E~)*
~ xEZ 2
where Ei runs over the components of the exceptional divisor E. Note t h a t S K i ( U ) ~- HI(U, 1C2,u) since d i m U = 1.
9. Localization for Singular Varieties
223
We now relate these remarks to the Chow group of 0-cycles. We give an ad hoc t r e a t m e n t for the case when the surface has one singular point; a more systematic approach can be found in Levine's paper (loc. cit.), and the sources cited there. Let X be a normal quasi-projective surface. The Chow group of 0cycles CH2(X) is defined by
CH2(X) =
Free abelian group on smooth point of X ((f)c{C C X a curve, C n Xsing = 0, f 6 k(C)*)"
Suppose Xsing = {P}, and let R Op, x. Let ~r" Y ; X be a resolution of singularities, l r - l ( P ) = E the reduced exceptional divisor. Then Z = Y x x Spec R is a resolution of singularities of Spec R. Let K = k(X) be the function field of X. Let (a), (f~), (7) denote the following complexes=
.
o , , o
,
k(C)"
0
P~.C
(/~)"
0
: K2(K)
T
(~
k(D)* 0~. ( ~ Z
DCY
(7)"
z
, o
QEX-{P}
CCX
0 ~ ~-K2(K) T___, ~ )
,0
QEY
k(D)" o
DCY DnE#O
~Z
~0
QEE
where in (a), C runs over the irreducible curves in X which do not pass through P , and in (fl), (7) D runs over appropriate sets of irreducible curves in Y (namely, all curves, in (f~), and curves meeting E, in (7))- The map T is the sum of tame symbol maps in (f/), (7), and cOis the divisor map on rational functions in all three complexes. We have an exact sequence of complexes 0
, (~)
, (~)
, (7)
,0.
Since Y, Z are regular k-schemes whose local rings satisfy Gersten's conjecture, (f~), (7) compute the cohomology groups Hi(Y, ]C2,y) and Hi(Z, K:2,z) respectively. We have a long exact cohomology sequence
nl(y, lC2,y) ---, HI(Z, IC2,z) ---, CH2(X) ~
CH2(y)--,O II
H2(y, IC2,y). If N C H 1(Z, K:2,z) is the subgroup defined earlier, then clearly N lies in
224
Algebraic K-Theory
the image of Hi(y, K:2,y), since we have a commutative diagram
o @z QEE
l
( ~ k(D)*
F
~
(~ Z
DCY
QEY
so t h a t the kernel of the top row maps to Hl((f~)) = Hi(Y, K:2,y) such t h a t the composite map to H l ( Z , ~ 2 , z ) has image N. Let N ' c HI(Y, IC2,y) be the subgroup defined by the kernel of the top row; t h e n we have an exact sequence
HI(y, IC2,y)/N ' -----~H I ( z , IC2,z)/N ~ , C'H2(X) Let
, CH2(y)
; O.
SKo(CR) = ker(K0(CR) J , FoKo(E)) ~- HI(Z, IC2,z)/N.
T h e n we can rewrite the above sequence as
H'(Y, IC2,y)/N'
, CH2(X)
, SK0(CR)
, CH2(Y) ----, 0
where SKo(CR) is an analytic invariant, i.e., depends only on the completion of R, and Hi(Y, K2,y)/N' depends on the given algebraic local ring. (9.14). Let Y be a smooth, quasi-projective surface over a field k, k-rational point, Y , Y the blow up at x, E C Y the exceptional divisor. Then HI(Y,K:2,?) ~ HI(Y, 1C2,y) (9 k*, where the summand k* is the image of the natural covariant map H~ ICI,E) ~ k* --~ Hi(Y, K:2,?).
Lemma x E Y
a
P r o o f . Let K = k(Y) be the function field of Y. Consider the complexes (a')"
0
, K2(K) CCY
(~')
.
0--~K2(K)
T
QEY
( ~ k(D). ___. ( ~ Z DCY
(7')
"
0
,0
,k(E)*
,0
QEY , ~Z
-0
QEE
where (~0Q E E Z is the group of 0-cycles of degree 0 on E "~ - - IP19 We can identify (7 t) with a subcomplex of (fY), giving an exact sequence of complexes o
,
(-/) - ~
(~')
,
(~') - - ~
o,
9. Localization for Singular Varieties
225
which yields an exact sequence (since H1((7')) ~ k* ~- H~ , H ~ (]:r, K:2, f. ~" , H~
0
IC2,y)
IC1,E))
k*
H 1(l7, K:2,?) ~', H 1(Y, K:2,y)
,0.
One checks immediately that the natural maps lr* : Hi(Y, IC2,y) Hi(IT',K:2,~.) give right inverses to the maps lr.. Hence we have a split exact sequence 0
, k*
, H l ( ] f , K]2,f,)
We note t h a t the map k*
*
Ht(Y,]C2,y)
, O.
, H i ( y , ]C2,f.) is induced by
k* C k(E)* C ~
k(D)*.
DC~"
This proves (9.14). For any smooth variety Y/k, there is a natural map (Pic Y ) | H I ( y , IC2,y), given as follows: HI(Y, IC2,y) is H 1 of the complex
0 ~
K2(k(Y))
, 1 ~ k(x)*
, ~
xEY l
Z
k*
, O.
xEY 2
Let Div Y be the group of divisors on Y. Then (~
k* ~- (Div Y)@z k* C ( ~ k(x)*, xEy t
xEY 1
and clearly 0((Div Y)| = 0. Hence we have an induced map (Div Y)| k* ~ H i ( Y , K:2,y). We claim this factors through the quotient map (Div Y) | k* --~ (Pic Y) | k*. To see this, consider the diagram
T xEY 1
k(Y)*.zk*
Gk*
(Div Y) |
k*
xEY 1
where 0y 9 k(Y)* , Div Y is the divisor map on rational functions. Since (coker 0y) = Pic Y, this proves the claim. C o r o l l a r y (9.15). Let Y be a smooth, projective surface over an algebraically closed field which is birational to a ruled surface. Then the natural map (Pic Y ) | k* : ~.Hi(Y, K:2,y) is onto.
226
Algebraic K-Theory
Proof.
K1 ( C ) |
This is true if Y = C x IP1, where C is a s m o o t h curve, since Ko(Y) ~. KI (Y) is onto, from (5.18), and since k* ~ g 1(C, IC2,c) ~ SK1 (C)
(Pic C) |
(as ~ ) k(x)* ~- ~ ) k*). But by (9.14), xEC 1 xEc l k*
coker((Pic Y) |
~ Hi(y,/C2,y))
is unchanged under the blow up of a point on a s m o o t h surface; hence it is a birational invariant. This proves the corollary. We now return to our situation of a normal surface singularity. Let X / k be a normal projective surface with a unique singular point P E X , and let ~ : Y ". X be a resolution of singularities, ~ r - l ( P ) - E the reduced exceptional divisor, R = Op, x , Z - Y x x Spec R. We assume below t h a t k is algebraically closed. (9.16). Under the above conditions, let Y be birationally ruled, and suppose that 7r* : C H 2 ( X ) ----, C H 2 ( y ) is an isomorphism. Then we have an exact sequence
Proposition
Cg(R) | where Cs
k*
, K0(CR)
". FoGo(E)
,0
is the ideal class group.
P r o o f . The exact sequence (constructed earlier)
HI(Y,/C2,y)/N'
~ C H 2 ( X ) ~*, C H 2 ( y )
* SK0(CR)
~0
and the surjection (9.15) Pic Y |
.'; H 1(Y, ]C2, Y )
k*
give a surjection Pic Y |
k*
;-"SKo(Ca).
This clearly factors as Pic Y |
k*
HI(y, Ic2,y)/N'
,
Pic Z |
k*
HI(Z, K2,z)/N SNo(CR)
(where P i c Z | k* ~ HI(z,]C2,z) is analogous to the left hand vertical arrow). One sees immediately t h a t if E 1 , . . . , E n are the irreducible components of E, then
9. Localization for Singular Varieties
227
maps to N c H* (Z, ]C2,z), so that we have a surjection
But (Pie Z ) / ( ~ - ~ l S i < n Z[Ei]) ~ P i c ( Z - E) ~- Cg(R). This proves (9.16). T h e o r e m ( 9 . 1 7 ) . Let G be a finite group acting as a group of linear transformations on a two-dimensional k-vector space. Let R = k[[x, y]]a be the ring of invariants for the induced action on k[[x, y]] via linear substitutions. Then Ko(CR) = Z. (In short, Ko(CR) = Z if R has a two-dimensional quotient singularity). P r o o L Let X - Spec(k[x,y] G) for the action of G on A 2 induced by the given two-dimensional linear representation, and let P 6 X be the image of the origin under the natural quotient map f 9A 2 ; X. T h e n X is a normal affine surface, and P 6 X is the unique singular point (since k[x, y]a is a two-dimensional normal graded ring). Let )~ D X be a projective surface with )(,ing = {P}, and let 7r" Y ~ )( be a resolution of singularities. We then have a diagram h V , W
91
1,
?
X" ~t
where V, W are smooth, projective rational surfaces, h is birational, W x.g X - A 2 (i.e., f - l ( X ) - A2). Thus W is a smooth, projective surface containing A 2 as an open set, and V is a resolution of singularities of Y x.g W. We have an induced diagram of maps of Chow groups of 0-cycles
z ~=
CH
2(v)
CH2(?)
~-
,
CH
2
(w)
~-
Z
CH2(2)
There are also natural transfer maps f . 9 C H 2 ( W ) ~ C H 2 ( ) f ) , g, 9 CH2(V) .~ CH2(17") such that f . o f*, g. o g* both equal multiplication by d = IGI = deg f = deg g. This is clear for g as V, Y are smooth, and is also true for f because f - l ( p ) = {0} consists of only the origin in A 2 C W; now to construct f . we use the isomorphism Free abelian group on points of W - {0} C H 2 ( W ) '~ ( ( f l ) v I V C W curve, 0 r C, and fl 6. k(C)*)"
228
Algebraic K-Theory
This isomorphism follows from the diagram below, whose rows are complexes, and columns are exact, 0
Z
0
or
0
--.
l
K2(k(W))
--.
(~
1
k(x)"
--.
xEW l
0--*
1
K2(k(W))--*
--,
0
-,
0
-,
0
xEW~-{0}
zEW 1
~
1
Z
xEW 2
1
(~ k(x)* xE(Spec S) 1
---~
I
Z
where S = Oo,w, and the facts t h a t H~(SpecS,]C2) = 0 for i = 1,2 (the middle row computes H'(W,K:2), and the b o t t o m row H ' ( S p e c S , K:2)). Thus the kernel of the degree map deg 9 C H 2 ( ) ~ ) ; Z is annihilated by d. But ker(CH2(.~) , Z) is divisible, since it is a quotient of a direct sum of jacobians of smooth curves. Hence C H 2 ( ) f ) ~ Z ~ C H 2 ( Y ) . Now the function field k ( X ) - k(x, y)G so t h a t k(x, y ) / k ( X ) is separable. Hence X is a rational surface (see Hartshorne, Algebraic Geometry, Ch. V, Remark (6.2.1)). Finally, Cg(R) is finite, since it has exponent d, and is finitely generated (being a quotient of Pic Y, where Y is a smooth rational surface). T h u s Cg(R) | k* = 0, since k is algebraically closed, and hence k* is divisible. Hence Ko(CR) ~- FoGo(E) ~ Z since E is a connected tree of rational curves. It remains to note t h a t the composite
Ko(CR)
,
FoG0(E)
II
Z
,
Fogo(?)
deg
Z
II
C H 2 ( ~ ")
is just the length map KO(CR) - - ~ Z, and FoGo(E) an isomorphism; hence K0(CR) -~ Z given by length.
FoKo(Y)
; Z is
(9.8). Let X be a normal quasi-projective surface with only quotient singularities, i.e., for each P e Xsing, (~P,X = R is as in (9.17). Let r 9Y ~ X be a resolution of singularities. Then or* 9 C H 2 ( X ) C H 2 ( y ) is an isomorphism.
Corollary
P r o o f . If 7r 9 X p ---, X is a resolution of the singularity at a given point P E Xsing (so that X p - 7r-l(P) ~ X - {P}, and 7 r - I ( P ) n (Xp)sing -- 0), then one shows t h a t there is an exact sequence SKo(CR) ~C H 2 ( X ) , CH2(Xp) ~ 0. (See Levine's paper, cited earlier, for details; see also
9. Localization for Singular Varieties
229
V. Srinivas, "Zero Cycles on a Singular Surface I," Grelle's J. 359 (1985), pg. 97, Prop. 4). This immediately implies the corollary, from (9.17). There has been further recent progress on the topic of localization for singular varieties. Marc Levine has generalized his results on surfaces, proved above, to arbitrary dimensional isolated Cohen-Macaulay singulaxities (see M. Levine, "Localization on Singular Varieties, Invent. Math. 91 (1988), 423-464). Using the results in this paper, Levine gives a construction of two modules over the local ring kIIX, Y,z, Wll/(XY - z w ) with negative "intersection multiplicity." This was originally proved by S. Dutta, M. Hochster and J. E. McLaughlin in their paper "Modules of Finite Projective Dimension with Negative Intersection Multiplicity," Invent. Math. 79 (1985), 253-291. Levine's results depend on a new (equivalent) definition of Ki given by H. Gillet and D. Grayson, and stronger forms of Theorems A and B proved by them, leading to a generalization of Theorem (9.1). R. Thomason and T. Trobaugh have recently obtained the "most general" localization theorem. Their results were announced in "Le Theoreme de Localization en K-Theorie Algebrique," C. R. Acad. Sci. Paris 307 (1988), 829-831. The detailed proofs are in their paper "Higher Algebraic K-Theory of Schemes and of Derived Categories," in The Grothendieck Festschrift Vol. III, Prog. in Math. Vol. 88 (1990), pp. 247-435.
Appendix A Results from Topology A general reference (cited below as [W]) is: G.W. Whitehead Elements of Homotopy Theory, Grad. Texts, No. 61, Springer-Verlag (1978). Our discussion of homotopy groups, etc. is based on: P.A. Griffiths, J.W. Morgan, Rational Homotopy Theory and Differential Forms, Prog. Math. 16, Birkh/iuser (1981).
( A . 1 ) C o m p a c t l y G e n e r a t e d S p a c e s . A topological space X is compactly generated if X is Hausdorff, and a subset A c X is closed if (and only if) A N K is closed, for every compact subset K C X. The category of compactly generated spaces is a convenient one in which to do algebraic topology (see [W], I, Chapter 4). We will assume that all spaces under consideration are compactly generated, and all categorical constructions of spaces (e.g., products, quotients) are made within this category, unless specified otherwise (however, we do not impose any such conditions on the underlying topological space of a scheme; this exception should cause no confusion). Clearly any locally compact Hausdorff space is compactly generated. Given any Hausdorff space X, let k ( X ) be the topological space with the same underlying set X, and the following topologyma subset A C X is closed in k ( X ) if and only if A N K is closed in K for every compact subset K C X. T h e identity map of sets gives a continuous map k(X) -----, X , and one verifies the following properties: (i) k ( X ) is compactly generated, and k ( X ) - X if and only if X is compactly generated; (ii) k ( X ) and X have the same compact subsets; more generally, if Y is compactly generated, then composition with the natural map k ( X ) ; X gives a bijection between the continuous maps Y ; k(X) and the continuous maps Y ~ X; (iii) as a particular case of (ii), k(X) ~ X induces isomorphisms on homotopy groups, and on the complexes of singular chairs, hence on homology and cohomology with arbitrary coefficients. The product in the category of compactly generated spaces is given by k ( X x Y), where X x Y is the usual product (i.e., has the standard product topology). However, if either X or Y is locally compact then
Appendix A: Results from Topology
231
X x Y is compactly generated; hence the notion of homotopy is unchanged within this category. Next, if X is compactly generated and Y is a Hausdorff quotient (with the quotient topology), then Y is compactly generated. If X is a Hausdorff space with an increasing sequence of closed subsets X n which are compactly generated, and if X - lim X n (i.e., a subset A c X is closed if and only if A N X n is closed for all n; we also say X has the weak topology with respect to the collection { X n } ) , then X is compactly generated. In particular, any CW-complex (see (A.8) below) is compactly generated. Any closed subset or open subset of a compactly generated space is compactly generated. If f 9X , Y is a function such that fl K is continuous for every compact set K C X, and if X is compactly generated, then clearly f is continuous. Let X, Y be compactly generated spaces, and let C ( X , Y ) denote the space of continuous maps X , Y with the compact-open topology (a basis of open sets for this topology is given by
U(K, V) = {f e C(X, Y) I f(K) C V}, for K c X compact, V C Y open). Let F(X,Y) = k(C(X,Y)). Then the evaluation map e 9 k ( X x F ( X , Y ) ) ~ Y is continuous. This is one pleasant consequence of forming the function space and product in this category. In the sequel, "space" will mean "compactly generated space" unless specified otherwise; similarly "X • Y" stands for the compactly generated product, etc. (A.2) H o m o t o p y G r o u p s . Let X be a space, with a base point x G X . The n t h h o m o t o p y group l r n ( X , x ) is defined to be the set of h o m o t o p y classes of maps f ' ( I n , OI n) ~. ( X , x ) (i.e., maps f " I " .~ X with f ( O I n) = x), where I n is the unit n-cube in R n, and OI n is its boundary, (for n = 0, I ~ is a point, and OI ~ = 0). In fact for n = 0, ~r0(X, x) is only a set with a distinguished point; the elements of lr0(X, x) are in bijection with path components of X, and the distinguished element corresponds to the path component of x. For n - 1, we obtain the fundamental group of homotopy classes of loops based at x E X. For n >_ 2, 7rn(X,x) is an Abelian group, with the group operation given by juxtaposition of n-cubes--if f, g are maps (In, OI n) ~ ( X , x ) , and s l , . . . , s n are coordinates on I n, then [f] + [g] = [hi in ~rn(X, x ) where h " ( I n, OI n) ; ( X , x) is the continuous m a p defined
by h ( s l , . . . , sn) =
f ( 2 s l , s 2 , . . . , sn) g ( 2 s l - 1, s2, . . . , sn)
if 0 _~ 81 _~ if 89_< sx _< 1.
232
Algebraic K-Theory
Pictorially, this may be described by
Ifl§ C o m m u t a t i v i t y follows immediately (for n >_ 2) from the sequence of diagrams (,-, denotes a homotopic map, and x denotes the constant m a p ) g
The fundamental
follows:
l t/.(I
group
01
~(X,x)
x
acts on the right
on ~.(X,
, (X,
let g" (I, 01) , (X,x) represent a class in ~rl (X,x). the subset ( I n x {0})U ((OI n) x I) of I "+1, by
h(s,O) = f ( s ) , h(s, t) = g(t),
x), n >
I,
and Define
a m a p h on
for (s,0) e I " x {0} for (s,t) e OI" x I.
Since (I n • {0})tJ ((OI") • I) is a deformation retract of I "+1, (as it consists of all faces of OI "+1 except I n • {1}), we can extend h to a m a p h : I "+1 , X. Let [f][a] e l r n ( X , x ) denote the class given by hli . • ( 1 " One verifies that this class is independent of the particular extension , and depends only on the homotopy classes [f], [g]; this gives the action of r l on r , . If n = 1, we can represent this by the picture:
Here, the arrows represent orientations of the intervals; the fourth side represents [f][gl. But the square gives a homotopy of this loop to the loop g - 1. f . g (where 9 is composition of loops). Hence the action of ~'1 on itself is by conjugation. We can generalize the above to the case of pairs. Let X be a topological space, A a subspace, x E .4 a base point. T h e n t h relative homotopy group (for n _> 1) ~r, (X, A; x) is defined to be the set of homotopy classes of maps f " (I n, OI n, 01I n) " ( X , A , x ) , where 01I n = (OI "-1 • I ) tJ (I n-1 • {0}) consists of all of the faces of the n-cube I n except one (thus, f 9I n , X is a m a p such that f ( O I " ) c A, and f(01 I " ) = {x}, and the homotopy classes are with respect to homotopies through maps I n , X of this type). In fact, lrl (X, A, x) is only a set with a distinguished point (but we m a y abuse terminology and call it the first relative homotopy group). If n > 2, we can again define a group law on r n ( X , A, x) by "juxtaposition" of maps, i.e., if
Appendix A" Results from Topology
233
f , g are maps (In, oIn, 01I n) ~ ( X , A , x ) we define IIl + [g] ~ =,,(X,A,~.) to be the class of h" (Ir',Oln,Olln) , ( X , A , x ) given by
h(s~,...,s,)
=
f(2sl,...,sn) g(2s~- 1,...,s,)
if0 < sl < 1/2 if 1/2 < s~ < 1.
One easily sees, by an argument analogous to t h a t used for absolute homotopy groups, t h a t l r n ( X , A , x ) is Abelian for n > 3. The fundamental group 7rl(A,x) acts on 7rn(X,A,x), for n > 1, as follows--let f " ( I n , o I n , 01I n) , ( X , A , x ) represent a class in 7rn(X,A, x), and let g 9 (I, 0I) , (A,x) represent a class in 7rl(A,x). We can define a map h" ((OlI n) x I ) O (I n x {0}) ,X by h [ l . x {0} = f ' and hl(o~t.)x I = gop2 (where P2 is the second projection). Now ((OxI n) x I) O (I n x {0}) is a deformation retract of I n+l, since it consists of all faces of I n+l except two adjacent ones; further, we can choose such a deformation retraction which maps the face I n- 1 x { 1} x I into itself at all times, so t h a t the final retraction maps this face into t h e "top" face I n- 1 x {1 } of I n (this face is precisely the one omitted in 011 n). Using this retraction, we can extend h to a map h 9 I n+l ~ X , which will then satisfy h(OI n x {1}) C A, h(OlI '~ x {1}) = {x}. Thus h[l,~x{1} represents a class in rrn(X,A,x), which we define to be [f]Ia], the result of [91 acting on Ill. We leave it to the reader to verify (or see [W]) t h a t this does give a well defined action. There are natural maps (homomorphisms, if n > 2) i 9 n n ( X , x ) ---* I r n ( X , A , x ) and 6 " r , ~ ( X , A , x ) ~ 7rn-l(A,x) (the latter is obtained by restriction to the top face I n-1 x {1} of In). Since we have a n a t u r a l hom o m o r p h i s m 7rx(A,x) ~ 7rl(Z, x), we have a hi(A, x)-action on , r n ( X , x ) for n > 1. One can show t h a t the maps i, 6 are 7rl(A)-equivariant. We use the following terminology--given sets SI, 5'2, 5'3 each of which has a distinguished base point, and maps f 9 $1 , S~, g 9 $2 ~ Sa preserving the base points, we say t h a t $1 ~ 5'2 , 5'3 is exact if g - l ( s 3 ) = f ( S l ) , where s3 E 5'3 is the base point. If the Si are groups and the base points are the respective identity elements, and f, g are homomorphisms, this agrees with standard algebraic terminology.
Theorem (A.3). triples (X, A, x), -, ~,,+,(X,A,~)
There is a long exact sequence, natural for maps of --, , . ( A , ~ )
--, , . ( X , ~ )
. . . ,r, ( A , ~ ) - , ,~, ( X , ~) --, ~ , ( X ,
-4 ,r,,(X,A,x)
A, ~)
-, ...
--, ~ o ( A , ~) --, , ~ o ( X , ~ ) .
The maps in the sequence are compatible with the 7rl(A,x)-actions.
234
Algebraic K-Theory
The proof of this result is left as an exercise to the reader (compare [W], IV Chapter 2). We define the Irl (A, x)-actions on n0(A, x) and ~ro(X,x) to be trivial. The sequence may be completed to the right --. Ir0(A, x) ~ lr0(X, x) --~ 7r0(X, A, x) - , 0 if we define lr0(X, A, x) to be the pointed set obtained by identifying all the points of lro(X,x) which correspond to the path components of X which meet A. Given a pair ( X , A ) and a map f . : (In,OIn, 01I n) } (X,A,x), representing a class in lrn(X, A, x), we have an induced map
f" Ha(I n, oIn; Z)
, H~ (X, A; Z).
Since Hn(I n, OIn; Z) -~ Z, and the homomorphism f . depends only on the homotopy class of f, we have a map ~rn(X,A,x) , H n ( X , A ; Z ) given by [f] ~-, f.(6n), where 6n e Hn(In, OIn;Z) is the standard generator (corresponding to the standard choice of orientation--see [W], IV, pg. 169). This is called the Hurewicz map, and is a homomorphism for n > 2 (for n > 1, if A - {x}). Since the base point plays no role in the definition of the homology groups Ha(X, A; Z), the nurewicz map kills the ~rl(A, x)-action (the ~rx(X,x)-action, if A = {x}), i.e., it factors through the coinvariants for this action (see [W], IV(4.9), (4.10)). The first computation of homotopy groups is given by the result of Brouwer, that homotopy classes of maps from an n-sphere into itself are classified by their degree, if n > 1. This can be restated as Theorem
(A.4). The Hurewicz maps
~rn(Sn, x) ~ lrn(I n, DI n) ~
g n ( S n , Z) (n > 1), g n ( I n, oIn; Z) (n > 2)
are isomorphisms of groups. (For the proofs, see [W], IV(4.5), (4.6)). Next, we state the Hurewicz theorems. Recall t h a t a space X is nconnected if lri(X, x) = 0 for i < n (in particular, 0-connected is the same as path connected). A pair (X, A) is called n-connected if ~r,(X, A , x ) = 0 for i < n. T h e o r e m (A.5). (i) For any O-connected space X , the Hurewicz map gives an exact sequence 1
' [~h (X, x), 7rl (X, x)]
} ~rl(X, x)
* Hi (X, Z)
; 1.
(ii) For any n-connected space X , n > 1, the Hurewicz map
lrn+ l (X, z)
} Hn+ I (X, Z)
Appendix A: Results from Topology
235
is an isomorphism. (iii) For any n-connected pair (X, A) such that A is 1-connected, and n > 1, the Hurewicz map Irn+ 1(X, A, x)
, Hn+ I (X, A; Z)
is an isomorphism.
(For the proofs, see [W], IV, Chapter 7). C o r o l l a r y ( A . 6 ) . Let f 9X ----, Y be a map between simply connected spaces which induces isomorphisms on integral homology groups. Then f induces isomorphisms on homotopy groups. P r o o f . We first replace f by an inclusion, using the mapping cylinder My. Regarding f as a map X x {1} ---, Y, we define M I to be the pushout
x•
1
XxI
f,
Y
--o
Mf.
1
Clearly, a deformation retraction of X x I onto X x {1 } induces one of Mf onto Y, and the composite X x {1} ; Y , Mf is homotopie to the inclusion X x {0} c M I. Hence, replacing Y by M f , we are reduced to the ease when f is an inclusion. Now the hypothesis on f and the exact homology sequence for ( X , Y ) give H n ( Y , X ; Z ) = 0 for all n. Hence, by induction, (A.5)(iii) gives ~rn(Y,X,x) = 0 for all n (and any base point x E X) (to start the induction, note that ~rl (Y, X , x ) - 0 since X, Y are simply connected). Hence, by (A.3), f induces isomorphisms on homotopy groups. (A.7) P r o d u c t s . If ( X , x ) , (II, y) are spaces with base points, and X V Y -(X x {y}) U ({x} x Y) C X x Y, the smash product X A Y is defined to be the quotient ( X x Y ) / ( X A Y). Let x A y E X A Y be the image of (x, y), which we take to be the base point. Given maps f : (In, cOIn) ~ (X,x); g : (I m, c9Im) ~ (Y, y) the product map f x g : I m+n : X x Y maps cOIm+n - ( 0 I n x ira) U (In x OI m) into X V Y. Hence the induced map to the smash product f A g : (Im+", c9Ire+n) : ( X A Y, x A y) represents a class in the (m + n)th-homotopy group of X A Y. One checks that this gives a product (which is bilinear, for m, n > 1) •
y)
,
^ Y,
^ v).
Thus, given a space ( X , x ) and a map (X A X , x A x) have an induced product •
,
:
(x,x)
we
236
Algebraic K-Theory
This p r o d u c t satisfies x . y = ( - 1 ) t o n y 9x for x E 7rn(X,x), y E 7rm(X,x) (this follows from the fact t h a t the map I m+n ~ I re+n, induced by the switch m a p I m • I n , I n x I m, induces multiplication by ( - 1 ) mn on H'~+'~(Im+n,oIm+n; Z), i.e., has degree ( - 1 ) ran, and from (A.4)). ( A . 8 ) G i r d - C o m p l e x e s . (See [W], II): Let A be a compactly generated topological space, X a topological space containing A as a closed subset. A CW-decomposition of the pair (X, A) is a nested sequence of closed subsets Xn c X , n > 0, such t h a t (i) A c X0, and X0 - A is a discrete closed subspace of X0. (ii) X = UXn and X has the weak topology with respect to {Xn} (i.e., B C X is closed ~ B M Xn is closed V n). (iii) Xn is an n-cellular extension of Xn-1 (for n > 0), i.e., for some index set Sn (possibly empty), which we regard as a discrete space, there is a continuous m a p fn : (An x Sn, OAn x Sn) -'--* (Xn, Xn-1) such t h a t the diagram below is a pushout: 0A~•
X~_I
---.
,
A~•
X~
(here An = { ( x 0 , . . . ,xn) e R n+l I xi > 0, ~-'~xi = 1} is the s t a n d a r d n-simplex, and OAn is the subset of points with at least one vanishing coordinate, i.e., 0 A , = A n M (xoxl...Xn = 0)). Since Xn is a Hausdorff quotient of Xn-1 ll(An x Sn), it follows by induction t h a t Xn is compactly generated for all n. Hence X = lim Xn is compactly generated. T h e image of An • {s}, for s e Sn, is called a (closed) n-cell of (X, A) (for the given CW-decomposition). A pair (X, A) consisting of a compactly generated space and a closed subspace is called a relative CW-complex if it has a CW-decomposition; X is a C W - c o m p l e x if (X, 0) is a relative C W - c o m p l e x . A closed subset A of a C W - c o m p l e x X is called a subcomplex of X if A is also a CW-complex, and X, A have CW-decomposition such t h a t every n-cell of A is also an n-cell of X. Then (X, A) is also a relative C W - c o m p l e x . A space X is said to have the homotopy type of a CW-complex if there is a C W - c o m p l e x Y together with a h o m o t o p y equivalence f : Y X. Two useful facts about spaces with the h o m o t o p y type of a CWcomplex are given by the following results, due to W h i t e h e a d and Milnor, respectively.
Appendix A: Results from Topology
237
T h e o r e m ( A . 9 ) (Whitehead theorem). Let l : X , Y be a continuous map between connected spaces each of which has the homotopy type of a CW-complex. Suppose f induces isomorphisms on homotopy groups. Then f is a homotopy equivalence. (For the proof, see [W], V(3.8)). C o r o l l a r y ( A . 1 0 ) . Let X , Y be simply connected spaces, each of which has the homotopy type of a CW-complex, and let f 9X , Y be a map which induces isomorphisms on integral homology. Then f is a homotopy equivalence. P r o o f . This follows at once from (A.6) and (A.9). If X, Y are spaces, let IF(X, Y) denote the function space of all (continuous) maps from X to Y, topologized as in (A.1). T h e o r e m ( A . 1 1 ) (Milnor). Let X be a compact space, Y a space with the homotopy type of a CW-complex. Then F(X, Y) has the homotopy type of a CW-complex. (This is proved in Milnor's article, J. Milnor, On spaces having the homotopy type of a CW-complex, Trans. A.M.S. 90 (1959), 272-280). We recall the standard method used to compute the homology of a CW-complex, using the complex of cellular chains. If (X, A) is a relative CW-complex, and 5'. denotes the set of n-cells, let C n ( X , A) be the free Abelian group on Sn; then H ~ ( X n , X n - 1 ; Z)
f Cn(X,A)
(0
ifi=n otherwise.
The homology sequence of the triple (X~+I, Xn, X n - l ) (where we define X_ 1 - A) gives an exact sequence ...
, Hi(Xn, X n - I ; Z ) 0
, Hi(X.+I,X.-I;Z)
; Hi(Xn+I,Xn;Z)
~.H i - l ( X n , X n - 1 ; Z )
' ....
In particular, taking i = n + 1, we have a map d,,+ 1" C , + , (X, A)
, (7, (X, A),
for each n > 0; one sees easily that dn o dn+l - 0, where we define do = 0 (this follows from the construction of the exact sequence of the triple from the three exact sequence of the pairs (Xn, X n - t ) C ( X n + l , X n - 1 ) C (Xn+l,Xn); in particular the map (9 above is defined to be the composite of the boundary map Hi(Xn+I,Xn;Z) and the natural map Hi_ 1(Xn, Z)
, H i - 1(Xn, Z) : H i - 1( X , , X=_ 1; Z)).
238
Algebraic K-Theory
(A.12). For any Abelian group G, the homology groups H~(X, A; G) of a relative CW-complex (X, A) are given by
Theorem
H~(X,A; G) = H,(C.(X,A) |
G,d. | 1),
the homology groups of the relative cellular chain complex with coeI~cients in G. P r o o f . We have
H~(X. Xn-x; G) = { Cr`(X,A) | '
G
0
if i = n otherwise
(this follows from the case G = Z by the universal coefficient theorem). Hence, by induction on p, and the long exact homology sequence of a triple, we obtain H i ( X , + p , Xr`; G) = 0 if i > n + p or i <_ n. Since any compact set in X lies in some Xm, the complex of singular chains with coefficients in G is the direct limit of the corresponding complexes for the Xm. Hence
Hr`(X, A; G) = lim Hn(Xm, A; G) for any n >_ 0. We deduce that the natural maps
Hn(X,A; G) gn(xn+,,Xr`-2; G)
, gr`(X, Xr`_2; G), , Hr`(X, Xr`_2; G)
are isomorphisms for all n >_ 0 (here X~ = A for i < 0). Next, from the exact sequence of the triple Xr`-2 C Xn-1 C Xn, we have an exact sequence
0
, Hn(Xr`,Xr`_2;G)
' , Hr`(Xr`,X,.,_l;G) o Hr`-x(Xn-l,Xr`-2;G)
where 0 -- dr, | 1 " Cr`(X, A) | G
, C,~_x(X, A) @ G. This computes
ker(dr` | 1) ~ Hn(Xn, Xn-2; G). Finally, from the exact sequence of the triple X n - 2 c X,~ c An+ 1, we have an exact sequence
Hn+x(Xn+x,Xn;G) o, Hr`(Xr`,X,_2;G)
, Hn(Xn+I,Xn_2;G)---,O.
But the composite
gn+l(Xr`+t,Xr`;G)
o , Hr,(X.,X._2;G)
~ ~ Hr`(Xn, Xn-x; G)
is just dn+ 1 | 1, by the naturality of the exact sequence of a triple, for t h e map of triples (Xn+ 1, Xn, Xr`_ 2) ~ " (Xn+ l, Xr`, Xr`_ 1). Thus image 0 i m a g e d n + l | 1.
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239
(A. 13) L o c a l C o e f f i c i e n t s . (See [W], VI). Let X be a topological space. A local system on X consists of the following data: (i) for each x E X we are given a group Gx (ii) for each path 7 from x to y, we have an isomorphism 7" : Gx ~ Gy, which depends only on the homotopy class of 7 (among paths joining x to y).
(iii) if ~/1, "/2 are paths joining x to y and y to z, respectively, and "/1, ~/2 is the composite path joining x to z, then (~1,"/2)" ---- (~2)* O (')'1)* " G x
, Gz.
These three conditions are equivalent to the condition t h a t x ~-. Gx is a group valued contravariant functor on the fundamental groupoid of X. Clearly, a local system on a path connected space X is determined by the following data: if x E X is a base point, it suffices to be given the group Gx, and an action of Irl(X,x) on Gx. For any space X, and any integer i >__1, x H 7ri(X, x) is a local system on X. We can define homology and cohomology groups H . ( X , A ; G ) , H*(X, A; G) for a pair (X, A) and a local system G on X. For our purposes, it suffices to do this under the hypothesis that each path component of X and of A is open, and has a universal covering space; if {Xa [ c~ E J} are the path components, we take
H,
H.(X, A; G) -
A n Xo; GIxo),
o~EJ
and
H'(x,A;c)
= 1-[
(~EJ
Hence it suffices to define homology and cohomology with local coefficients of pairs (X, A) where X is path connected and X, A have universal covering spaces. Let p" X - - . X be the universal covering space. If we fix a base point x E X, we have a natural free action of H = 7rl (X,x) on )(, with quotient X. If a 9 An ---4 X is a singular n-simplex, then X • x An , An is a covering space with group H, i.e., is a disjoint union of copies of An permuted freely by H. Thus, the complex S.()~) of singular chains is a complex of free Z[H]-modules, which in degree n has a basis in bijection with the set of n-simplices in X. If A c X, and A = p-l(A), the singular complex S.(A) is a subcomplex of free Z[H]-modules of S.(X), which in degree n has a basis in bijection with the set of n-simplices in A. Hence Z[II]-bases for 8~(,4), S,~(,~) can be chosen so that the basis for Sn(/]) is a subset of that for Sn()~). Thus, the complex of relative singular chains
240
Algebraic K-Theory
S,(.~, A) = S,(X)/S,(fi.) is a complex of free Z[II]-modules, such t h a t for each n, the short exact sequence
0
,s.(R)
,0
is a split exact sequence of Z[H]-modules. If G is a local coefficient system on X, let G~ ~ Go, which is a (right) H-module in a natural way; we can also regard Go as a left H-module (using the anti-automorphism g ~-. g-1). We define
Hn(X,A;G) = U~(Go Xztn] S . ( X , J])) H'~(X,A; G) = H " ( U o m z [ n ] ( S . ( ) ( , A), Go)). If Ao is a path component of A, let Ao - X x x Ao be the induced covering (possibly not connected) of Ao, and let fi-o denote the universal covering of A0. Let YI~ = 7rl(Ao,xo), and choose a path joining xo and x (in X ) , to give a homomorphism [It ; H. Then S.(Ao) is a complex of free W-modules. The universal covering Ao } Ao factors through (some path component of) fi-0. This gives a map S.(A0) } S(fi.0). One sees easily t h a t under this homomorphism, S.(fi.o) ~- Z[YI] | S.(fi.o). If G is a local coefficient system, corresponding to the H-module Go, then G[ Ao is the local coefficient system corresponding to the H'-module structure on Go induced by the map H' } II, and so Go |
S. (A0) ~ Go |
S. (/]0).
From this, one sees at once that the homology groups of Go | S.(A) are precisely the groups H.(A, G[A). A similar argument works for cohomology. Since the exact sequence of complexes of H-modules 0
, S.(A)
, S.(X)
, S.(X,A)
,0
consists of a split exact sequence of free H-modules in each degree, 0
} Go |
S.(A)
, Go |
S. (X)
, Go |
S.(R, A)
,0
is an exact sequence of complexes, which yields the long exact sequence of the pair (X,A) } Hi+ 1(X, A; G)
, H,(A, G)
H~_I(A, G)
, H~(X, C)
Hi(X, A; G)
;...
for any local coefficient system G on X (where Hi(A,G) denotes Hi(A, G[A)). A similar argument yields the exact cohomology sequence for a pair (X, A) with coefficients in G. Suitable reformulations of the Eilenberg-Steenrod axioms hold for homology and cohomology with local coefficients (see [W], VI, Chapter 2). In
Appendix A: Results from Topology
241
particular excision holds. Hence, we can compute homology with local coefficients using a CW-decomposition. Let (X, A) be a relative CW-complex such that X is path connected; let p" )( ; X be the universal covering, .~ = p-1 (A) the induced covering space of A. Then we have isomorphisms (from excision) for n > 0
Hi(Xn, Xn-1; G[x., ) -
Hn(f(n,)~'n-1; Z),
{ ~ Go | - O,
if i = n otherwise
(where Xn = A if n < 0). Here rl = lrl(X, x) for a base point x E A, and Go is the H-module associated to G. Thus, the complex of relative cellular chains C , ( ) ( , / ] ) (see (A.12)) is a complex of free Z[Hl-modules, and we can form the complex of relative cellular chains on (X, A) with coefficients in G, denoted C . ( X , A ; G), by
C , ( X , A ; G ) = H , ( X , , X n _ I ; G ) ~ Go |
Cn(X,A).
The proof of (A.12) goes through in this context to prove T h e o r e m (A. 14). We have natural isomorphisms
Hn(X,A;G) ~ H,~(C.(X,A;G)) H , (G0 |
C. (X, A) ).
for all n >_ O. An analogous result is valid for cohomology with local coefficients. Let C " ( X , A ; G ) = Homz[n](Cn(X,A),Go) ~- H"(Xn, Xn-I;G). As in the case of homology, we also have the vanishing result H ~(X~, X~_ 1; G) = 0 for i ~ n. However, the proof of
g n ( x , A; G) = lim g n ( x m , A; G) m
requires some care (see [W], VI, (4.3)). This yields, by the standard argument, the result T h e o r e m (A.15). We have natural isomorphisms
Hn(X, A; G) ~- Ha(C *(X, A; G)) ~- H"(Homz[n] (C. (X, A), a0))
for all n >_ O. (A.16) O b s t r u c t i o n T h e o r y . (Reference: P. Olum, Obstructions to extensions and homotopies, Ann. Math. 52 (1950), 1-50).
242
Algebraic K-Theory
Let (X, A) be a relative CW-complex, where A is non-empty, and fix a base point x0 E A. We will assume t h a t X, A are p a t h connected. Consider the following two extension problems: (i) Given a continuous map f " (A, xo) ' (Y, yo) for a p a t h connected space Y, when does f extend to a map F ' ( X , xo) ~ (Y, yo) (with FIa = f ) ? (ii) Given two continuous maps Fo, F1 " (X, xo) ' (Y, Yo) such t h a t FOlA -- F11A = f, when is F0 homotopic to F1 relative to A, i.e., when does there exist a map H " X • I .~ Y such t h a t H(x, O) - Fo(x), H(x, 1) = Fl(x), g ( x , t) = f ( x ) for (x, t) e A x I ? Obstruction theory gives a set of necessary conditions for the solutions of these problems. First, consider the question (i). If f : A , Y is given we can always find extensions of f to the 1-skeleton X1 of (X, A). Indeed, we can assign arbitrary values to the points of X0 - A, and then extend to X1 using the fact t h a t Y is path connected. If it is possible to extend f to the 2-skeleton X2, then since 7rl(X2,x0) ~- 7rl(X, x0), we would be able to find a homomorphism 0 9 7rx(X, Xo) , rl(Y, f(xo)) such t h a t
7rl (A, zo)
.l(X, zo)
commutes. It turns out t h a t this is the only obstruction to finding some extension of f to the 2-skeleton X2. Next, fix a homomorphism 0 9 ~rx(X, xo) , lrl(Y, f(xo)) as above, which is compatible with f . , and consider the problem of extending f to a m a p F 9 (X, xo) ~. (Y,f(xo)) such t h a t the h o m o m o r p h i s m on lrl induced by F is 9. Now lrl(Y,f(xo)) acts on 7ri(Y,f(xo)), i > 1, giving a local system lri(Y) on Y with local group ~q(Y, f(xo)) at f(xo). Given the homomorphism 8, we can form the pullback local system O*lri(Y) on X with local group ~r,(Y, f(xo)) at x0, and ~rl(X, x0)-action induced by 0. Suppose that one can find an extension fn " Xn , Y of f to the n-skeleton Xn of (X, A), for some n > 2, such t h a t the induced homomorphism on lrl is 6. Then one can define an obstruction cocycle, giving an
obstruction class O"+l(O,f,.,) ~_ H"+X(X,A;O*1r.(Y)) in the cohomology group with coefficients in the local system O*Trn(Y), whose vanishing is a necessary and sufficient condition t h a t fn[x._ 1 extends to a m a p fn+l " Xn+l --* Y where Xn+l is the (n + 1)-skeleton (i.e.,
Appendix A: Results from Topology
243
on+l(O, fn) = 0 r after possibly modifying fn on X n - X n - l , we can extend fn to fn+l defined on Xn+l). More generally, consider all possible extensions fn of f to the n-skeleton which induce 0 on fundamental groups, and let the obstruction set on+l(O,f) C Hn+I(X,A;O*zrn(Y)) be the subset consisting of all the obstruction classes. Then for any n > 2, f extends to Xn+l (compatibly with 0) r 0 E 0 "+1 (0, f) r 0n+2(0, f ) is non-empty. This gives a set of necessary conditions for the solution of (i). In the special case when Hn(X, A; O*Trn_l(Y)) = 0 for n > 3, f extends to X. The question (ii) can be regarded as a special case of (i), with X replaced by X • I, and A replaced by (X • {0, 1} U A x I) C X x I. By van K a m p e n ' s theorem, 7rl(X x {0, 1} U A x I) is the amalgamated free produ(~t 7rl(X) x~1(A ) ~rl(X), and the map lrl (X x {0, 1 } U A x I)
,7rl (X x I) ~ 1rl (X)
is the natural quotient. L e t f 9X x { 0 , 1 } t J A x I , Y be given by ][xx(0} = F0, ] [ x x { l ) = F1, ](x,t) = f(x) for (x,t) e A x I. T h e n /70 is homotopic to F1 relative to A if and only if f extends to a m a p X x I ; Y. Clearly the condition t h a t f extends to the 2-skeleton of (X x I, X x {0,1} U A x I) is just t h a t (F0). = (F1). 9 7q(X, xo) ; ~rl(Y, f(xo)), from the above calculation of fundamental groups. Hence, we need only consider the following special case of (ii). Let f " (A, xo) , (Y,f(xo)) be given, and let 0 " Trl(X, xo) ~ 7rl(Y,f(xo)) be a homomorphism compatible with f. We consider maps F0, F1 " X ; Y extending f and inducing 0 on lrl, and ask for conditions t h a t F0 be homotopic to F1. Then for each choice of a homotopy between Folx.
and F l l x . , where n > 1, which we regard as an extension of ] to the (n + 1)-skeleton of (X x I, X x {0, 1} U A x I), there is an obstruction class in Hn+2(X • I, X • {0, 1} W A • I; 0*Trn+l(Y)), where/~" 7rl(X • I, (xo,0)) ---* zrx(Y,f(xo)) is induced by 0 and the natural isomorphism 7rl(X, x0) ~ 7rl (X, x I, (x0, 0)) (given by g x {0} c g x I). There is a suspension isomorphism
H"+I(X,A;O*1rn+I(Y)) ~- H n + 2 ( X • I , X • {0,1} tO A • I; O*Irn+l (Y)), so t h a t we can regard the obstruction class as an element of the former group. Thus, for any n > 1, we have an obstruction set on+I(O;Fo, F1) C Hn+I(X,A; O*rrn+l(Y)), which is non-empty ~ Folx.. is homotopic (rel-
F0lx.+,
FIIx.+,
ative to A) to F~Jx., and 0 e On+l(e;Fo, F1) ~ and are homotopic relative to A. If Hn(X, A; O*Trn(Y)) = 0 for n > 2, then F0, F1 are homotopic relative to A.
244
Algebraic K-Theory
( A . 1 7 ) F i b r a t i o n s . A surjective map p" E , B is called a fibration if it has the homotopy lifting property, i.e., for any d i a g r a m Z x {0}
ZxI
It
[-I
(where f " Z : Eisamap, H- ZxI Z , B), there exists a m a p H " Z x I H[z
_
E
;
B
.~ B a h o m o t o p y of p o f 9 E such t h a t p o H = H , with
4r
~i~
p is a fibration, then for any d i a g r a m Z x {0}
ZxI
n
f,
ft
E
~ B
where _0 = ( 0 , . . . , 0) E I n, there is a m a p H 9Z x I '~
, E with I t l z x {~} =
f , such t h a t p o H = / - t . This implies t h a t if x E B , Fx = p - l ( x ) , y E Fx, then for any n >_ 1 the natural maps 7rn(E, Fx; y) , ~rn(B,x) are bijeetive (see [W], IV, (8.15)). T h e long exact h o m o t o p y sequence for the pair ( E , F=) yields:
Theorem
yeF~.
(A.18).
Let p : E ~
B be a fibration, x E B , Fx - p - l ( x ) ,
Then there is a long exact homotopy sequence ~r,.,+l(B,x)
~
~r,.,(F=,y)
--4 l r , . , ( E , y )
~
7r,.,(B,x)
---. I r , . , _ l ( F = , y )
--4 . . .
. . . ~ lro(F=, y) ~ 7ro(E, y). If p : E ~ B, f f : E ~ ~ B ~ are two fibrations such that there is a fiber preserving map E ~ E', then the sequence f o r p m a p s to that f o r p'.
Next, we remark t h a t if p : E , B is a fibration, f : B ' , B any map, E ' - B ~ x B E, p' : E ' , B' the n a t u r a l map, t h e n p' is a fibration, since the h o m o t o p y lifting property for p immediately yields the h o m o t o p y lifting p r o p e r t y for p'. T h e h o m o t o p y lifting property implies the following more general lifting property: L e m m a ( A . 1 9 ) . Let Y be a space, A a closed subspace such that (Y, A) is a D R - p a i r (i.e., A is a deformation retract of Y, and there is a map
Appendix A: Results from Topology u :Y
245
, I w i t h u - 1 ( 0 ) = A). Let p : E
a given map and g " A there exists a lifting h : Y
, B be a fibration, f : Y ". B , E a lifting of f l A (i.e., p o g = f i t ) " T h e n ~ E of f which extends g (i.e., p o h = f a n d
hit =g). P r o o f . Let (I) : Y x I ~ Y be a d e f o r m a t i o n r e t r a c t i o n of Y onto A, i.e., (I)(Y x {0}) = A, ~ ( y , t ) = y V(y,t) ~_ A x I, (I)(y, 1) = y Vy E Y. L e t :Y x I , Y be defined by
ql(Y, t)
f r
t/u(y))
! ~)(y, 1)
if t < u(y) if t > u(y).
O n e easily checks t h a t ~ is continuous. T h e n f o ~ : Y x 1 , B and k : Y , E , given by k(y) = g(~o(y,O)), are the d a t a for a h o m o t o p y lifting p r o b l e m , since p o k = f o ~ l r ' x (0)" If H 9Y x I , E is a lifting of f o ~ w h i c h e x t e n d s t h e lifting k of f o ~ l r x (0), t h e n h 9Y ---. E given by h(y) = H ( y , u(y)) is t h e desired lifting of f e x t e n d i n g g. We can use this l e m m a to prove a c e r t a i n local t r i v i a l i t y p r o p e r t y of a fibration. If p : E , B, p~ : E ~ , B are fibrations w i t h t h e s a m e base B, we say t h a t p a n d p~ are fiber homotopy equivalent if t h e r e exist maps f : E , E ~, g : E t , E and homotopies H : Ex I ; E, H': E' x I , E ' such t h a t (i) p o H ( x , t) = p(x) for all (x, t) E E x I , and H ( x , O) = x, H ( x , 1) - g o f ( x ) V x E E; a n d (ii) p' o H ' ( x ' , t ' ) = p'(x') V (x',t') E E' x I, g ' ( x ' , O) = x', H ' ( x , 1) = f o g(x ~) V x ~ ~_ E ~. In p a r t i c u l a r , f, g are h o m o t o p y equivalences; we call such a m a p f : E ~ E ~ a fiber h o m o t o p y equivalence from E to E ~ ( a n d similarly for g : E r , E ) . We also say t h a t E and E ~ have t h e s a m e fiber h o m o t o p y type. ( A . 2 0 ) . Let p : E , B be a fibration over a contractible base B. Let bo E B , Fo = f - 1 (bo ), and let p' : Fo x B ~ B be the projection. Then there is a fiber homo topy equivalence f : E ~ Fo x B such that flFo " Fo , F0 x {b} is homotopic to the identity.
Theorem
Proof. Let h : B x I , B be a d e f o r m a t i o n r e t r a c t i o n from B t o {b0}, so t h a t h(b,O) = bo, h ( b , l ) = b, h(bo, t) = bo V b E B , t E I. Let f0 " F0 x B ~ E be the c o m p o s i t e of p r o j e c t i o n F0 x B ~ Fo w i t h t h e inclusion Fo C E , so t h a t p o f0 : F0 x B * {b0}, a n d let ho " F0 x B x I , B be t h e c o m p o s i t e of p r o j e c t i o n Fo x B x I , B x I with h:B x I ~. B. T h u s ho(x, b, t) = h(b, t), V x E Fo, b E B, t E I, a n d fo, ho are d a t a for a h o m o t o p y lifting problem. Let Ho : Fo x B x I )E be a lifting of ho, i.e., p o Ho = ho, such t h a t HolF(,xBx{O} = fo.
Let
g : Fo x B ) E be given by g(x, b) = Ho(x, b, 1), so t h a t p o g = p~, i.e., g is a fiber p r e s e r v i n g m a p E ~ ; E, where E ~ = Fo x B.
246
Algebraic K-Theory
Similarly the identity map 1E : E , E and the m a p h l : E • I ~ B, given by the composition hi = h o (p x 1i) (where p x 1i : E x I ,B x I is the obvious map), are d a t a for a h o m o t o p y lifting problem. Hence there is a m a p H I " E x I , E with p o HI - hi, and HllEx{1) = 1E. Let f : S --. E ' ( = F0 x B) be defined by f ( x ) - (Hi (x, 0), p(x)) Vx E E; t h e n p' o f = p, so t h a t f is fiber preserving. We claim t h a t f, g give a fiber h o m o t o p y equivalence between E and E ~. Indeed, let
h2"FoxBxlxI re'F0
x
B
,B
x
{1}
x
I U Fo
x
B
x I
x
{0,1}
~E
be defined by h2(x, b, s, t) = ho(x, b, s) = h(b, s), and
m(x,b,s,t) =
Ho(x,b,s) g o ( x , b , 1) gl(g(x,b),s)
if t = 0, s e I if s = 1, t e I if t - 1, s e I.
To see t h a t m is well defined and continuous, we need only verify t h a t m ( x , b , l , 1 ) = Ho(x,b, 1) = g l ( g ( x , b), l); but H i ( x , 1) = x Vx e E, and g(x, b) = Ho(x, b, 1) by definition. We claim t h a t there is a diagram Fo x B x {1} x l U Fo x B x l x {O, 1}
m
FoxBxIxI
~E
~. B
i.e., for ( x , b , s , t ) E F0 x B x {1} x I U F o x B x I x {0, 1}, we must verify t h a t h 2 ( x , b , s , t ) - p o m ( x , b , s , t ) . Now h 2 ( x , b , s , t ) = h(b,s). On the other hand, p o re(x, b, s, O) - p o Ho(x, b, s) - ho(x, b, s) - h(b, s), and a similar calculation works for points (x, b, 1, t); finally p o re(x, b, s, 1) --p o H l ( g ( x , b), s) - hi (g(x, b), s) = h(p o g(x, b), s) - h(b, s). Hence t h e above d i a g r a m commutes. Since (I x I, {1} x I U I x (0, 1}) is a D R - p a i r , so is ( E ' x I • E ' x { 1 } x I U E ' x I x {0, 1}). Hence by (A.19), b2 can be lifted t o / / 2 9 Fo • B • I x I , E extending m. L e t / / 2 " F o x B x I * Fo • B be given by H2(x, b, t) = (/~2 (x, b, 0, t), b). T h e n H2(x, b, O) = (H2(x, b, O, 0), b) = ( m ( x , b, O, 0), b) = (Ho(x, b, 0), b) = (fo(x, b), b) = (x, b) for (x, b) e Fo x B, and
H2(x,b, 1) = (H2(x,b,O, 1),b) = (m(x,b,O, 1),b) = (Hi (g(x, b), 0), b) -- f o g(x, b). Finally p' o H2(x, b, t) - b V(x, b, t) E F0 x B x I.
Appendix A: Results from Topology
247
An analogous a r g u m e n t works for the other composite g o f . We define
h3"E•215
,B
n " E x {0} x I U E
x I x {0,1} - - , E
by h3(x, s, t) = hi (x, s) = h(p(x), s),
HI(x,s) Hi (x, 0) Ho(f(x),s)
n(x, s, t) =
if t = 0, s E I if s = 0, t E I ift=l, sEI.
T h e n n is well defined, since
n(x, O, 1) = Ho(f(x),O) = H o ( U l ( x , O ) , p ( x ) , O) = f o ( H l ( x , O ) , p ( x ) ) = Hl(x,O). Further we have a c o m m u t a t i v e diagram Ex{O}xlUExIx{O,
1}
'~
t
ExIxI
; E
t"
ha
.... ~. B
(as before, we immediately reduce to checking p o H o ( f ( x ) , = h(p(x), s); but p o H o ( f ( x ) , s ) = h o ( f ( x ) , s ) = h o ( H l ( x , O ) , p ( x ) , s ) = h(p(x),s) by definition of h0). As before, a lifting H3 " E • I x I .~ E extending n determines a fiber-preserving homotopy between the identity map of E and g o f , by restricting Ha to E • { 1 } x I. This completes the proof of (A.20). C o r o l l a r y ( A . 2 1 ) . If p " E , B is a fibration over a path connected base B, then any two fibers of p are homotopy equivalent. P r o o f . Ifb0, bl E B, let 7 " I .~ B be a path joining b0 to bl. T h e n 7*E = I x B E .~ I is fiber homotopy equivalent to the p r o d u c t fibrations p-l(bo) x I ---~ I, p - l ( b l ) x I , I. Hence these two fibrations are fiber homotopy equivalent, and in particular p-l(bo) is h o m o t o p y equivalent to p-l(bl). C o r o l l a r y ( A . 2 2 ) . Let p" E , B be a fibration, 7" I ~ B a path with 7(0) = b0, 7(1) = bl. Let F~ = p-l(b~), i = O, 1. Then there are canonical isomorphisms
7"" Hn(Fo, G) ----. H,,(F1, G),
7 . " g " ( F ~ , G)
; H"(Fo, F)
for any coej~cient group G and n > O. Further, 7", 7. depend only on the homotopy class of 7 (among paths joining bo to bl), and are compatible with composition of paths. ((7 o 6)* = 6" o 7", (7 o 6). - 7. o 6.). In particular, b H Hn (p- l (b), G), b ~-, H n (p- l (b), G) give local systems on B.
248
Algebraic K-Theory
P r o o f . Let 7 * E - I • B E - I be the pull-back fibration. T h e n by (A.20) there is a fiber homotopy equivalence f " F0 • I * 7 * E , where f is a h o m o t o p y equivalence on each fiber, such t h a t fifo• " F0 * F0 is homotopic to the identity map of F0. T h e restriction of f to F0 • {1} is a m a p f 9 F0 " F1 which is a h o m o t o p y equivalence. Let 7", 7. be t h e m a p s on homology and cohomology induced by f . To check t h a t t h e y d e p e n d only on 7, it suffices to show t h a t if f l , f2 are two fiber h o m o t o p y equivalences F0 • I * 7 * E as above, then f l , f2 "F0 ." F1 are homotopic. Let gl, g2 " 7* E ; F0 • I be the "inverse" m a p s to f l , f2 respectively, so t h a t the gi are fiber preserving, and the composites fi o gi, gi o fi are homotopic to the respective identity maps t h r o u g h fiber preserving homotopies. T h e n we have maps g2 o f l , gl o f2 " F0 • I , F0 • I such t h a t the two composites g2 o fl o gl o f2 and gl o f2 o gl o f l are each homotopic to the identity by fiber preserving homotopies, and such t h a t the m a p s F0 • {0} , F0 • {0} induced by restricting g2 o f l and gl o f2 are each homotopic to the identity. If we prove g2 o f l , gl o f2 are each homotopic to the identity by fiber preserving homotopies, then we have fiber preserving homotopies f l ~ (f2 o g2) o f l : f2 o (g2 o f l ) "" f2, and ~
o (12 o g 2 ) =
o/2)
o g: ~ g2.
Hence it suffices to remark t h a t if f : F x I ; F • I is a self fiber-homotopy equivalence of the p r o d u c t fibration pl : F • I , F, such t h a t fo = f]F• : F ; F is homotopic to the identity, t h e n f is homotopic to the identity, by a fiber preserving homotopy. Let H 9 F > I > I ~ F • I be defined by H ( x , s , t ) = (Pl o f ( x , st),s); t h e n H ( x , s, 1) - (pl o f (x, s), s) = f (x, s) since f is fiber preserving, while H ( x , s , O ) - (pl o f ( x , O ) , s ) = (fo • l t ) ( x , s ) , and clearly H is a fiber preserving homotopy. But since f0 : F , F is homotopic to the identity, f0 • 1i : F • I , F • I is homotopic to the identity through a fiber preserving homotopy. This completes the proof t h a t 7-, 7* are well defined. If "Yl, ~/2 are homotopic paths in B joining b0 to bl, and H 9I • I ;B is a h o m o t o p y from "yl to "Y2 preserving end points, then H * E = (I • I) • B E , I > I is also homotopy equivalent to a product, so t h a t we obtain fiber h o m o t o p y equivalences of F0 > I with 7~ E and -y~E such t h a t the induced m a p s F0 ; F1 are homotopic. Hence (71)* ---- (72)*, (~'1)* -- (72)*- We leave to the reader the verification t h a t 7 ~-' 7-, 7 ~-* 7* are compatible w i t h composition of paths (the latter assignment is, of course, contravariant).
Appendix A: Results from Topology
249
(A.23) The L e r a y - S e r r e S p e c t r a l S e q u e n c e . Let p " E , B be a fibration, where B is a connected CW-complex. If 7r 9 B , B is the universal covering,/5" E , B the pullback fibration, then by (A.22), for any two points bl, b2 o f / 3 there are canonical isomorphisms Hi(F1, G) ~Hi(F2, G) where Fj =/5-1(bj), j = 1,2, for any i >_ 0 and any coefficient group G. Let Bp be the p-skeleton of B for a fixed CW-decomposition, /3p = 7r-1 (Bp) the p-skeleton of/3 for the induced CW-decomposition, and let E v = p - 1(Bp), Ep = ~-1 (~p). If Sp denotes the set of p-cells of Bp, the set of p-cells of Bp can be identified with S v x 17, where H = 7rl(B, bo) for some fixed base point b0 e B, such that the natural action of 17 permuting the/~cells of B v becomes the action of II on S t, • 17 induced by the trivial action of II on Sp and the left regular representation of H on itself. We have a pull-back fibration pl " (Ap • Sp • I-I) •
E
; (A v x Sp x H)
induced by the natural map Ap • Sp x II , Bp associated with the C W decomposition of Bp. Fix a base point _0 ~ Ap. Then by (A.20) Pl is fiber homotopy equivalent to
II
pr'(o.s.~) • a.
( s , g ) E S p x Il
'
H
a..
( s,g)e S1, • II
since Ap is contractible. Since
(((a~ • s . • n) • ~ ~) H ~ - 1 . ((0a. • s~ • n) • ~ ~) H ~ - . ) . (~.~._.) is a relative homeomorphism of NDR-pairs (see (A.42); f " ( X , A ) , (Y, B) is a relative homeomorphism if f is onto, Y has the quotient topology from f, and f 9X - A ; Y - B is a homeomorphism), the excision theorem gives isomorphisms for each n > 0
Hn(F-,p,/~p-,; G) ~ H,,,((Ap x Sp x H) x h "E, (OAr, x Sp x II) • z} E'; G)
@
Hn(p'[l(O,s,g) x Ap, pll(O_,s,g) x cOAp;a)
@
H , _ p ( p l ' ( 0 , s, g), G) (Kunneth theorem)
(s,g)ESvxH ,..,,./
(s,g)ESp x l]
~-
Hn-r(Fo, G) '~ Hn-p(Fo, G) |
Cp(B)
(s,g)ES,, x rl
where F0 = p-l(b0) p-l(b0) for some fixed base point b0 E/5-1(b0), and we have used the canonical isomorphism =
H.-pCPi- ~(0_,s, g), G) u H._,,(Fo, G)
250
Algebraic K-Theory
given by (A.22). Here Cp(/}) is the group of cellular p-chains on /~ (see
(A.12)).
Now II acts on (/)p,/~p-l), hence on Cp(/~) = Hp(/~p,/)p_l;Z) and on Hp(/)p,/~p-1; G). If we let II act on itself through the left regular representation and act trivially on Ap • Sp, the induced action on Ap • Sp • H makes
a H-equivariant relative homeomorphism. Also, H acts o n (Ep,Ep_l) so that (/~p,/~p-1) * (/}p,/)p-1) is II-equivariant. Hence H acts on Hn(Ep, Ep_ 1; G). Under the isomorphism constructed above, we claim the induced H-action on Hn-p(Fo, G ) | Cp([~) is the diagonal H-action on the tensor product corresponding to the natural action on Cp(B) described above, and the H-action on Hn-p(Fo, G) corresponding to the local system b H Hn(p-l(b), G) on B (from (A.22)). To verify the claim, we observe that if g e H, b e / ~ , b = ~r(b) e B, then/5-1(b) =/5-1(g 9b) = p-l(b), but the two homotopy equivalences of Fb = p-l(b) with F0 = p-l(b0) = i5-1 (b0) (determined by paths joining b, gb respectively to b0) differ by a self homotopy equivalence of F0 which induces the action of g E 7r on H.(Fo, G) given by (A.22). In fact the union of the images in B of the paths joining b, gb to b0 give a loop at b0 representing g E H = 7rl (B, b0). Thus for x e Hn_p(Fo, G), y e Cp(/)), g e H we have g - ( x | y) = (g-x) | (g-y) where- denotes the H-action). Regarding H,_p(F0, G) as a right H-module by x- g = g - 1 . x for x 6 Hn_p(F0, G), we can rewrite the formula for the action as g(x . g | y) = (x | g . y). Next, consider the long exact homology sequence for the triple (/~p, Ep-1, Ep-2) which yields the (H-equivariant) boundary map
Hn (.Ep, JF_Jp_, ; G)
' Hn-1
(/)p-l,/)p-2; C).
By our earlier calculations this can be viewed as a map H,_p(F0, G) |
Cp(/})
, Hn-p(Fo, G) |
Cp_: (/~).
It can be shown (see [W], XIII, (4.8)) that this boundary map is of the form 1 | dp, where dp'Cp(B) * Cp-I (B) is the cellular boundary map
(see (A.12)).
Finally, the natural map (/~p,/~p_ 1) ---* (Ep, Ep_:) is just the quotient map modulo the free H-action on the former, so that Hn(/~p,/~p-y ;G) ; Hn (E p, Ep_ 1; G) factors through the II-coinvariants
Hn(F,p, Ep-1;G) / { x - g - I x Ix e Hn(Ep, Ep-1;G),g e YI}.
Appendix A" Results from Topology
251
We have a diagram of pairs whose horizontal maps are relative homeomorphisms, and vertical maps are quotients modulo a free H-action,
(((Ap x Sp x H) x ~ E,) L[ Ep-l, ((OAp x Sp x H) x B E) I_I/~p-1))
1
1
(((Ap x S.) x B E) LI Ep_,, ((0A, x SIn) x s E) I_I Ep-1) Computing with the left-hand vertical arrow, one sees that in fact H . (Ep, Ep-1 ;G) is identified with the II-coinvariants of H.(/~p,/~p-l ;G), i.e., we have an isomorphism
Hn(Ep, Ep-1; G) ~ H-coinvariants of Hn-p(Fo,G) | = Hn-p(Fo, G) |
Cp(/3)
Cp(/~),
since the kernel of the surjection onto the coinvariants
g,,_r(Fo, G) |
C.([~)
, g n ( E , , E . - I , G)
is precisely the subgroup generated by all elements of the form - x|
g-(x|
= xg -1 |
x|
Thus, the long exact homology sequence for the pair (Ep, Ep-l) can be written as
... ---. gn(Ep-x, G)
; gn(Ep, G) ~ Y n - , ( f o , G) | - , gn-l(Ep-l,V) '...
C.(B)
and the composite
Hn(Ep, Ep_x;G)
o, Hn-I (Ep_ 1, G)-----* H._ 1(Ep_ 1,Ep-2;G)
(the boundary map in the exact sequence of the triple (Ep, Ep_ 1, Ep_ 2)) is identified with
1 | dp" gn-p(Fo, G) |
Cp([t)
, gn-p(Fo, G) |
Cp-1 (/}).
Since lim H,~(Ep, G) = Ha(E, G), the method of exact couples (see ApP
pendix C) yields a spectral sequence (of homological type) whose complexes of E l terms are identified (by (A.14)) with the complexes of cellular chains used to compute the homology groups with local coefficients H.(B, H.(Fo, G)). Thus we have shown: (A.24). Let p " E ~ B be a fibration, where B is a connected CW-complex. Then there is a first quadrant spectral sequence of homological type
Theorem
E~, o = H . ( B , g ~ ( f , a ) ) ~
H.+~(E, a )
252
Algebraic K-Theory
for any coefficient group G, where Hq(F, G) denotes the local coefficient system on B given by b H Hq(p-1 (b), G) (and (A.22)). An analogous argument (but some care is required in passing from Ep, for large p, to E = lim Ep) yields T h e o r e m (A.25). Let p : E ~ B be a fibration, where B is a connected CW-complex. Then there is a first quadrant spectral sequence of cohomological type
E2p,q = g p (B, g q (F, G)) ~
H p+q (E, G)
for any coefficient group G, where Hq(F, G) is the local coefficient system on B given by b H S q ( p - l ( b ) , G ) and (A.22)). We state a lemma about fibrations needed below. L e m m a (A.26). Let p" E ; B be a fibration, b E B, F = p-1 (b), x e F. Then with the natural action of 7rl(F,x) on 7ri(F,z), i > 1, 7rl(F,z) acts trivially on ker(Tri(f,x) , rri(E,x)). P r o o f . The exact homotopy sequence for the pair (E, F) shows t h a t we have a 7rl (F, x)-equivariant isomorphism
ker(. (F,
,7r,(E,x)) "~ image(O" Tri+l(E,F;x)
; 7ri(F,x)).
So it suffices to prove that under the above hypothesis, 7rl (F,x) acts trivially on 7rn(E,F;x) for n > 2. But the map of pairs ( E , F ) ~ (B, {b}) yields an isomorphism p. 9 7rn(E,F; x) ~- 7rn(B,x), (since p is a fibration), which is 7rl(F, x)-equivariant, where r l ( F , x ) acts on the homotopy sequence for (B, {b}) through the map 7rl(F,x) , 7rl({b},b) = 0. This follows from the naturality of the homotopy sequence for a pair and the naturality of the rl-action of the subspace on the sequence. ( A . 2 7 ) H o m o t o p y F i b e r s . Let f : X , Y be an arbitrary (continuous) map of spaces. Let P ( Y ) denote the free path space of Y, i.e., P ( Y ) = F(I, Y) is the space of maps I , Y (topologized as in (A.1)). There are maps fi : P ( Y ) ~ Y, i --- 0, 1 given by associating to any path its initial and final point, respectively. The evaluation map e : P ( Y ) x I ~ Y gives a homotopy between the fi. Let M ( f ) be defined as the pullback
M(f)
X
I
/
I
,
P(Y)
,
Y
Appendix A: Results from Topology I
253
o
T h e n the composite p 9 M ( f ) ~. p ( y ) I, , Y is called the mapping path fibration of f . T h e r e is a section A : X t M ( f ) of P0 given by A(x) = (x, wf(x)), where for any y e Y, w u denotes the constant p a t h
z
,
{u}.
Lemma.
With the above notation
(i) P0, A are homotopy inverses (ii) p o A: X
~M ( f )
, Y is just f
(iii) p = f l o f ' ~ f0 o f ' = f o P0 ( ~ denotes a homotopy) (iv) p : M ( f )
, Y is a fibration.
P r o o f . (i), (ii), (iii) are left to the reader as easy exercises ([W], I, C h a p t e r 7). To prove (iv), let g : Z ; M ( f ) be a map, g : Z x I , Y a h o m o t o p y beginning with p o g i.e., H ( z , O) = p o g(z) V z E Z. Let H' : Z x I , M ( f ) be defined as follows: if g(z) = (x, 7) for some p a t h -y: I , Y with 7(0) = f ( x ) , then p o g(z) = 7(1); let H ' ( z , t ) = (x,~/t) where 7 ( 2 s / 2 - t) if 0 _< s _< 1 - t / 2 7t(s) = H ( z , ( 2 s + t - 2)) i f l - t / 2 < _ s < _ l . (i.e., j u x t a p o s e ? and the p a t h beginning at x = p o g(z) given by H ) . Clearly p o U ' ( z , t ) = ~/t(1) = U ( z , t ) . We leave it to the reader to verify t h a t H ~ is continuous. D e f i n i t i o n . T h e fiber p - l ( y ) of p : M ( f ) ". Y is called the homotopy fiber of f at y, denoted by F ( f , y) (or just F ( f ) ) . By (A.21), we see t h a t if Y is p a t h connected, the h o m o t o p y t y p e of F ( f , y) is independent of Y. Explicitly, F ( f , y) is the set of pairs (x, 7) E X x P ( Y ) with 7(0) = f ( x ) , 7(1) = y. S i n c e p i s a fibration, and P0 : M(f) ; X is a h o m o t o p y equivalence such that f o P0 : M ( f ) ~ Y is homotopic to p, we have a long exact homotopy sequence
7ri(F(f , y), (x, wu)) (P'); , rri_l(F(f,y), (x, wu))... --, where x e f - - l ( y ) is a X = {y}, F ( f , y) is just we obtain isomorphisms If Y is a connected quences ((A.24), (A.25)) 2p,q
base point of X (and wu : I , {y}). Taking f~(Y), the space of loops based in Y based at y; lri+l (Y, y) ~- lri(f~(Y),wu). C W - c o m p l e x , we have Leray-Serre spectral sefor homology and cohomology
= Hp(Y, HqCF(f), G)) ~
Hp+qCX, G)
E~ ,q = HP(Y, H q ( F ( f ) , G)) ==, HP+q(x, G)
254
Algebraic K-Theory
for any coefficient group G, where Hq(F(f), G), Hq(F(f), G) are the local coefficient systems on Y associated to p (by A.22). In this case, the h o m o t o p y class of the self homotopy equivalence of F(f, y), induced by the class of a loop a 9I } Y based at y, has the explicit representative ~" F(f, y) } F(f, y) given by 5(x, 7)= (x, a. 7) where
a.~/(t) =
7(2t)
if 0 _< t _< 1/2 if 1 / 2 < t < _ 1 .
a(2t-1)
The homotopy fiber F(f, y) has the terizes it: to give a map Z } F(f, y) g : Z } X (ii) a homotopy of f o g Z } {y}. (A.26) immediately yields acts trivially on
following property which characis equivalent to giving (i) a map : Z ; Y to the constant map the fact t h a t rl(F(f,y),(x, wu))
ker(~rn(F(f, y), (x, wy)) P*, ~rn(X,x)) for any n >_ 1. Finally if f 9X } Y is a map to a path connected space Y, and if g" Y } Y is a covering space, f " )( - X x ~ Y } Y the induced map, and y E Y, then the natural map, F(f , ~l)
, F(f , y)
(where y = g(~)) is a homeomorphism. Indeed, if (x, 7) E F(f, y) where 7 is a p a t h in Y with 7(0) = f(x), 7(1) = y, then there is a unique p a t h -~ in Y lifting 7 such that q(1) = ~; there is a unique point & ~ X with ~ ~-. x under X , X, and f(~) = ~). T h e n (x,7) ~-~ (~, q) gives the inverse homeomorphism F(f , y) } F ( f , ~l). ( A . 2 8 ) S p e c t r a l S e q u e n c e for a C o v e r i n g . Let p : Y } X be a Galois covering space with (discrete) covering group G. If f : An - X is a singular p-simplex, then Ap • x Y } A n is a trivial covering, since A n is simply connected. Hence the group of singular p-chains Sp(Y) is in a natural way a free Z[G]-module, such t h a t Sp(X) = Z | Sp(Y) where Z is regarded as a G-module with trivial G-action. Let ...
,P,,
,P,,-1--...-~Po
,z
,o
be a projective Z[G]-resolution of Z, and Apq -- Pp • Sq(Y) so t h a t {Apq} form the terms of the double complex associated to the tensor product over Z[G] of the singular complex of Y and the resolution of Z. T h e n the total complex Wot(Apq) has homology groups Hn(Wot(Apq)) = Hn(X,Z), since the spectral sequence with El,q obtained by taking the homology groups in the p-direction has E2p,q - 0 for p ~ 0, and E02,n - H n ( Z |
Appendix A" Results from Topology
S.(Y)) = H.(S.(X))
255
= H n ( X , Z ) . The second spectral sequence for the
double complex has
E~,q = Pp |
Ha(Y, Z),
so t h a t E2p,q = TorZ[G](z, Hq(Y, Z) = Hp(G, Hq(Y, Z)). Hence we obtain a spectral sequence of homological type
S~,q=
Hp(G, Hq(Y,Z))
~
Hp+q(X,Z).
Similarly, one has a spectral sequence of cohomological type
E~ 'q = HP(G, H q ( Y , Z ) ) ~
HP+q(x,z).
(A.29) Quasi-Fibrations. A continuous surjective map p " E , B is called a quasi fibration if for each x 6 B, y 6 p - l ( x ) and i > 0 the natural m a p p. 9~r~(E, p - l(x); y) ; ~r~(B, x) is a bijection. (See A. Dold, R. Thom, Quasifaserungen und unendlische symmetrische produkte, Ann. Math. 67 (1958)). Note t h a t p. is an isomorphism of groups for i _> 2, and a bijection of sets with a distinguished point for i - 0, 1. We can use p. to give 7 r l ( E , p - l ( x ) ; y) the structure of a group, so that p. becomes an isomorphism of groups even for i - 1. The long exact homotopy sequence for a pair yields a long exact sequence
; lq+l(B,x)
, lr,(p-l(x),u) : ~ l q ( E , y ) ~,
~~o(B,x)
. . .
, lr,(B,x)
,0.
Thus for each x 6 B, the natural map p - l ( x ) ~ F(p, x) to the homotopy fiber at x (see (A.27)) induces a bijection on homotopy groups, i.e., is a weak h o m o t o p y equivalence. Thus if B is path connected, so t h a t the homotopy fibers of p at any two points of B are homotopy equivalent, we see t h a t for any x l , x 2 6 B the fibers p - l ( X l ) , p - l ( x 2 ) are weakly h o m o t o p y equivalent. Given two quasi fibrations p : E ; B, p' : E ' , B', a map f : E ----. E ' is called fiber preserving if there is a map of sets g : B , B' making E , E'
B
~ B' g
commute (note t h a t g need not be continuous). If f 9E , E ' is fiber preserving, x 6 B, x' - g(x), y 6 p - l ( x ) , y' : f ( x ' ) 6 p ' - l ( x ' ) then we have a m a p of pairs ( E , p - l ( x ) ) , ( E ' , p ' - l ( x ' ) ) which induces a m a p
256
Algebraic K-Theory
between their exact homotopy sequences; hence we have a commutative diagram --*Tri+l(B,x)
~
---~Tr,+l(B',x')--*
7r,(p-l(x),y)
---*
lr,(E,y)
---,
7ri(B,x)
---4
lri(p'-'(x'),y')
--.
7ri(E',y')
--~
7ri(B',x')
---4
9-- ~
9
7r0(B,x)
--.
x')
. .
0
0
where g. is the unique arrow (for each i) making the diagram below commute: r,(E,p-l(x);y)
P';
rz(E', p ' - l ( x ' ) ; y')
~
7ri(B,x)
lr,(B', x) i
P.
If p " E - - 4 B is any (continuous) map, a subset U C B is called distinguished for p if p - l ( U ) ~. U is a quasi fibration. T h e o r e m ( A . 3 0 ) . Let p : E B. A s s u m e that
, B be a map, L / = {U~} an open cover of
(i) each Ui is distinguished for p
(ii) f o r each x e Ui t3 Uj there exists Uk E LI with x e Uk C Ui tO Uj. Then B is distinguished f o r p, i.e., p is a quasi fibration.
The proof of this theorem requires the following results. L e m m a ( A . 3 1 ) . Let p" E ~ B be a map, U C B a distinguished subset f o r p. Then the following two statements are equivalent: (a) p. " l r i ( E , p - l ( x ) ; y) ~- 7rl(B,x) for all x e U, y e p - l ( x ) , i > O. (b) p. " r i ( E , p - l ( U ) ; y) "~ 7ri(B, U ; x ) f o r all x E U, y E p - : ( x ) , i > O. P r o o f . We have a diagram whose rows are the exact homotopy sequences of the triples (E, p - : ( U ) , p - l ( x ) ) and (B, U, {x}) 7ri(p-l(U),p-l(x);y)
1
7ri(U,x)
---, l r i ( E , p - l ( x ) ; y )
---,
1
7r,(B,x)
---. 7 r i ( E , p - l ( U ) ; y ) - - - ~ . . .
---.
1
lr,(B,U;x) ~ ...
Appendix A: Results from Topology
257
By hypothesis, 7ri(p-l(U), p - ' ( x ) ; y) ~ 7ri(U, x) for all i >_ 0. Hence the equivalence of (a) and (b) follows from the 5-1emma, except that a little care is required for i - 0, 1 (for i - 2 we are dealing with possibly non-Abelian groups, but the proof of the 5-1emma goes through). In case i = 0, recall that 7r0(X, Y) is obtained from the set lr0(X) of path components of X by identifying the points corresponding to path components meeting Y. From this, (a) =~ (b) (for ~r0) and the surjectivity of p. on 7r0 in (b) ~ (a) follow easily. To prove injectivity of p. "Tro(E,p-l(x);y) ~ 7ro(B,x)in (b) =~ (a), we have to prove that if El, E2 are path components of E lying over the same path component of B, then either E1 = E2 or El, E2 both meet p-l(x). Since [El] = [E2] in ~ro(E,p-l(U);y); let V1, 112 be two path components of p - l ( U ) with V~ N Ei # q) (so that V~ C Ei). Then p(Vl), p(V2) lie in the same path component of B, namely that which contains p(E1) and p(E2). Hence for some z E p(V1) we can find a path in B beginning at z and ending in p(V2), giving a class a e ~rl (B, U; z). Since by (b)
p. " lrl(E,p-l(U);w) '~ 7rl(B,U;z) for any z E U, w E p-1 (z), we may choose z as above in p(V1) and w E V1, and find a path in E joining II1 to some path component 173 of p-1 (U), such that p(V2), p(V3) lie in the same path component of U (since the image of this path under p must be homotopic to the original one used to define ~). Since we have an isomorphism
p. " lro(p-l(U),p-l(x)) ~ 7ro(U,x), 112 and II3 give the same class in rro(p-l(U),p-l(x)), i.e., either V2 = V3, or 112, V3 meet p-l(x). Since V1 and V3 are joined by a path in E, and Vl C El, we also have V3 c El. Hence either 112 C E l , i.e., E1 = E2, or El, E2 meet p - 1(x). Similarly to prove the statement about 7rl in (a) =~ (b) or (b) =~ (a), we again use the trick of shifting the base point. For example, if 0~1,0t2 e 7rl(E,p-I(U);y) have the same image in zrl (B, U; x), let wl,w2 be paths in E beginning at y and ending in p-1 (U) such that [w~] = a~ e lrl (E, p-l (U); y). Then p o w~(0) = x and there is a homotopy between p o Wl and p o w2, fixing p o w~(0) = x, and moving p o w l (1) to p o w2(1) along a path in U. We can regard w ~ l . w 2 (the composite path) as a path in E beginning at y' = wl (1) and ending in p-1 (U), i.e., as giving an element of ~rl(E, p - l ( U ) ; y'), and the homotopy between p o wl and p o w2 can be regarded a nullhomotopy of p o ( w i -1. w2). Thus to prove injectivity of p. on Irl in (a) =~ (b) or (b) =~ (a), one reduces to proving that the inverse image of the class of the constant path (loop) is the class of the
258
Algebraic K-Theory
constant path, for all possible choices of the base points. This follows from a straightforward diagram chase. We will prove the surjectivity of p. on 7rl in (a) =~ (b), and leave (b) =~ (a) to the reader. Suppose (a) holds; let c~ E ~rl(B, U; x). Then by a diagram chase, one can find f~ e 7 r l ( E , p - l ( V ) ; y) (for any given y E p - l ( x ) ) such t h a t p.f~, c~ have the same image in ~r0(U, x). Choose paths wl " I , B, w2 " I ~. E representing ~,f~ respectively. T h e n Wl(0) = x, w1(1) E U, w2(0) = y, w2(1) E p - I ( u ) , and p o w2(1) and wl(1) lie in the same p a t h component of U (this expresses the fact t h a t c~, p.f~ have the same image in 7ro(U,x)). Let y' = w2(1), x' = p(y'), x" = Wl(1). Let w3" I , U be a path with w3(0) = z", w3(1) = x'. T h e n 7 = ( ( p o w2) -1 9 Wl) 9 w3 is a loop in B based at x' (here 9 denotes composition of paths). By (a), p. " T r l ( E , p - l ( x ' ) ; y ') "~ 7rl(B,x') so t h a t we can find a path w4 " I ~ E with w4(0) - y', w4(1) E p - l ( x ' ) , such t h a t p o w4 is a loop in B based at x' which is homotopic to 7 through a homotopy fixing x'. Now w 2 . w 4 is a path in E beginning at y and ending at w4(1) e p - l ( x ' ) c p - l ( U ) , so t h a t it represents a class in ~rl (E, p - 1(U); x). Further, the homotopy of p o w4 with 7 can be regarded a homotopy between Wl and p o (w2. w4) keeping the first end point fixed at x, and moving the second end point Wl (1) to p o (w2-w4)(1) = x' along the path w3, i.e., p.[(w2, w4)] = c~ (a picture of the homotopy is given below; the top and b o t t o m sides are m a p p e d to x') x t
x t
w3 x v! pow4
Wl pow2 x I
T h e other result needed to prove (A.30) is a weak form of the homotopy lifting property. T h e o r e m ( A . 3 2 ) . Let p " E ~ B be a map, ld - {Ui} an open cover ol B satisfying the hypothesis of (A.30). For some n > O, assume given a diagram
/"-~ •
1
I n-1 X I
~
h
---* fl
E
l
B
Appendix A: Results from Topology
259
and an open set U E lg such that [-I(OI n - 1 x I U I n - 1 X { 1 }) c U. T h e n there , E with H l l . _ , x { 0 } = h, and a d e f o r m a t i o n
exists a m a p H " I n - I x I D " I n-1 x I x I
, B such that
i) D ( z , s, O) = H ( z , s), D ( z , s, 1) = p o H ( z , s), D ( z , O, t) = p o h ( z ) f o r all z E 1 " - 1
s 9 t 6 I.
ii) D ( O I " - 1 x l x l U I
"-1 x { 1 } x I )
cU.
We first prove (A.30) using (A.31) and (A.32). P r o o f o f ( A . 3 0 ) . F r o m (A.31) (since the Ui cover B) it suffices to show t h a t for x E U E/d, y E p - l ( z ) , i _> 0 the m a p p . " 7ri(E, p - l ( U ) ; y) ~ 7ri(B, U; x). (a) p . is onto. let c~ 6 l r , ( S , U ; x ) , i > 0 (i = 0 is trivial s i n c e p is clearly onto). Let Ho 9 ( I ' - I x I , I i-1 • {1}) ; ( B , U ) be a m a p representing c~9 such t h a t H 0 ( I '-1 • {0} U 01 '-1 x I) = {x}. We can find a h o m e o m o r p h i s m 8 - I i - 1 • I ----, I ~- 1 x I such t h a t O(I i-1
x
{0})= I i-'
8 ( I i-1 X {1}
U
x
{0}
U
OI i-1
X
I,
OI ~-1 x I) = I i-1 x {1)
(see the figure below for 8-1; the lines are 8({s} x I) for s E i i - 1 ) I i-1
x
{1}
O i ~- I x I
I i-1
x
{0}
We apply (A.32) with n - i,/~r = HooO, h ( I ' - 1 • {0}) = {y}; for the given U e b / , / ~ ( I i-1 x { 1 } U 0 1 '-1 x I ) = H o ( I '-1 • {i}) c U. T h e n i f g , D are as given by (A.32), H o O-l . ( I '-1 x 1 , 1 ' - 1 x {1}) , (E,p-l(U)) gives a class f~ E 7 r ~ ( E , p - I ( U ) ; y), and D gives a h o m o t o p y between Ho and p o H o O- 1, so t h a t p. (fl) = a. (b) p . is one-one. Let a 6 l r i ( E , p - l ( U ) ; y ) such t h a t p.(c~) = 0 (it suffices to consider this situation, and show t h a t a = 0, to prove the result for i > 1--see the proof of (A.31); the case i = 0 is left to the reader). Let h'(I',OI') , (E,p-l(U)),f-I'(I'xI, OI'• ". ( B , U ) be maps such t h a t h represents a , and H is a null homotopy of p o h to the constant m a p I~ , {x}. By (A.32), with n = i + 1, we can find H " ( I ' x I , OI' • I )
, (E,p-l(U))
with H I , , •
} = h,
260
Algebraic K-Theory
and H ( I '
x {1}) C p - l ( U ) .
T h e n H[~,x(1 } is also a representative for c~,
i.e., a = 0. This completes the proof of (A.30). T h i s proof of (A.32) uses the following l e m m a . L e m m a ( A . 3 3 ) . Let p " F ~ U be a m a p , V c U, G = p - l ( V ) . m > 0 be a n integer such that f o r each x E V , y E p - 1 ( x ) , the m a p p . " Iri(F, G; y)
Let
, r i ( U , V; x)
is injective f o r i = m and surjective f o r i - m + 1. A s s u m e given m a p s
(i) FI" ( I m x I , I m x {1})
, (U,V)
(ii) h" ( I m x { O } U O I m x I, OI m x { 1 } ) : = ~ ( F , G ) (iii) d" ( I m x {O} x I U O I m x z x I, OI m x { 1 } x I ) d ( z , s , O ) -- g ( z , s ) , d ( z , s ,
1) = p o h ( z , s )
, ( U , V ) with Vz E Z,s E I
(such t h a t (z, s) E I m x {0} U O I m x I ) . T h e n we can f i n d m a p s
(a) g " ( I m x I, I m x {1})
~ (F, G) w h i c h e x t e n d s h
(b) D " ( I m x I , I m x {1} x I ) ----. (U, V ) w h i c h e x t e n d s d, and satisfies D(z,s,O) = fI(z,s),D(z,s,
1) = p o g ( z , s )
V ( z , s ) e I "~ x I.
P r o o f . h determines a class a E 7rm(F, G) and d, H give a h o m o t o p y ofpoh to a m a p whose image lies in V, i.e., p.(c~) = 0 in 7rm(U,V). Since p . is injective on lrm, a = 0, so t h a t we can find a m a p H r 9 ( I m x I , I m x {1}) , (F, G) which extends h; we m a y assume w i t h o u t loss of generality t h a t H ' is constant on the subset K x [3/4, 1] C I m x I , where K = [1/4, 3/4] m C I m (we can deform t h e identity m a p of I m x I to a m a p shrinking K x [3/4, 1] to a point b u t which is the identity on I m x { O } U O I m x I , and maps I m x {1} into itself). Let y = g ' ( g x [3/4, 1]) so t h a t y E G (as g ' ( g x {1}) C g ' ( l m x {1}) C G). Let /~ = I m x l x {O, 1 } U I m x {0} x l U O I m x l x I = c l o s u r e o f 0 ( I m x I x I ) - I rn x {1 } x I, so t h a t O K = Z m x {(1,0),(1, 1)} U O I m x {1} x I.
Define D ' 9 (/~, OR) , (U, V) by D ' ( z , s, O) p o g ' ( z , s ) , for ( z , s ) e I m x I, and D ' ( z , s , t ) = I m x {0} x I U O I m x I x I. Further D ' c o n t r a c t s g point x = p ( y ) . N o w / ~ is homeomorphic to i r n + l , a class ~ E lrm+l(U, V ) .
f I ( z , s), D ' ( z , s, 1) d ( z , s , t ) for ( z , s , t ) e x [3/4,1] x {1} to t h e so t h a t D r d e t e r m i n e s
Since p. is onto on lrm+l, we can find a m a p
Appendix A" Results from Topology
261
f'(Im+l,0I re+l) - ~. (F, G) with f ( I m x { 0 } U O I m x I) = {y}, such t h a t p . [ f ] = - t 3 in lrm+l(U, V , x ) . Define g ' ( I m x I, I m x {1}) , (F, G) by H ( z , t)
j' f o r t) H ' ( z , t)
for (z, t) e g x [3/4,1] otherwise,
where r 9K • [3/4, 1] } I m • I is the h o m e o m o r p h i s m obtained from the obvious linear h o m e o m o r p h i s m s x ~ 2 ( x - 1/4) of [1/4, 3/4] with 1, and x ~ 4 ( x - 3 / 4 ) of [a/4,1] with I. We note t h a t f or
• {3/4} O O K • [3/4, 1]) = {y}
so t h a t H is continuous. Next, let DI " (/~, OR) ~
D l ( z , s , 1) = p o g ( z , s )
(U, V) be defined by
for ( z , s ) e g • [3/4, 1] and
D l (z, s, t) = D'(z, s, t) elsewhere o n / ~ . T h e n by construction, D1 represents ( - / 3 ) + 13 in 7rm+l (U, V; x) so t h a t D1 extends to a m a p D " ( I m • I • I, I m • {1} • I) ~. (U, V). T h e pair H, D have all the desired properties. P r o o f o f ( A . 3 2 ) . We can find an integer N > 0 which is so large t h a t if we divide I = [0, 1] into N equal s u b i n t e r v a l s / j = [2-~N1,~ ] , 1 < j < N , then for any m > 0, and any m-cube a which is a face of the resulting subdivision of I n - 1 • I into n-cubes of size l / N , we can choose an open set U a E H such t h a t (i) H ( a ) C
U a for all such a
(ii) if a is a face of a ~, then U a C U ~' (iii) if a meets cOIn - ~ x I U I n - ~ x {1 }, then U a C U. T h e existence of such an assignment follows from a s t a n d a r d a r g u m e n t , whose details are left to t h e reader (work by descending induction on m, using successively finer subdivisions; this uses hypothesis (ii) of (A.30) a b o u t the cover L(). Now each m - c u b e a of the subdivision has the form a l • ~r2 where a l is a cube in t h e subdivision of I n - l , and a2 C I is either a point or a subinterval Ij. Clearly I n is the union of all m-cubes a = a l • a2 with a2 = Ij for some j . We'll order the cubes of this type, so t h a t a = a l • a2 precedes T = ~'1 • T2 if a2 precedes T2 in the obvious ordering of t h e I j , or else a2 = r2 a n d d i m a l < d i m T1; we choose an a r b i t r a r y ordering of the mcubes a l x a2 w i t h a fixed interval a2 and a l ranging over all ( m - 1)-cubes in I n - 1. We will successively construct the desired maps H, D on the above cubes a l x a2 (with a2 an interval) taken in order. If a is one such m-cube,
262
Algebraic K-Theory
so t h a t a = a l • a2 is homeomorphic to I m - 1 x I, then from H, D on the earlier cubes we are given maps (i) [ t , 7 " ( I m - l x I , I m - l x { 1 } ) ~ ( U a , U a~x(l}) ( w h e r e a l x { 1 } denotes the product of al with the second end-point of a2 c I; al x {0} has a similar meaning)
(ii) H~,
9 (I m-1
p-l(va, •
x {0} U OI m-1
X I, OI m-I
(iii) D~, " ( I m - ' x { O } x l u O I m - I x l x I ,
x
{1})
OIm-lX{1}XI)
; (v-l(Ua), ' (Ua, U al•
satisfying the hypothesis of (A.33) with V = U ~• U = U ~. Hence we can extend H, D continuously over a, as desired, proving (A.32) by induction on the position of a in the above ordering. We give two more results about quasi fibrations which are needed in the main text. ( A . 3 4 ) . Let p" E ; B be a surjective map, B' C B a subset which is distinguished for p, and E' = p - I(B,). Assume given deformations Dt " E , E, dt " B : B, t E I such that Do = 1E, do = 1B, Dt(E') C E', dt(B') C B', DI(E) C E', d l ( B ) C B', and p o DI = dl o p. Suppose that for each x E B, y E p - l ( x ) , and i >> 0
Lemma
(D1). " lh(p-l(x), y) ~- lr,(p-l(dl (x)), Dl (y)). Then p is a quasi fibration. P r o o f . Since dr, Dt are deformations of the respective identity maps, ( d l ) . " lh(B, x) ~ ~h(B', x'),
x' = dl (x)
( O 1 ) . " lr,(E, y) ~ l h ( E ' , y'),
y' = D1 (y).
Since p o D1 = dl o p, D1 (p- 1(x)) C p - * (x'), and we have a m a p from the exact homotopy sequence of ( E , p - l ( x ) ) to t h a t of ( E ' , p - l ( x ' ) ) . Since (D1). " lr,(p-l(x),y) "~ l h ( p - l ( x ' ) , y ') by hypothesis, the 5-1emma gives isomorphisms (O1). 9 l h ( E , p - l ( z ) ; y ) ~ 7 h ( E ' , p - l ( x ' ) , y ' ) . Hence in the diagram below, the horizontal and right-hand vertical maps are isomorphisms, so the left-hand vertical map is one too.
~r,(E,p_l(x);y )
"'1
lh(B,x)
(ol);
,
(dl).
lh(E,,p_l(x,),y,)
l
lr,(B',x')
Appendix A: Results from Topology L e m m a ( A . 3 5 ) . Let p : E ~ B be a map, BI C B2 C . . . C increasing sequence of subsets that (i) each Bi is distinguished for p has the weak topology relative to the Bi (i.e., a subset of B is closed only if its intersection w i t h each Bi is closed in Bi) (iii) points are in B. Then p is a quasi fibration.
263 B an (ii) B if and closed
P r o o f . From (ii) and (iii) one sees easily that any c o m p a c t subset of B lies in some Bi. Hence any compact subset of E lies in some p - 1 (Bi). If x E Bi, y E p - l ( x ) , t h e n 7 r n ( E , p - l ( x ) ; y) ~ lim Trn(p-1(Bj),p-1(x); y) j>l
lira 7rn(Bj,x)
(by (i))
u ~(B,z). ( A . 3 6 ) C o f i b r a t i o n s a n d N D R - P a i r s . Let i 9A ; X be the inclusion of a closed subspace. T h e n i is called a cofibration if i has the h o m o t o p y extension property, i.e., any map F 9X • {0} U A x I ~ Y extends to a map G 9X x I ; Y. Taking Y = X • {0} U A x I, F = identity, we see t h a t i is a cofibration ~ X • {0} U A x I is a retract of X x I. Let i 9 A .~ X be the inclusion of a closed subspace. Following N. Steenrod (A convenient category of topological spaces, Mich. Math. J. 14 (1967), 133-152), we call ( X , A ) a n N D R - p a i r ("neighborhood deformation retract pair") if there exist maps u" X ~ I, h" X x I ---, X such t h a t u - l ( 0 ) = A, and h(x, 0) = x, h(x, 1) E A if u(x) < 1 and h ( x , t ) = x if x E A, t E I. Thus, A is a deformation retract of the open set u - l ( [ 0 , 1]). If h(x, 1) E A V x E X , we say t h a t (X, A) is a DR-pair. L e m m a (A.3r). If (X, A), (Y,B) are NDR-pairs, so is ( X x Y , X x B U A x Y). If one of them is a DR-pair (and the other an N D R - p a i r ) then the product pair is a DR-pair. Theorem (A.38). If i : A the following are equivalent
; X is the inclusion of a closed subspace,
(i) ( X , A ) is an N D R - p a i r (ii) X x {0} U A x I is a (iii) X x
deformation retract
of X x I
{0}OAxIisaretractofXxI
(iv) i is a cofibration. P r o o f . Since (I, { 0 } ) i s a DR-pair, (i) =~ (ii) follows from (A.37); clearly (ii) =~ (iii), and (iii) ~ (iv) as remarked above. So it suffices to show
264
Algebraic K-Theory
(iii) =~ (i). Let R" X • I ; X x {0} U A • I be a retraction, pl " X • I - ; X , p2 " X x I , I the projections, h = Pl o R, k = P2 o R, where i0i is the r e s t r i c t i o n of p~ to X x {0} U A x I. T h e n h(x, O) = pl o R(x, O) = p l ( x , 0) = x, and if x e A, h(x,t) = Pl o R ( x , t ) - Pl(x,t) - x. Let vm " X --, [0, 1/2 m] be the m a p vm(x) - m i n ( 1 / 2 m, k(x, 1/2m)) a n d let u " X --, I be given by the (uniformly convergent) series oo
u(x) = 1 - E
oo
vo(x) vm(x) = E
m--1
( ( 1 / 2 m ) - vo(x)vm(x)).
m=l
T h e n u(x) -- 0 ~
vo(x)vm(x) - 1/2 m V m :> 1. If x E A, t h e n k(x, 1/2 m) = p2oR(x, 1/2 m) = p2(x, 1/2 m) = 1/2 m, so t h a t vm(x) = 1/2 m V m _> 0, a n d u(x) = O. If x r A, t h e n R(x,O) = ( x , 0 ) E ( X - A) • {0} which is o p e n in X x {0} U A x I, so t h a t R(V) C ( X - A) • {0} for some o p e n set V C X • I with ( x , 0 ) E V. In particular k(V) = {0}, so t h a t for all large m, k(x, 1/2 m) - 0, a n d vm(x) = 0; thus u(x) > O. Finally u(x) < 1 ~ Vo(X)Vm(X) > 0 for some m > 0; in p a r t i c u l a r vo(x) = k(x, 1) > 0, i.e., R(x, 1) E A • I c X • { 0 } U A x I. H e n c e h(x, 1) E A if u(x) < 1. T h e pair (h, u) represent (X, A) as an N n R - p a i r . P r o o f o f ( A . 3 7 ) . Let u : X - - ~ I, h : X x I , I represent (X, A) as an NDR-pair, and let v : Y ~. I, j : Y x I ~ Y represent (Y, B ) as an NDR-pair. Let w : X x Y --~ I be given by w(x, y) = u(x).v(y) (pointwise p r o d u c t ) so t h a t w - l ( 0 ) = X x B U A x Y. Define q : X x Y x I - - , X x Y
by
if ( x , y ) e A x B
q(x,y,t)=
( h ( x , t ) , j ( y , t ~ y ) ))
if v(y) >_ u(x) and v(y) ~ 0
(h(x, t ~--~),j(y, t))
if u(x) >_ v(y) and u(x) ~ O.
One checks t h a t q is continuous, and w, q represent ( X • X xBUA• as an NDR-pair. If u, h represent (X, A) as a D R - p a i r , a n d u ~ = 89 t h e n u', h also r e p r e s e n t (X, A) as a DR-pair; if we c a r r y out t h e above c o n s t r u c t i o n w i t h u', h t h e n w' = u ' . v < 1 on all of X • Y, so w', q represent ( X • Y, X x B U A x Y) as a D R - p a i r . L e m m a ( A . a 9 ) . Let f " ( X , A ) ~ ( Y , B ) be a relative homeomorphism (i.e., f induces a h o m e o m o r p h i s m X - A ~- Y - B, f is onto a n d Y has t h e q u o t i e n t topology from X). Then if (X, A) is an NDR-pair, so is (Y, B). P r o o f . Let u " X ; I, h - X • I ~ X represent (X, A) as an NDR-pair. T h e n v " Y ---, I, j " Y x I ~ Y given by v(y) = u ( f - l ( y ) ) ,
j(y,t) = { f y ~
if y c~ E B' B, t E I
Appendix A: Results from Topology
265
are continuous, since f is a relative homeomorphism, and represent (Y, B) as an NDR-pair. C o r o l l a r y ( A . 4 0 ) . /f ( X , A ) is a relative CW-complex, then ( X , A ) is an NDR-pair, i.e., A ~-~ X is a cofibration. P r o o f . If A c B C X are inclusions of closed subspaces such t h a t (B, A) and ( X , B ) are N n R - p a i r s , then from (A.37) (iii), ( X , A ) is an NDR-pair. If (X, A) is an NDR-pair, ( X L[ Y, A L I Y) is an NDR-pair for any Y, and (An, 0An) is an N D R - p a i r for any n _> 0. Hence (Xn, A) is an N D R - p a i r for any n > 0, where Xn is the n-skeleton of (X, A), since by the earlier remarks ( X , , X n - 1 ) is an N DR-pair V n >_ 0 (where X _ 1 = A). T h e r e is a retraction Rn " Xn x I ~ X , x {0} U Xn-1 x I; extend this to retraction S . . X x {0} u X . x I - - , X x {0} u X , _ ~ x I
by letting Sn(x,O) = (x,O) V x E X . Let Tn " X • { 0 } U X n • I X • {0} O A • I be the composite retraction Tn = So o - - . o Sn. T h e n Tn+l restricts to Tn on X • {0}UXn x I; since X • I = lim ( X • { 0 } U X n • I) has --4
the weak topology, the set theoretic retraction X x I ~ X • {0} U A • I, which restricts to Tn on X • {0} U Xn • I, is continuous. Hence (X, A) is an N D R - p a i r . As a simple example of the use of the above notions, let (X, xo) be a space with a base point. T h e base point is called non-degenerate if (X, {x0}) is an NDR-pair. Given any pointed space (X, xo) let X be the quotient space of X I_[ I obtained by identifying x0 with 0 E I; let ~ E X be the image of 1 E I. T h e n ()(, ~) has a non-degenerate base point, and (X, ~) - (X, x), given by collapsing the image of I in X to a point, is a homotopy equivalence. T h e process of forming )( is sometimes called "adding a whisker." L e m m a ( A . 4 1 ) . Let f, g " (X, xo)
' (Y, Yo) be two maps which are freely homotopic, i.e., there is a homotopy H; X x I ---, Y with H ( x , O) = f ( x ) , H(x, 1) = g(x) V x E X (but which may not satisfy H(xo, t) = yo V t E I). Let w " I --. Y be the loop at yo given by w(t) = H(xo, t), and assume [w] = 0 in Zrl(Y, Yo). Then there exists a homotopy H " X • I - - . Y with U(x, O) = f ( x ) , U ( x , 1) = g(x) and U(xo, t) = yo V x E X , t E I, provided xo E X is non-degenerate. P r o o f . Since X x {0}U{x0} x I is a retract of X x I, X x {0} x I U { x 0 } x I x I isaretractofXxlxI, so t h a t any map F " X x {0} x I n { x 0 } x l x I --~ Y extends to a map G " X x I x I , Y. Let F be given by
F(x, s, t) = ( H ( x ' t) Hl(s,t)
ifs=0,(x,t) EXxI ifx=x0,(s,t) EIxI
266
Algebraic K-Theory
where H1 " I x I , Y is a null h o m o t o p y of w, i.e., H l ( 0 , t) = w(t), H1 (s, 0) = H1 (s, 1) = H i ( l , t) = Y0 for all s, t E I. Let G " X • I x I - , Y e x t e n d F , and let H " X x I -4 Y be given by
I G(x, 3s, O) H(x, s) =
if0_< s_< 1/3
G(x, 1, 3s - 1)
if 1/3 < s <_ 2/3
G ( x , 3 - 3s, 1)
if 2 / 3 < s <_ 1.
One sees t h a t H has the desired properties. ( A . 4 2 ) . Let f " (X, A) ~ (Y, B) be a relative homeomorphism, where (X, A) is an NOR-pair. Then f . 9 Hn(X, A; G) ~ Hn (Y, B; G), f* 9 Hn(Y, B; G) ". H " ( X , A; G) are isomorphisms for any coefficient group G, V n >_ O. Lemma
P r o o f . T h e r e is a neighborhood U of A in X such t h a t A is a d e f o r m a t i o n r e t r a c t of U, and f(U) = V is a similar neighborhood of B in Y. T h e n we have isomorphisms (for any coefficient group)
Hn (X, A) "~ Hn (X, U) ~- Hn ( X - A, U - A)
(excision)
H,(Y-B,V-B) "~ H,(Y, V)
(excision)
H,(Y,B)
and a similar argument works for cohomology. ( A . 4 3 ) H S p a c e s . A space (X, x0) with a non-degenerate base point is called an H-space if there is a m a p lz : ( X x X , (xo, xo)) ~ (X, xo) such t h a t / z l ( x ) = #(x, xo), #2(x) = #(Xo,X) are each homotopic to the identity m a p of X (through homotopies fixing x0).
L e m m a (A.44). Let ( X , xo) be an H-space. X x X
Then there is a m a p ~ 9 , X w i t h ~ ( x , xo) = ~ ( X o , X) = x V x ~ X .
P r o o f . Let pt : X x X ~ X be a m a p giving an H - s p a c e s t r u c t u r e on X , i.e., i~i (x) - td(X, Xo), p'2(x) = Izt(xo, x) are each homotopir to the identity m a p of X . Equivalently, i f X V X = X • {x0}U{x0} • X c X • X, k : X V X r X x X, V : X V X ~ X t h e folding m a p (i.e., V(x, xo) V(xo, x) = x V x E X), then #' o k ~ V (where ~ denotes a h o m o t o p y preserving the base points). By (A.37), (X • X, X v X) is an NOR-pair, a n d so by (h.as) (iv), X v X , x x x has the h o m o t o p y extension property. Hence, given #' : X x X - X a n d the h o m o t o p y / z ~ o k ~ V of ~t Ix v x, we can find a h o m o t o p y ~' ~-- ~, where ~ X x X ~ X satisfies # o k - V, i.e., #(x, xo) =/z(x0, x) - x Vx E X.
Appendix A: Results from Topology Lemma
(A.45).
267
/ f (X, xo) is an H-space, then (X, xo) is simple, i.e.,
7rt (X, xo) acts trivially on rrn (X, xo) for n > 1. P r o o f . Let /z 9 X • X , X be a m a p giving an H - s p a c e s t r u c t u r e such t h a t #(x, x0) = #(x0, x) = x V x E X , as in (A.44). T h e n if f 9 (In, OI n) ~ (X, xo), g " (I, OI) ' (X, xo) represent given classes c~ e 7rn(X, xo), [3 e 7rt(X, x0) then h(s,t) = # ( f ( s ) , g ( t ) ) gives a m a p h " I n x I , X satisfying h(s, O) = f ( s ) V s e I n, h(s, t) = g(t) for all s e OI n (so t h a t f ( s ) = xo) and all t E I. Hence by definition of the action of 7rl on lrn, the m a p h(s, 1) 9 (I n, OI n) ~ (X, xo) represents the result of/3 E 7rl acting on a e 7rn; b u t clearly h(s, 1) = f ( s ) since g(1) = x0. This proves the lemma. An H - s p a c e is homotopy associative if the maps # o (# x 1), # o (1 x #) 9 X x X x X --~ X are homotopic (1 denotes the identity m a p of X , # o (# x 1)(x, y, z) = # ( # ( x , y), z), etc.). It is homotopy commutative if /z o s is homotopic to #, where s 9X • X , X • X is the switch m a p s(x, y) = (y, x). Finally, (X, x0) is an H-group if it is h o m o t o p y associative, and there is a m a p j " (X, xo) ; (X, xo) (an inverse up to homotopy for t h e product) such t h a t # o (1 x j ) o A, # o (j x 1)o A - X ~ X are b o t h homotopic to the c o n s t a n t m a p X ; {x0}, where A ( x ) = ( x , x ) is t h e diagonal. ( A . 4 6 ) . A homotopy associative H-space is an H-group if and only if the shear map r 9X • X ~ X • X, r = ( x , # ( x , y ) ) is a homotopy equivalence.
Lemma
P r o o f . If (X, xo) is an H-space, with product # : X x X ~X, j : X X an inverse up to h o m t o p y for #, then #(x, y) = (x, # ( j ( x ) , y)) is clearly a h o m o t o p y inverse for r Conversely if r is a homotopy equivalence, and r :X x X , X x X is a h o m o t o p y inverse for r and pi : X x X , X, i - 1, 2 are the projections, then let j be defined by
j ( x ) - p2 o ~)(x, Xo) - p2 o r o i l , where il(X) -- (X, Xo),i2(x) -- (X0, X). We have homotopies Pl~PlOr162162 so t h a t p l o r #o(1
P2~p2or176162176162 andso
x j ) o A ~-- # o ((pl o 0 o i l ) x (p2 o~) o i l ) ) o A =Do(pl
•176176
=#o(pl
xp2) oAor
=Dor
--~p2oil = . ,
• (~boil))oA
268
Algebraic K-Theory
the constant map (with value x0). Since j is a right inverse up to homotopy for p, and ~u is homotopy associative, j is also a left inverse up to homotopy for ~u. C o r o l l a r y (A.47). Let (X, xo) be a path connected, homotopy associative H-space with the homotopy type of a CW-complex. Then X is an H-group. P r o o f . Let # : X x X , X be a map giving an H-space structure such t h a t #(x, x0) = #(x0, x) = x, and let r : X x X ~ X x X be the shear map (of (A.46)). Since r xo) - (x,x) and r x) - (xo, x), one sees t h a t under the natural direct product decomposition • x,
• -.(x,
(induced by the inclusions X ~ X x X , x H (x, xo) and x ~ (xo, x)) r : ' nn(X X X , (xo, xo)) becomes the group theoretic shear map (a, fl) - - ~ ( a , a + ~ ) on lrn(X, xo) x ~n(X, xo). Hence r induces isomorphisms on homotopy groups, so by the Whitehead theorem (A.9) r is a homotopy equivalence.
r n ( X x X , (xo, xo))
C o v e r i n g S p a c e s of S i m p l i c i a l S e t s . In this section we follow the treatment given in: P. Gabriel, M. Zisman, Calculus of Fractions and Homotopy Theory, Ergeb. Math. 35, Springer-Verlag, 1967). We will use the notation and terminology of Chapter 3 of the main text. A map p" E ---, B of simplicial sets is said to be locally trivial with fiber F (also a simplicial set) if for each simplex a e B(n_) (for any n), if A(n) .~ E is the associated map of simplicial sets, and A(n) x B E is the fiber product (so that (A(n) x B E)(p) -- A(n)(p) x B(p)E(p) for all p >_ 0), there is an isomorphism g of simplicial sets making the diagram below commute: A(n) x B E
9 . A(n) x F
zx(n) If f : Y ~ X is a map of topological spaces, we say t h a t f is locally trivial with fiber Z (a topological space) if X has a covering by open subsets U with the property that there exists a homeomorphism g making the diagram below commute:
UxxY
;
U
/
UxZ
Appendix A: Results from Topology
269
Note t h a t by convention of (A.1) all the spaces are compactly generated, and products, fiber products are formed in the category of compactly generated spaces, so t h a t the above notion of local triviality differs from the "standard" one. However, the two notions do agree if Z is locally compact, and in particular if Z is discrete. T h e o r e m ( A . 4 8 ) . Let p 9 E ~ B be a locally trivial m o r p h i s m of simplicial sets with fiber F, a simplicial set. Then the geometric realization Ipl" IEI--, IB] is a locally trivial map of topological spaces with fiber IFI . Before proving this result, we describe an open neighborhood Ux of a given point x E ]B I, for any simplicial set B (and any x E IBI). These neighborhoods will be used to prove (A.48). First, if m >_ 1 and V C OAm is any (relatively) open subset, let V* C Am denote the open subset (the "open cone over V") V* = { t x + ( 1 - f)xo [ x 6 V, O < t _< l } ,
where we regard Am as the subset of R m+l
Am ~---{ ( 8 0 , . . . ,$m) E R m+l [8 i > O, ~-]Si -~ 1}, and x0 = ( r e1+ l , ' ' " re+l) 1 is its barycenter (the sum in the definition of V* is as vectors in R m+l where t, 1 - t are scalars). Clearly V is a deformation retract of V*. Next, given a CW-complex X, with n-skeleton X n for each n > 0, and a point x E X, then we can find a unique integer n >_ 0 such t h a t x E X n - X,~_ 1 (where X_ 1 is empty), and hence we can find a unique n-cell a - A,, , X such t h a t x E a ( A n - O A , ) (where 0A0 is empty). Let [In = a ( A n - 0 A n ) ) so t h a t Un C X n is open. For m > n, we inductively define open subsets Um c Xm by =
u..,-, o (U,,u O"
where a runs over the m-cells of X. Then Um is indeed open in Xm, and Um n X m - 1 = U,n-1. Let Ux = Um>n Urn. Since X has the weak topology with respect to the Xm, and U~ Q Xrn = Urn is open in Xm for all m > n, Ux is open in X. This works, in particular, for X = [B[ where B is any simplicial set. Note t h a t U~ has the weak topology relative to the Urn. T h e above construction has the following property. Let f 9Y , X be a m a p of topological spaces where X is a CW-complex and let x E Xn - X,~_ 1. Assume t h a t there exists a space Z and homeomorphisms gm " Urn X X Y ~Um x Z, for m >_ n, such t h a t gm+l[u,~xxY = gin, and U.., x x Y
9,. ~ U.., x Z Um
270
Algebraic K-Theory
,U=xY commutes. Then the resulting set theoretic map g" U= • x Y = gin) is a homeomorphism making the diagram below (given by g]u,,,x x Y commute:
Um x x Y
9
~ U. x Z
u= This is one advantage of working with compactly generated spaces. P r o o f of ( A . 4 8 ) . From the discussion above, if x E IBI lies in the interior of an n-cell of the C W - c o m p l e x IBI, it suffices to construct a compatible family of homeomorphisms
gm-urn
; u,. x IFI
x,B, IEI
for m > n, such that the diagram below commute:
u ~ x,~ I IEI
"~ .~ U~ x IFI U,~
T h e existence of such a map g , follows from the fact t h a t for any m-cell ~,. A~ ~ I n l w e are given a homeomorphism g(a) (compatible with projection to A,~),
g(a)'AmXlBI I E I ,
'AmxlFI,
which is obtained by realizing a corresponding isomorphism of simplicial sets; in particular this is valid for the n-cell whose interior contains x. Given gin" UmxisllEI ~ UmxlFI, w h i c h w e w i s h t o e x t e n d t o g m + l " Um+lxIBI IEI ---* Um+l x IFI, we see t h a t for any (m + 1)-cell a 9 Am+l -. IBI, gm determines the restriction of gm+l to ( a ( A m + l ) Q Urn+l) xIB I IEI. T h e maps g(a), gm thus give two homeomorphisms g(a)', gin(a)'
a - l ( u m ) XlB I JE I
, a - l ( U m ) x IF{
compatible with projection on the first factor. Thus, if Aut(IFI) is the subspace of IF(IF I, IFI) consisting of homeomorphisms, we can find a map 8"a-l(Um) , Aut(IFI) , such t h a t
gin(a)'= ~ o g(a)', where ~" a - l ( U m ) x
JFI
, a - l ( U m ) x IF]
is given by 8(x,y) = (x, 9(x)(y)). Now a - l ( U m ) is a deformation retract of a - ' ( V m ) * , so we can extend 8 to a m a p 8" 9 a - l ( U m ) * , Aut(IFI) and so define 8"" a - l ( V m ) * x Ifl ~ a - l ( V m ) * x IFI
Appendix A: Results from Topology
271
by O"(x,y) = (x,O*(x)(y)). Next, let
gm+,Ca)"
•
IEI
,
a-'(u..)* •
IFI
be the composite g,r,+l(a) = 0" o #(a) (where #(a) is the restriction of g(a) to a - l ( U ~ ) . Since Um+l = U m U ([.Ja a(a-l(Um)*)), the reader can verify t h a t gm+l (a) do patch together to give an extension of 9m to a m a p gm+l : Um+l • IBI [El ~ Um+l • [FI, as desired. This completes the proof of (A.48). We define a morphism p : E ~ B of simplicial sets to be a simplicial covering if E ( 0 ) ~ B(0) is surjective, and for every diagram of maps of simplicial sets A(0) .~ E
l
l"
, B there is a unique m a p of simplicial sets A(n) , E making the diagram commute. If [B I is connected (equivalently, any two 0-simplices of B are j o i n e d by an edge-path formed from 1-simplices) then one easily verifies that p : E , B is a simplicial covering if and only if it is locally trivial with a n o n - e m p t y discrete simplicial set as fiber (a discrete simplicial set is one which has no non-degenerate positive dimensional simplices). Since the realization of a discrete simplicial set is a discrete space, we deduce from (A.48):
C o r o l l a r y (A.49). Let p : E --, B be a simplicial covering. IBI is a covering of topological spaces. realization [p[: [E I
Then the
In particular, the construction in Example (3.10) in the main text does give a construction of the classifying space B G of a discrete group G. L e m m a ( A . 5 0 ) . Let f , g : GI , G2 be two homomorphisms between (discrete) groups G1, G2. Assume that f, g are conjugate, i.e., for some 5 E G2, f ( x ) - 5 - 1 g ( x ) 5 for all x E G1. Then there is a homotopy h : BG1 x I ~. BG2 between B f and Bg such that the loop in BG2 traced out by the image of the base point of BG1 under h, represents the class E ~rl (BG2) ~ G2. P r o o f . T h e homomorphisms f, g induce functors ], ~ ' G _ 1 , G__2 (in the notation of E x a m p l e (3.10), Chapter 3) such t h a t B ] = B f , B~ = Bg. Now __G2 is a category with a single object ,, such t h a t Hom_G2 (,, , ) -- G2. Hence 5 E G2 gives a morphism 5 : 9 , 9 in _G2, such t h a t for any x ~_ G1, 5 o f ( x ) -- g ( x ) o 5, where f ( x ) , g(x) ~_ G2 are again regarded as morphisms 9 , 9 in G2. Thus 5 defined a natural transformation of
272
Algebraic K-Theory
functors f , ~, whose realization gives the desired homotopy (we take the 0-cells corresponding to the unique objects of G1, __G2 to be the base points of B__G1, B_G2 respectively). ( A . 5 1 ) T h e H u r e w i c z T h e o r e m for H - S p a c e s . If f " X , Y is a map of path connected topological spaces which induces isomorphisms on integral homology groups in dimensions < n, then if X, Y are simply connected, f induces isomorphisms on h o m o t o p y groups lri for i < n, and a surjection on ~r, (see (A.5), (A.6)). This may not be true if we drop the hypothesis of simple connectivity. However, if X, Y are H-spaces which have universal covering spaces, then the result holds. Our proof of this in (A.53) is based on discussions with M.S. R a g h u n a t h a n . L e m m a ( A . 5 2 ) . L e t X be a p a t h - c o n n e c t e d H - s p a c e w h i c h has a u n i v e r s a l c o v e r i n g space zr 9 X ~. X . T h e n lrl(X) acts t r i v i a l l y o n the integral
homology g ouw P r o o f . Let x0 E X be the base point. By (A.43) we may assume t h a t the multiplication # 9X x X ~. X, giving the H-space structure on X , satisfies/~(xo,x) = l t ( x , xo) = x V E X , i.e., x0 is a two-sided identity for #. A point y e X is given by a pair (x, ['y]) where x e X , [3"] a homotopy class of paths in X joining x0 to x; the map 7r" X , X is then ~r(x, [3"]) = x. In particular, the fundamental group lrl(X, x0) is identified with the fiber 7r-l(x0) by [w]-~-. (x0, IT]) for any class [w] e 7rl(X, x0). The action of r l (X, x0) on X is described by
= and w -3, is the composite path I
, X,
w(2t) 7(2t_1)
(w-7)(t)=
if 0 _~ t _< 1/2 if 1 / 2 _ < t _ < 1 .
The identity element of 7rl (X, x) gives a natural base point Y0 E X. If w" I ~. X is a loop based at x0, and 3"- I , X a path from xo to x in X, define Wl " I , X, 3"1 " I ---o X by
= { (2t) 0
"r, (t) = {
-y(2t - 11
if0
Then clearly [Wl] = IT] in 7rl (X, x0), and [3,1 - [3'11 in the set of homotopy classes of paths j o i n i n g x 0 t o x in X. The path (Wl,3"1) " I , X x X satisfies # o (Wl, 3"1) = w-3,, the composite path.
Appendix A: Results from Topology
273
Let/2-X x X ; X be given by ~((Xl, [')'11), (x2, [ ' ) ' 2 1 ) ) - " (]2(Xl, X2), [/Z 0 ('/1,72)]) where (71, "12)" I ; X x X. One sees at once t h a t / 2 is well defined and continuous, and satisfies /2(yo, y) =/~(y, Y0) = Y V y e X , i.e., 12 has a two-sided identity Yo. Thus (X,/2) is an H-space. From the above computations, if y E X, ~ E 7rl(X, x0), then a(y) = fz((xo, oO, y ). Since X is p a t h connected, left multiplication with respect to/2 by (x0, a) is homotopic to left multiplication by yo, i.e., translation by cz on )( is homotopic to the identity map of X. This proves (A.52). ( A . 5 3 ) . Let f 9X - - , Y be a map between path connected H-spaces which have universal covering spaces. Suppose f induces isomorphisms on integral homology groups in dimensions < n. Then f induces isomorphisms Theorem
f,
, -,(Y, f(xo)),
i <
and a surjection for i = n. P r o o f . Replacing f by its mapping cylinder M ! (see (A.6)) we may assume f is an inclusion. T h e theorem is then equivalent to the statement 7r~(Y,X, xo) = 0 for i < n. This is trivial for n = 0, so assume n > 1. Since X, Y are H-spaces, their fundamental groups are Abelian, by (A.44); now by (A.5) (i), 7rl (X, xo) '~ 7rl(Y, Xo) ~- G, say. T h u s if Y , Y is the universal covering, X = Y x y X, then X , X is the universal covering and X ~-, Y. As in (A.28), one has a spectral sequence of homological type
EUv,q = Hp(G, Hq(Y, f(; Z)) ~
Hp+q(Y, X; Z).
Since f . 9 H i ( X , Z ) ~ Hi(Y,Z) is an isomorphism for i <_ n, Hi(Y, X; Z) - 0 for i < - - n; hence E p~176 , q = 0 for p + q -< - n. Further, from (A.52), E 2O , q - -
Ho(G, Hq(~r, f(; Z))
--
Hq(Y, f(; Z)
(actually, this follows from the proof of (A.52)). Let q be the smallest integer such t h a t Hq(Y, Jr; Z) # 0 (if q does not exist, then Hq(Y, f(, Z) = 0 V q); we claim q >_ n + 1. Indeed, for r _> 2, we have the complex of E r terms E rr, q - - r + 1 r E [ r,q+r- 1 ~ Eo,q ~
where E r2, q - - r + l --- 0 since q - r + 1 < q, while E -2r , q + r - 0 since - r < 0 9 1 -H e n c e E 02, q ~ E 0~176 T h u s q > - - n + l " Hence, in any , q , i.e. , H q ( Y , X ; Z ) # 0 . case H i ( Y , X ; Z ) = 0 for i <_ n. Since X, Y are simply connected, (A.5) (iii) implies vri(Y, X; x0) = 0 for i <_ n, as desired.
274
Algebraic K-Theory
C o r o l l a r y (A.54). Let f : X ~ Y be a map between connected H spaces, each of which has the homotopy type of a CW-complex. Suppose f induces isomorphisms on integral homology groups. Then f is a homotopy equivalence. P r o o f . This follows at once from (A.53) and the Whitehead theorem (A.9). Finally, we prove a result of Milnor, which is the basic homeomorphism used to show that geometric realization of simplicial sets commutes with products, provided the topological product is formed in the category of compactly generated spaces. Consider the simplicial set A(r) • A(s), given by the functor A ~ ~. Set, p ~ H o m h ( p , r ) x Homzx(p, _s). The projections A(r) x A(s) A(r), A(r) x A(s) =~; A(s) induce maps on the geometric realizations IA(r) x A(s)l - At, IA(r) x A(s)l , As, giving the product map f'la(r) x A(s)l , A,. x A,,. T h e o r e m (A.55). f is a homeomorphism. P r o o f . We begin by making some general remarks about geometric realizations. Given any simplicial set T, any point x E IT[ lies in the interior of a unique non-degenerate simplex, since any point in a C W complex lies in the interior of a unique cell (here a 0-simplex is its own interior). Thus, given x E IT[, there exists a unique integer n > 0, a unique non-degenerate simplex a E T(_n), and a unique point P E A~n (where A~n = { ( t o , . . . , t n ) E R n+i [ t i > 0, Eti = 1} is the interior of An), such that x is the image of (a, P) under the quotient map H T(n) x An
;; ITI.
n>0
(a, P) is called the non-degenerate representative of x (any (a', P') mapping to x under the quotient map will be called a representative of x). For any n-simplex a E T(n), let A(a) be the corresponding nondegenerate simplex (it is uniquely determined by a). If A(a) is an ( n - p)simplex, then there is a unique sequence j l , . . . , jp of integers such that A(a) = T(Sj,,) o . . . o T ( S j , ) ( a ) ,
0 < jl <""
< jp < n,
where the Sj are the degeneracy morphisms in A. Since the arrows (S i ) . : Am ; Am_l (for any m > 1) induce maps A ~ , A~ 1, one sees that if x E ]T], and (a, P) is any representative of x, with a E T(n) and P E A ~ then the non-degenerate representative of x is (A(a), (S~)., (P)), where =
(Sj,).
for the above sequence j 1 , . . . , jp.
o 999o
(Sj,,).
Appendix A: Results from Topology
275
We now show t h a t f is bijective. Suppose x E IA(r) x A(s)t has nondegenerate representative ( ( a l , a 2 ) , P ) where (hi,a2) is a non-degenerate simplex of A(r) x A(s). Then f ( x ) = ( f l ( x ) , f 2 ( x ) ) where f l ( x ) e ]A(r)l = Ar has non-degenerate representative (A(al), ( S a l ) . ( P ) ) , and f2(x) e As has non-degenerate representative ()~(a2), (Sa2). (P)). On the other hand, suppose xl e IA(r)l, xs e ]A(s)l have nondegenerate representatives (al, PI) and (as, P2), respectively. Let ax be an g-simplex, as an m-simplex, P1 = ( s o , . . . , st) E ~ , P2 = ( t o , . . . , tin) E A ~ Let uo < -'- < uk be the sequence obtained by putting the distinct elements of
{~0 + . - . + ~}0<~<~ u {to + . - . + tj}0<j<~ in increasing order. Let Vo = uo, Vl = u l - U o , . . . , vk - Uk -- Uk-1. Then P = ( v 0 , . . . , v k ) E A~. Let Pl < "'" < Pk-e be the integers p such that up tg {so + . . . + Si}o
o~ = a ( s ) ( S ~ ,
o...o .... ) o . . . o
A(~)(S~,)(o~) a(s)(S~,)(o2).
By construction, the indices p, q run over disjoint sets, so t h a t a is a nondegenerate k-simplex of A ( r ) x A(s). Define g ' l A ( r ) l x I A ( s ) l ~ [A(r) • A(s)l by g ( ( a l , P1), (a2, Ps)) = ((01,02), P )
where 0~, P are given by the above formulas. By definition, A(O~) = a~, ( S a ) . ( P ) = P~, i = 1,2. Hence f o g . l A ( r ) l x IA(s)[ : -IA(r)[ x [A(s)] is the identity. By a straightforward calculation, one checks (left to the reader) t h a t g o f is the identity on IA(r) x A(s)l. Hence f is a continuous bijection of compact, Hausdorff spaces, i.e., is a homeomorphism. This proves (A.55).
Appendix B Results from Category Theory A small category is a category with a set of objects. Thus, if r is a small category, we have a set Ob(7 of objects, and a collection MorC of sets indexed by Ob (7 x Ob (7, MorC = {Morc(A,B) I ( A , B ) e O b C x ObC}, which satisfy the usual axioms for a category. Let C, 79 be small categories, F , G functors from (7 to 79. We say t h a t , G such F is isomorphic to G if there is a n a t u r a l t r a n s f o r m a t i o n T : F ~. G(C) is an isomorphism. Clearly t h a t for each C E Ob(7, T(C) : F ( C ) isomorphism of functors is an equivalence relation. A functor F : C ----, 79 is called an equivalence of categories if there is a functor G : 79 .~C such t h a t the two composite functors G o F : C ~ C, F o G : 79 ,79 are isomorphic to the identity functors of C,/:) respectively. L e m m a ( B . 1 ) . Let F : C ~ 79 be a functor between small categories C, D. Then F is an equivalence of categories ..t---e. i) every object of l) is isomorphic to an object of the form F ( C ) , for some C E ObC, and ii) for any pair of objects C1, C2 E ObC, F . : Homc(C1, C2) ---* H o m v ( F ( C 1 ) , F ( C 2 ) )
is bijective.
P r o o f . (=~) Let G : 29 , C be a functor such t h a t there exist isomorphisms of functors r/c : lc ~ G o F , r/v : l v ~ F o G. T h e n given D e Ob 79, we have an isomorphism fly(D) : D ~. F o G(D). Hence i) holds. Next, given C1, C2 E Ob(7, there is a d i a g r a m
Morc(C1,C2)
F.
~. Morz~(F(CI),F(C2)) ~ identity
Morc(GF(C1),GF(C2)) ~
Morc(C1, C2)
Hence F . is injective on morphisms. A similar a r g u m e n t shows t h a t G . is injective. Hence, in the above diagram, all the arrows are injective, while the composite "identity" is surjective. Hence F . , G . are also surjective.
Appendix B- Results from Category Theory
277
( ~ ) For each D E Ob :D, consider the (non-empty) set (C, T) are small categories) X(D) = {(C,r
I C E ObC, r
D
~. F ( C ) an isomorphism}.
By the axiom of choice, we can choose one such pair (CD, r for each D E Ob :D. We define a functor G 9 :D ~ C by letting G ( D ) = C o ; if D 1 , 0 2 E Ob :D, r ) 0 2 a morphism, let G(r = F.-1 (r ), where F. " Homc(CDI, Co2) ~. H o m v ( F ( C D , ) , F ( C D 2 ) )
O00r
is the bijection given by ii). Clearly G is a functor, and fizz(D) = CO gives an isomorphism of functors lz~ - - ~ F o G. Given C E O b C , we have an isomorphism CF((::) " F ( C ) ~ F(CF(c.)). Since F. " H o m c ( C , CF(C)) .... ; Homz~(F(C),F(CF(a))) is bijective, we have an arrow t i c ( C ) " C -; CF(C), where CF(C) = G o F ( C ) . One verifies at once t h a t ~7c " l c ~ GoF is an isomorphism of functors. This proves (B.1). We say t h a t a category C has a set of isomorphism classes of objects if there exists a set S, a collection of objects of C indexed by S, (C, E ObC [ i E S}, and for each C E O b C we are given an index i E S and an isomorphism. r " C , Ci. Let T) be the full subcategory of C with the Ci as objects. T h e n there is a functor F " C ~ T~ given by F ( C ) = Ci, where i E S is associated to C as above, and given an arrow u " C ~. C', we let F ( u ) = r ou or T h e n if G " T) , C is the inclusion functor, clearly F o G 9 T~ ~ T), G o F 9C , C are isomorphic to the respective identity functors. Hence G 9 T) ; ~: is an equivalence of categories. In particular, C is equivalent to a full small subcategory. If C is equivalent to two small full subcategories T~I, T~2 then clearly ~)1 and :D2 are equivalent. Any of the categories we need to deal with (below and in t h e m a i n text) are either small, or have a set of isomorphism class of objects, and hence are equivalent to small full subcategories. In the latter situation we often tacitly assume t h a t a given category has been replaced by an equivalent small full subcategory. ( B . 2 ) A b e l i a n C a t e g o r i e s . A category C is called an additive category if it satisfies the following conditions: i) there exists a 0-object 0 E O b C (i.e., 0 is both an initial and a final object) ii) for any M, N E O b C , the direct sum and direct p r o d u c t exist iii) H o m c ( M , N ) has a structure of an Abelian group, for any M , N E C, such t h a t composition of morphisms in C is bilinear. Let C be an additive category. An arrow g" K kernel of an arrow f 9M ~ N if for any arrow h 9P
, M is said to be a ; M with f o h = 0,
278
Algebraic K-Theory
there exists a unique arrow k : P , K with h - g o k. Equivalently, for any object P of C, we have an exact sequence of Abelian groups 0 - - . n o m c ( P , K) g'~ n o m c ( P , M )
f', nomc(P, g).
If a morphism f has a kernel, it is unique up to unique isomorphism. Similarly, an arrow gl : M , C is a cokernel of f : M , N if for any object P in C, we have an exact sequence 0
.~Homc(C, P ) g;, H o m c ( N , P )
f*~ Homc(M, P ) .
Suppose every morphism f in C has a kernel and a cokernel. Let ker f , coker f denote the kernel and cokernel respectively (we fix one representative for each arrow within the respective isomorphism class). Let im f = ker(coker f), coim f = coker(ker f ) be the image and coimage of f , respectively. By the universal properties defining kernels and cokernels, there is a canonically defined arrow coim f , im f. An additive category C is called an A belian category if C also satisfies i) every morphism has a kernel and a cokernel ii) for any morphism f, the canonical arrow coim f morphism.
" im f is an iso-
Note t h a t condition ii) does not depend on the choices of the various kernels and cokernels within their isomorphism classes. If ,4 is a small Abelian category, we will assume below and in the main text t h a t fixed choices have been made of representatives within the isomorphism classes of arrows representing the kernel and cokernel of any given arrow f : M ~ N. These representatives will be called the kernel and cokernel of f, respectively. Similarly, consider monomorphisms (arrows with kernel a 0-object) f : M , N. For fixed N, consider the isomorphism classes of pairs (M, f ) , where (M, f ) ~ ( M ' , f ' ) if there is an isomorphism g : M ~ M ' making M g
L
f N
MI
commute. Within each such isomorphism class of pairs, we fix a choice of one pair, and we call the collection of chosen pairs the set of subobjects of N. Similarly we fix choices of all quotient objects of each object M E ,4. T h e collection of subobjects of an object N forms a partially ordered set in a natural way, with respect to the partial order (denoted C)
Appendix B: Results from Category Theory
279
if and only if there is a commutative diagram
M N
1% for a (necessarily unique) monomorphism j. If i:N'
}N
is a monomorphism, then for any subobject ii :N1
~N'
there is a unique subobjec[ of N isomorphic to f oil :N1---*N.
This gives an injective order-preserving map from the subobjects of N ' to that of N. In considering the partially ordered set of subobjects of a given object N, we identify the partially ordered set of subobjects of any given subobject i 9 N ' ; N with its image under the canonical order-preserving injection; loosely speaking, "a subobject of a subobject is a subobject." Similarly, when we speak of "layers" of length r NI
} N2
,."
' Nr
of subobjects of an object N, we mean the corresponding linearly ordered chain in the partially ordered set of subobjects of N. A filtration by subobjects has a similar meaning. Let ij = N j } N , j = 1, 2 be subobjects of N. We define NI NN2 N and NI + N2 " N as the unique subobjects isomorphic to ker(N
} N/N1 @ N/N2)
and im(N1 ~B N2
; N)
respectively. These have the standard properties. Finally, given any morphism f 9M ; N, there is an order preserving map from the partially ordered set of subobjects of M to that of N, induced by ( M ' - ~ M ) : - i m ( f 0 i). Similar remarks apply to the partially ordered set of quotients of a given object. (B.3) Q u o t i e n t A b e l i a n C a t e g o r i e s . Let B C ~4 be an inclusion of an Abelian category B as a full additive subcategory of a small Abelian
280
Algebraic K-Theory
category ,4. In particular, the inclusion functor is assumed to preserve finite direct sums, 0-objects and the Abelian group structure on Hom-sets. T h e n B is called a Serre subcategory (or an dpaisse subcategory, or thick subcategory) if i) any object of A isomorphic to an object of B lies in B ii) B is closed under taking subobjects, quotients and extensions in ,4, i.e., if 0 ~ M ' -~ M --, M " --~ 0 is exact in A, then M E B r M r, M " E B. If B C ,4 is a Serre subcategory, we construct below the quotient Abelian category .A/B. Our t r e a t m e n t follows the article by P. Gabriel, Des categories abdliennes, Bull. Math. Soc. France 90 (1962), 323-448. T h e category A / B has the following description:
i) Ob A / B
=
Ob,4
ii) let M, N E Ob A = Ob A/13, and let M r c M , N ' E N be subobjects such t h a t M / M r E B, N ' E B. There is a natural h o m o m o r p h i s m H o m A ( M , N) , H o m A ( M r, g i N ' ) . As M ' , N ' range over such pairs of subobjects, the groups
HomA(M',N/N') from a directed system of Abelian groups, and we define
HomA/B(M, N) =
lim
HomA(M', N/N').
(M~,N ~)
One easily verifies t h a t the composition law for morphisms o f , 4 yields a bilinear composition law for morphisms of .A/B, so t h a t .A/B is an additive category. Let T " .4 ~ .A/B be the canonical additive functor. L e m m a ( B . 4 ) . Let u E H o m A ( M , N ) . Then T u is null ~ imu E Ob 13. Similarly T u is a monomorphism ~ ker u E Ob B, and T u is an epimorphism ~ coker u E Ob B. P r o o f . If im u E Ob B, then u ~-, 0 in H o m A ( M , N / i m u) so t h a t u ~-, 0 in H o m A / 6 ( M , N ) (which is the direct limit). Hence T u = O. Conversely if T u - O, then for some pair M r, N ' of subobjects of M, N respectively, with M / M r , N r E O b B , the image of u in H o m A ( M ' , N / N r) vanishes. T h u s u ( M ' ) c N ' so t h a t u ( M ' ) e O b B . Also M / M ' E O b B . Hence M / ( M ' + ker u) E Ob B. We have an exact sequence
0
, u(M')
, im u
, M / ( M ' + keru) ~
with extreme terms in Ob B, so t h a t im u E Ob B.
0
Appendix B: Results from Category Theory
281
Next, we show T u is mono ~ ker u E Ob B. Suppose T u is mono. Let i 9 ker u , M be the canonical arrow; then u o i - O, so ( T u ) ( T i ) - O. Since T u is mono, T i - O. Hence by the first part of the lemma, proved above, im i -~ ker u is an object of B. Conversely, suppose ker u E O b B. To prove T u is mono, it suffices to show t h a t if f 9 T P ~ T M is any non-null arrow in . A / B , t h e n T u o f 9 T P ~ T N is also non-null. Now f is induced by some arrow f " P ' ~. M / M ' in A, where P I P ' , M ' E Ob B. Since ker u E O b B, we m a y assume without loss of generality t h a t ker u c M ~. T h e n u induces a m o n o m o r p h i s m u ' ' M / M ' ~ N / u ( M ' ) . Since f ~ 0, imf r hence i m ( u ' o f ) r H e n c e T ( u ' o f ) ~ O. W h u s T u o f ~ O. m
m
T h e proof t h a t T u is epi ~
coker u E Ob B is similar.
L e m m a ( B . 5 ) . Let u E H o m A ( M , N ) , i " K p " N - ~. C the cokernel, o f u. T h e n T i " T K Tp" TN ~ T C a cokernel, o f T u .
~. M the kernel, a n d ~ T M is a kernel, a n d
P r o o f . Clearly T i is mono, from (B.4). Let f " T X ~ T M such t h a t T u o ] = 0. T h e n j~ is induced by some f E H o m A ( X ~, M / M ~) w h e r e X / X ~, M ~ E Ob B. We have a diagram with exact rows 0
0
,
~
,
K
i
I
K/KAM'
.~
M
,
M/M'
i~
u .~
I
,
N
I
N/u(M')
u ~
where u', i t are induced by u, i respectively. Since T u o f = O, T ( u ~o f ) = O, and so i m ( u ' o f ) E O b B . Let X " = ] - l ( i m i ' ) ; then X ' / X " ~- i m ( u ' o f ) , so t h a t X ~ / X ", X / X " E Ob B. The restriction of f to X " factors t h r o u g h amorphismg'X" -~. K / K A M ~ since u ~ o f i X , , = O, a n d i ~ = k e r u ~. If E H o m A / B ( X , K ) is the m o r p h i s m determined by g, then T i o D = f . T h e s t a t e m e n t a b o u t cokernels is proved similarly.
L e m m a ( B . 6 ) . Let u E H o m A ( M , N ) . ker u, coker u E Ob B.
T h e n T u is an i s o m o r p h i s m < >
P r o o L Clearly T u is an isomorphism ::~ T u is mono and epi :=~ ker u, c o k e r u E O b B by (B.4). Conversely, if keru, cokeru lie in O b B , t h e n let q : M .~ coim u, j : im u ~. N be the canonical epi and m o n o ,4, respectively, a n d let ~ : coim u .~ im u be the canonical isomorphism. T h e n we have a d i a g r a m
282
Algebraic K-Theory M
u
coim u
0
N
.~ im u
So it suffices to prove Tq, T j are isomorphisms. Now coim u - coker(ker u) = M / i ( g ) , where i : K , M is the kernel, with K E Ob B. The identity map of coim u is thus an element of nomA(coim u, M / i ( g ) ) , and hence gives an element ~ E HomA/B(coim u, M). Clearly ~ is the inverse of Tq; hence Tq is an isomorphism. The proof that T j is an isomorphism is similar. P r o p o s i t i o n (B.7). A / B is an Abelian category, such that T " ,4 is an exact functor.
, A/B
P r o o f . Let f 9T M ~.T N be a morphism in A / B . We must show that f has a kernel, cokernel, coimage and image, and that the natural morphism from the coimage to the image is an isomorphism. Let f be induced by f E n o m . a ( M ' , g / Y ' ) , where M / M ' , Y ' E Ob B, and let i " M ' .~ M, q" N ~.N / N ~ be the natural mono and epi (in ,4) respectively. Then we have a diagram in ~4/B
TM
TM'
Y'~
TN
;
T(N/N')
T!
where by (B.6), Ti, Tq are isomorphisms. But by (B.4), (B.5) T I has ~ k ~ nel and a cokernel, namely T(ker f ) and T(coker $). Similarly T(coim f ) , T(im f ) are a eoimage and image for T f , respectively. Since the natural map T ( c o i m J ' ) . , T 0 m J ' ) has the form Tg, where g " c o i m f . , i m f is the natural map, and since 9 is an isomorphism (as .,4 is an Abelian category), Tg is an isomorphism. Hence A / B is an Abelian category. The exactness of T follows at once from (B.4) and (B.5). E x a m p l e (B.8). Let R be a Noetherian commutative ring, S c R a multiplicative set, ,4 the Abelian category of finitely generated R-modules, B the full Serre-subcategory of S-torsion R-modules (thus M E B ~ for some s E S, s M = 0). Let C be the category of finite S-1R-modules, L : .4 ; C the localization functor L ( M ) = S - 1 R | M. We claim there
Appendix B: Results from Category Theory
283
is an equivalence of categories U 9 .A/B ~ C such that U o T and L are isomorphic functors ,4 ." C. To see this, we first note that there is a natural isomorphism of rings R ~ Hom.4(R, R) (where the multiplication on the latter is given by composition). For any M E Obr HomA(R, M) is a module over HomA(R, R) in a canonical way, and is naturally isomorphic to M as an R-module (under the isomorphism R ~ Hom•(R, R)). Now consider the ring Hom.A/B(TR,TR). There is a canonical ring homomorphism
R ~-~ HomA(R,R) --~ Hom.4/B(TR, TR). If s E S, multiplication by s gives a map r :R , R whose kernel and cokernel lie in B. Hence T, o r is an isomorphism. One easily verifies from the definitions that Hom.A/B(TR,TR) is commutative; now by the universal property of the localization homomorphism R ~ S-1 R, the map T. o r 9R -----4 Hom.A/B(TR,TR) factors uniquely through R - ~ S - 1 R . One can now verify that S - 1 R -----, Hom4/B(TR, TR) is an isomorphism. For any M E Ob j4, we have a module Hom.A/~(TR,T M ) over the ring Hom.A/6(TR,TR). Using the above isomorphism of rings we can regard Hom.A/B(TR,T M ) as an S-1R-module. Further there is an R-module homomorphism M
---. ~-
Homjt(R, M)
By the universal property of M
T.
~. HomA/B(TR,TM).
~ S - 1M = S - 1 R |
M, there is a
unique S - 1R_module map S - 1M CM Hom.A/~(TR,T M ) compatible with the above map. One can check that this is an isomorphism. Now define U :.A/B ; C by sending T M to the S-1R-module Hom.A/B(TR,T M ) , and define
U, 9HomA/B(TM,T N ) ---, Homc(Hom.A/~(TR,TM), Hom.A/n(TR,T N ) ) to be the map induced by the composition law for morphisms in .A/13. Then the CM define a natural transformation r 9L ---, U o T which is an isomorphism of functors. E x a m p l e ( B . 9 ) . Let X be a Noetherian scheme, Z C X a closed subscheme, U = X - Z, ,4 = Abelian category of coherent Ox-modules, B = Serre subcategory of modules supported on Z, C = Abelian category of coherent O r - m o d u l e s . Let j " U ~ X be the inclusion. Then there is an equivalence of categories F 9 ..4/13 ~ C and an isomorphism of functors j* ~ F o T 9 ,4 ; C. The proof is left to the reader.
284
Algebraic K-Theory
( B . 1 0 ) A d j o i n t F u n c t o r s . Let Set denote the c a t e g o r y of sets. Given a functor F " C , D between categories, (M,N): gives a functor D ~ • C have a functor D ~ x C
; Homv(M, FN)
." Set. Similarly given a functor G " 19 ; Set,
(M, N ) : A pair of functors F 9C two functors above ~)op x bijections
C
~C, we
" Homc(GM, g).
, 19, G 9D , (7 form an adjoint pair if the ~ Se__!tare isomorphic, i.e, if we have n a t u r a l
H o m c ( G M , N) ~ H o m v ( M , F N ) . In this situation we also say t h a t G is a left adjoint of F , and F is a right adjoint of G. Suppose F " C ---, /9 has a left adjoint G 9 / ) ; (7. T h e n given M E Ob 19, we have a bijection
H o m c ( G M , G M ) "~ H o m v ( M , F G M ) . In p a r t i c u l a r the identity m a p of G M gives an arrow iM " M ~FGM. This arrow has the following universal property: given any arrow u 9M F N with N E ObC, there is a unique arrow v 9G M ~ N such t h a t M
iM
;
FGM
FN commutes. Indeed we have a diagram Hornc(GM, GM)
v.[
Homc(GM, N)
--~
Homv(M, FGM)
~
(Fv).
Homz) (M, F N )
where v E H o m c ( G M , N ) corresponds (under the bijection of the b o t t o m row) to u E H o m 9 ( M , F N ) , showing t h a t ( F v ) . i M -- u, which is j u s t the c o m m u t a t i v i t y of the above triangle. Conversely, if v r 9 G M ~N is such t h a t F v ~ makes the above triangle c o m m u t e , t h e n (Fv~).iM -- u, so t h a t v.'(1GM) e H o m c ( G M , N ) maps to u u n d e r H o m c ( G M , g ) ~H o m v ( M , F N ) , i.e., v.'(1GM) = V, i.e., v' = v. Conversely, suppose F " C - - ~ T~ is a functor w i t h the following universal property: given any M E Ob/:), there is an a r r o w i M " M ----, F M
Appendix B: Results from Category Theory universal a m o n g arrows u " M , F N , i.e., given u " M exists a unique v " M , N in C such t h a t M
iM
285 ~ FN,
there
~. F M-
FN
commutes. If C, :D are small categories, then arrow i M " M ~. F M for each M E D. M1 , M2, t h e composite i M2 o w 9 M1 arrow F ~ " F M I ~ F M 2 , for a unique arrow G ( M ) = M , G ( w ) = ~ gives a functor G" D F'C
we can choose a universal Then given an arrow w 9 ; F M 2 factors t h r o u g h an ~" Mt -~M2 in C. T h e n , C which is left adjoint to
,D.
A similar characterization exists for a functor G 9D ---4 C which has a right adjoint, in t e r m s of an appropriate universal p r o p e r t y for maps G M - ~. N , for each N E C. (B.11) Filtering if
Categories.
A small category Z is said to be f i l t e r i n g
i) Ob 2r is n o n - e m p t y ii) given M , M ~ E Z there exists N E Z and a diagram M
M' iii) given M , M ~ ~_ Z and arrows f , g 9 M ~ M ' , there is an arrow h" M ' .~ N in IZ" such t h a t h o f = h o g 9M ~ N. For example, let (S, <_) be a partially ordered set. T h e n we may regard S as a category, w i t h objects given by elements of S , and for any x , 9 E S , H o r n s ( x , y ) is e m p t y unless x <_ y, in which case there is a unique m o r p h i s m x --, y. T h e n S is filtering e==~ S is non-empty and directed. A directed family of sets {Ai I i E I } indexed by Z is a functor Z ~ Se__.ttfrom Z to the category of sets. T h e n lim A i iEI
is defined to be the q u o t i e n t of H A~ modulo the equivalence relation, t h a t iEz
if x E A,, y E Aj, t h e n x ..~ y ~ there exists k E Z and arrows i j ~ k such t h a t (i ~. k ) . ( x ) = ( j , k ) . ( y ) in Ak.
, k,
286
Algebraic K-Theory
A direct family of small categories {Ci I i E I } indexed by I is a functor 2" ---, Cat, the category of small categories. Then i : .~ ObCi is a family of sets indexed by Z. We define lim Ci to be a small category iET
C, together with a natural transformation of functors 2" , Cat from the given functor i ~-~ Ci to the constant functor i ~-~ C, with the following universal property: given any natural transformation between the functor i ~-, Ci and the constant functor i H / ) , for a small category 79, there is a unique functor (7 , ~D such that Ci
\/
-~ C
D commutes for all i E Z. One can easily cheek t h a t lira Ci = t2 exists, with ObC = lira ObCi, and with the following morphisms: given M , N E ObC, we can find an index i E I and objects Mi, Ni ~ Ci such t h a t Mi ~ M, Ni ~ N; if i \ Z denotes the full subcategory of objects j of Z such t h a t there exists an arrow i ~ j in Z, then we let Home(M, N) = whereMi:
l i m Homcj ( M j, N j )
; M j , N i . ~ ~ N j underCi
,Cj.
Appendix
C
Exact Couples
( C . 1 ) T h e S p e c t r a l S e q u e n c e o f a n E x a c t C o u p l e . Let ,4 be an Abelian category. An exact couple in ,4 is an exact triangle D1 ~
D1
E1 T h u s El, Dx E O b , 4 , and we have a long exact sequence in ~4 9..---} E1
----' al
D1
bl
-~ D1
cl
~ Ex
----' al
D1 - " . . .
Let dl " E1 ~ E1 be the composite dl = cl oal. Then d 2 = Cl oal ocl oal = 0 as al o Cl = 0, by exactness. Thus (El,dx) is a differential group. Let E2 = ker dl/im dl, D2 = im bl =~ D1/ker bl, and define a new triangle
D2 __2z_, D2 E2
with maps i) b2 = bllo2
ii) as is induced b y al]kerdl (this is well defined as al o dl = 0, so t h a t al (ira dx) - 0) iii) cl (ker bx) = cl (im a l ) = im dl, so t h a t cx induces a m a p D 1 / k e r bl ---, , E2. The reader can easily verify ker dl/im bl, i.e., a m a p c2 : D2 t h a t the new triangle is exact (c.f. [W], Ch. XIII). This exact couple is called the (first) derived couple of the original exact couple. We can iterate this construction, and by induction define (for n > 2) the ( n - 1)th-derived couple
Dn
b,,
E.
~. Dn
288
Algebraic K-Theory
to be the derived couple of the (n - 2)th-derived couple (where the 0thderived couple is the original one). The sequence of derived couples associated to the given exact couple is called the spectral sequence of the couple. One computes by induction that for any n _> 1, Dn = im b]'-1 Dx/ker b~- 1, En = a i- x(im b7-1)/cl (ker b7-1) a n d the maps an, bn, an are given by i) an(x + c l ( k e r b ? - l ) ) = al(x) for x e a]-l(imb? -1) ii) bn(y) = bl(y) for y e imb~ -1 iii) cn(b'~-l(y)) = el(y) + cl(kerb~ -1) for y e D1. Here and below we follow the convention that b~ is the identity map of D1. Let Zn = a i- l(im b~- 1), Bn = cl (ker b~- 1). Then
E1 =
Z1 Z) Z2 ~ Z 3 . . "
DBn D"'B2
[) Z n D Z n + l D " "
D Bn+l
DB1--0,
and En - Z n / B n . Define
U-.
z =nz~ n>l
n>l
(if they exist). Then 0 C Boo C Zoo c El. Define Eoo = Zoo~Boo, which we call the limit term of the spectral sequence. The exact couples which we need to consider are bigraded, and are all obtained by the following construction (or its version for homology, see below). Assume given two collections of objects of .A indexed by Z • Z {Am'n}(m,n)ez•
{Em'n }(m,n)ez•
and suppose that we are given long exact sequences for all p E Z __.
Av+ 1,q-- 1
~ f p,q
Av,q
___. gl',q
Ep,q
___. hv,'i
Av+ l,q
~
fv,q+z
Ap,q+ 1
We can combine all of these exact sequences into an exact couple D1
b~
~.
Dt
E1 by setting D1 = (~p,q A p'q, E1 -- (~p,q E p'q, so that D1, E1 are bigraded with D~ 'q -- A p'q, E[ 'q - E p'q, and defining maps
al - (~hp,q,
bt
"-- ( ~ ) f p - - l , q + l ,
C1 ----(~9p,q.
Thus, with the given bigradings, a l has bidegree (1, 0), bl has bidegree ( - 1 , 1) and cl has bidegree (0,0). Thus, from the formulae given above
Appendix C: Exact Couples
289
for the terms and maps in the ( n - 1)st-derived couple, we see t h a t Bn, Zn C E1 and Dn C D1 are graded submodules, so t h a t Dn, En = Z n / B n have natural bigradings with respect to which an has bidegree (1,0), bn has bidegree ( - 1 , 1) and an has bidegree (n - 1, 1 - n). T h u s dn = an o an has bidegree (n, 1 - n ) , so t h a t dn has (p, q)-component dPn'q 9 En~,q ----, nn+n'q-n+l A bigraded exact couple with the above choices of gradings gives rise to a spectral sequence of cohomological type, characterized by the fact that En, Dn are bigraded and the maps an, bn, cn, dn have the above bidegrees. Next, we discuss convergence of the spectral sequence. Consider the directed system indexed by q E Z ...
__, A n - q , q
__._,
- 1,q+l
An-q
}
f ,, - ,a - t ,,1+ t
An-q-2,q+2
..~ ...
I,~ - ,a - 2 , ,1+ 2
and let A n = lim A n-q,q. T h e r e is a decreasing filtration FPA n = i m ( A n'n-n
~.An),
p e Z, Fn+lA n c FPA n.
We make two further assumptions about our data, which will be valid in all our applications: i) for each n there is an integer ql (n) such that fn - q,q A n-q'q is an isomorphism for q > ql (n)
9 An-q+l,q-I
ii) for each n, there is an integer qo(n) such that A n-q'q = 0 for q < qo(n). T h e n the above filtration on A n is finite, for each n, and we claim t h a t there are natural isomorphisms for any p, q and for n = n(p, q) sufficiently large E~'q ~ E~,q ~ F P A ~+q / F p+ t Ap+q - - - -
o
In this situation we say t h a t the spectral sequence converges; { A n } n e z is called the abutment of the spectral sequence, { F n A n } n e z the filtration induced by the spectral sequence, and we write E~ 'q =,, A n+q (or E p,q =~ A n+q ) to express the fact t h a t {An } is the abutment. To prove the claimed isomorphisms, we note that the exact triangle Dn
b,~ ~. Dn
yields an exact sequence (taking into account the bidegrees of an, bn, c~) ...--4
D~ -n+2,q't-n-2
--~ D~ -n+l,q't'n-I
~
E ~ ,q ~
D~ "t-l'q ~
D~ 'q+l
.--~...
290
Algebraic K-Theory
Now a s s u m e p, q are fixed and n is sufficiently large. T h e n DPn - n + 2 ' q + n - 2
= D~ -n+2'q+n-2
CI
= im(D~ +l,q-1
imb~ -a
~ D~-n+2,q +n-2)
im(D~ +1'q-1 __~ An+q) = Fp+IAp+q DPn- n + l'q+n- I ---- " lI')P-nT l ' q + n - 1
_-im(D~,q
f"l
imb~ -1
~ ; D ~ - n + l , q + n - l ) ,~ FPAp+q
and t h e m a p ~Dn p-'~+2'q+n-2 - - - , D~ -'~+l'q+n-1 is identified with t h e inclusion F p+ 1Ap+q C F P A p+q. Further Dr+ 1,q = D~+ 1,q CI im b~- 1 = im (D~ +n'q-
n-4-1
~
D~+ 1,q)
=0 since D~ +n'q-n+l - 0 if n > q + 1 - q o ( P + q + 1). Hence t h e c l a i m e d i s o m o r p h i s m F P A p + q / F p+ 1Ap+q ,~ EPn,q follows from the above exact sequence, provided n is sufficiently large. To see t h a t E ~ 'q '~ E ~ q we see t h a t for sufficiently large n, we have a c o m p l e x of E n - t e r m s , whose h o m o l o g y is P,q En+l,
EPn-n,q+nB u t EI~+n'q-n+l _- sequences
0
1
p-n+l,q+n L
d,,
EPn, q . ~
EPn+n,q_n+ l
*"1 = ~.p-n,q+n-1 for sufficiently large n, from t h e e x a c t
D~+n,q-n+
D
I
~. E [ + n , q - n +
....
~. D P - n + l , q + n - 1
I .____~. D ~ + n + 1 , q - n + 1
-
12p-n,q+n- 1
i-~p--n,q+n ..... ~..t.., 1 f l ' - n,qTn
and t h e facts t h a t r~p+n,q-~+l _ r ~ n + = - l , q - - + l _ O, and fv-,~,q+n-X, J'p-n,q+n are isomorphisms, for sufficiently large n. T h u s /~P,q
q l ~"~ "'" ,~ "~ E pn, + --" E ~ q "., -- FVAp+q/FP+IAv+q
~-~
for a n y given p, q and sufficiently large n. We c o m p u t e explicitly, in t e r m s of the original d a t a of a family of long e x a c t sequences, the differentials d l of the spectral sequence. W e have , AV+l,q " Next Cl{A~+~." Er~ 'q = E p'q, and al IE,v,q is just h v ' q 9 E v'q is t h e m a p gv+l,q " Av+l,q
_ ~ Ep+l,q. T h u s dl 9 E [ 'q ----* E [ +s'q is t h e
Appendix C: Exact Couples
291
dashed arrow in the diagram below: . . . _.4
Ap,q
~ Ep,q
. . . --~Ep+l,q-1
~ An+ x,q
.... ~ Ap+2,q-1 ~
An,q+ l
~
Ap+l,q ..... ~.Ep+l,q
En,q+ l
~
...
~ Ap+2,q --~ . . .
In an analogous fashion one can construct spectral sequences of h o m o logical type from the following data: assume given objects {Am,n}(m,n)ezxz,
{Em,n}(m,n)ez•
together with long exact sequences for each p E Z 9.. --~ Ap-l,q+l --~ Ap,q --4 Ep,q --~ Ap-l,q --* Ap,q-1 ---* . . . . Then one can combine them into an exact couple of bigraded objects DI
b~ .~ D1
E1 with E1 ~p,q Ep,q, D1 = ~p,q Ap,q, bigradings Elp,q : Ep,q, Dlp,q : Ap,q, and maps a l, bl, cl constructed from the maps in the exact sequences in a natural way. Thus al has bidegree ( - 1 , 0), bl has bidegree ( 1 , - 1 ) and Cl has bidegree (0, 0). T h e n the ( n - 1)th-derived couple ----
Dn
b, .~ Dr,
E. consists of bigraded objects En = ~ E p n, q , Dr, = ~Dn~ q such t h a t a n , bn, c~, dr, respectively have bidegrees ( - 1,0), (1, - 1), (1 - n, n - 1), ( - n , n - 1). In particular dn has (p, q)-component d~'q'Enn, q ~ Enn-n,q+n- 1. We have subgroups B,~q C Zpn,q C E~,q such t h a t n n+l Up,q C Bp,q
n+l C Zp,q
n C Zp,q
for all n, p, q. Let =
n
--p,q
t~p,q
;
n
then Eco = S E ~ , q is the limit term of the spectral sequence. T h e spectral sequence will converge if we assume t h a t for each n E Z, there are integers p o ( n ) , p l ( n ) such t h a t (i) A p , n - n , A n + l , n _ p _ l is isomorphism for p > pl(n), and (ii) A n , n _ n = 0 for p < p o ( n ) . Then if A n = l i m A n , n _ n, F p A n = im(Ap,n_;, --at p
; AT,),
292
Algebraic K-Theory
{A,~},ez is the a b u t m e n t of the spectral sequence, {FpAn}pez is the induced filtration on the a b u t m e n t , which is a finite increasing filtration of An, for each n. Further, given p, q we have isomorphisms for all sufficiently large n E ~ ,q "- - E ~= ~ p n+l ,q
...
= E p~176 "" ,q ~ - " FBA p + q / F p
- 1Ap+q
We write E~,q ~ Ap+q ( o r Epn,q ~ Ap+q) to denote t h a t the a b u t m e n t is { A p + q } , and E~,q = F p A p + q / F p _ l A p + q . We discuss two examples below. ( C . 2 ) T h e B G Q S p e c t r a l S e q u e n c e . This is the spectral sequence of T h e o r e m (5.20) of the main text. It is a spectral sequence of cohomological type, with AP'q= K_p_q(.MP(X)),
~ K_p_q(k(x)),
EP'q=
xEX"
and the family of long exact sequences . . . Ap+ i,q- 1
~
Ap,q
~ Ep,q
.
Ap+ 1,q
is taken to be the family of localization sequences
...K_p _ q( . M
p+l
(X)) ~
K_p_q(A4P(X))--+ ~]~ K_p_q(k(x)) xEXn ; K_l_p
_ q (A4n+' (X))
, ....
Here, we make the conventions t h a t fl4P(X) -- fl4(X), X p -- X ~ for p < 0, and K n ( . A 4 P ( X ) ) - 0 for n < 0, p E Z. We have A p,q - 0 unless p d-q < 0, and A p'q '~ A p - l ' q + l ~ K _ p _ q ( . A 4 ( X ) ) for p < 0, q E Z, so t h a t the a b u t m e n t terms are { K - n ( . A 4 ( X ) ) } n e z ; the filtration on K _ n ( . A 4 ( X ) ) induced by the spectral sequence is F P K _ n ( . A 4 ( X ) ) = F P A n - - i m ( A p'"-p
= im(K_,(A4P(X))
, A) , K_n(.A4(X))
i.e., is the topological filtration on K _ , ( . A 4 ( X ) ) . Finally, if X has Krull dimension d, then fl4P(X) - 0 f o r p > d, so A p'q - 0 for p > d. Hence A n-q'q - 0 for q < n - d, i.e., the spectral sequence is convergent.
( C . 3 ) T h e Spectral S e q u e n c e o f a F i l t e r e d C o m p l e x . Abelian category, and let C ~ be a cochain complex in ~4; C~
9
0
~ C 0 d~ ~ C 1 dl
~
Let ~4 be an
C 2 d2 ~ . . . .
Assume given for each n _> 0 a finite, decreasing filtration C" = FeC " D F1C " D ... D F'nC "=(0),
m=m(n),
Appendix C: Exact Couples
293
such t h a t d n ( F P C n) C F P C ~+1 Vp, n > O, so that F p C ~ 90 ----, F p C o ~
F pC1
~ ...
is a subcomplex of C ~ Let grPF C~ be the quotient complex ; F pC O/Fp+ 1C O
grPF C ~ 90
~ F pC 1/F~+ 1C 1 .__+ . . .
so t h a t we have a short exact sequence of complexes 0 ---, F p+IC ~
~ FPC .
~.grPF C .
~ O,
giving rise to a long exact sequence of cohomology objects (with H " ( ) = 0 forn <0) --+ H n ( F P + I C ~ ---, H n ( F P C ~ --+ H"(grPF C~ --, H n + l ( f p + l c
~ --+
Thus if we define F P C ~ = C", grPF C~ -- 0 for p < 0, the above family of long exact sequences yields a bigraded exact couple D1
\/
.~ DI
E1 with A e,q = D~ 'q = H e + q ( F p C ~ E mq = E~ 'q = H p+q(grpF C ~ and a b u t m e n t A n = l i m A "-q'q = l i m H n ( F " - q c *) = H " ( C ' ) , and induced filtration
'-+'
,--2
F P A n = i m ( A p'n-n
; A") = im(Hn(FnC ~
; H~(C~
= Fngn(c~
say. Thus we have a convergent spectral sequence of eohomological type E~ 'q
=
HP+q(grPF C~ - - ~ H , + q ( c ~
and limit terms E ~ q = grPF H p+q ( C ~
Similarly, given a chain complex in .A
Co 9 . . . C 2
d2 C1 d , Co
; O,
and finite increasing filtrations O = FoCn C F1Cn c . . . C F m C n = Cn,
m-re(n),
such t h a t d n ( F p C n ) C F p C n _ , V n, p, we let g r F C ~ be the complex grFC~ ....-4
F,C:/F,_IC2
-, FpC1/F,_,C,
--* F p C o / F , - , C o
--, 0
for each p e Z, where we let FpCn = 0 for p < 0, FpCn = Cn V p > re(n). T h e n we obtain a convergent spectral sequence of homological t y p e (obtained from a suitable bigraded exact couple) El,q = H p + q ( g r ; C ~ E~,q = grpF gp+q (Co).
::~ Hp+q(C,), with limit terms
294
Algebraic K-Theory
The spectral sequence of a filtered complex includes as special cases the usual spectral sequences encountered in homological algebra (see Caftan and Eilenberg's book Homological Algebra, for example), e.g., the spectral sequences of a double complex, and the Grothendieck spectral sequence for the derived functors of a composite; thus these spectral sequences can be regarded as arising from suitable exact couples.
Appendix
D
R e s u l t s from Algebraic G e o m e t r y
A general reference for algebraic geometry from the point of view most useful to us is
tHI
R. Hartshorne, Algebraic Geometry, Grad. Texts in Math. No. 52, Springer-Verlag, New York (1978).
We omit most of the proofs in this appendix, but they can be found in [H] (or [n] cites an appropriate reference). Sheaves Let X be a topological space. The open sets in X are partially ordered by inclusion, hence may be regarded as a category Tx. (D.1) A presheaf of sets on X is a functor T)~p ~ S e t , where S e t is the category of sets. Thus if ~ is a presheaf on X, then for each open set U C X, we are given a set ~ ( U ) , and for any smaller open subset V C U, a restriction map Puv : ~-(U) --+ ~'(V), such that Pvw o Puv = Pvw for W C V c U. Elements of ~ ( U ) are called sections of ~ over U; if U = X, they are called global sections. We sometimes also use the notation F(U, ~') instead
ot
Morphisms of presheaves are just natural transformations of functors. A presheaf ~" is called a sub-presheaf of jc if ~'~(U) C ~'(U) for each U, and the restriction maps for ~-t are obtained by restricting those for ~ . If f : ~" --. G is a morphism of presheaves, then U H (image~'(U)) is a presheaf, which is a sub-presheaf of ~. For example, let A be a set. For any topological space X, let ~'(U) = A for all open sets U, and let Pvv be the identity for all V C U. T h e n ~- is a presheaf on X called the constant presheaf associated to A. (D.2) A presheaf Jr is called a sheaf if for any open set U of X and any open cover {Ua}aeA of U, the following conditions hold.
296
Algebraic K-Theory
(i)
For any sections s,t e Jz(U), if Puu. (s) - Puu~ (t) for all a E ~4, then s = t.
(ii)
Let Ua~ = Us n Uf~ for any a,/~ E ,4; then for any family of sections s~ E ~'(U~), a E A, such t h a t p u ~ u ~ ( s ~ ) = pu~u~(s~) for all a, fl E ,4, there exists a (necessarily unique, by (i)) s E ~'(U) such t h a t Puu. (s) = s~ for all a E A.
(iii)
If U - r is empty, then ~ ( U ) is a 1-point set (i.e., a final object in the category Set).
Morphisms of sheaves are defined to be morphisms of the underlying presheaves. Thus we can make sense of subsheaves of a sheaf. However, if f : ~" --. G is a morphism of sheaves, the image presheaf is not a sheaf in general. (D.3)
T h e stalk ~x of a presheaf ~- at x E X is defined as ~'x = lim ~ ( U ) . U~x
(D.4)
Define
= 1-[
xEU
(If U = r define ~ ( ~ ) ( U ) to be a final object in S e t . ) T h e n ~ ( ~ ) is a sheaf, such t h a t all the restriction maps P u v are surjective; a sheaf with this property is called flasque (or flabby). For each open set U, there is a natural map ~ ( U ) --, I ] x e u ~x, giving a morphism of presheaves jc __~ ~ ( ~ ) . If ~is a sheaf, this is injective, giving an isomorphism of ~ with its image. In general, the image of ~" is a sub-presheaf. Let a(~') be the intersection of all the subsheaves of ~ ( ~ ) which contain the image of ~ (since ~(~-) is one such, the family of subsheaves is non-empty, and clearly any intersection of subsheaves is a subsheaf). If f : ~ --, ~ t is a m o r p h i s m of presheaves, there is an induced morphism of sheaves ~ ( ~ ) --, ~ ( ~ ) compatible with f , and hence a morphism a(9r ) --. a(~'~). In particular, if ~-t is a sheaf, so t h a t ~'~ - , a ( ~ "~) is an isomorphism, we see t h a t f factors uniquely through 5~ --, a(~-). Thus a is a functor from presheaves to sheaves on X , which is left adjoint to the inclusion functor from sheaves to presheaves. We call a(3 c) the sheaf associated to the preshea] :7:. (D.5) A presheaf of Abelian groups (or rings, or modules over a ring ...) on X is a functor from T)~p to the category A b of Abelian groups (or rings, or modules over a ring, ...). It is a sheaf if the analogues of the conditions (i), (ii), (iii) above are satisfied. If ~" is a presheaf of Abelian groups, G(5~),
Appendix D: Results from Algebraic Geometry
297
a ( ~ ) are sheaves of Abelian groups; a similar claim holds for sheaves of rings, modules, etc. In particular, for any Abelian group A, we have the constant sheaf A x associated to A, which is the sheaf associated to the constant presheaf determined by A (discussed earlier in (D.1)). If A is a ring, A x is a sheaf of rings. An important example is the sheaf Z x of rings determined by the ring Z of integers. More generally, we may consider sheaves with values in any category with arbitrary products and finite inverse limits, and which has a final object, since the sheaf conditions may be rephrased using only these notions. (9.6) Let j r be a sheaf of Abelian groups on X, and s 6 ~-(U). Then the support of s is the set I s [ = {x 6 U lsx # 0}, where sx is the image of s in the stalk 9rx. One sees easily that [ s [ C U is closed. We define the support of ~- to be the union of the supports of its sections, which is the set [ 3r [ = {x 6 X [ 9rx # 0}. This need not be closed in general. However, we will see later that this is the case for coherent sheaves of Ox-modules on a 'reasonable' scheme X (see (D.45)). (D.7) Let O x be a presheaf of rings on a topological space X. A presheaf of Ox-modules is a presheaf ~" of Abelian groups together with an Ox(U)-module structure on each Abelian group ~-(U), such that if V c U, then Pug : ~ ( U ) --~ ~ ( V ) is Ox(U)-linear, where ~'(V) is regarded as an Ox(U)-module via the ring homomorphism Puv : O x ( U ) ~ O x ( V ) and the given O x (V)-module structure. If Ox is a sheaf of rings, a sheaf of O x modules is a sheaf of Abelian groups which has the structure of a presheaf of Ox-modules. A sheaf of Zx-modules is just a sheaf of Abelian groups. In another direction, if X = {x}, then all presheaves which satisfy the sheaf condition (iii) are in fact sheaves; a sheaf of rings O x is identified with a ring R (the stalk of O x at x), and the category of sheaves of Ox-modules is identified with the category of R-modules. Convention: whenever we consider sheaves of Ox-modules, we will assume that O x is a sheaf of rings. The category of presheaves of Ox-modules on a topological space X forms an Abelian category in a natural way. The category of sheaves of Oxmodules is a full additive subcategory, which is also an Abelian category; for any morphism f : ~- --, Jr', the sheaf kernel of f is the presheaf kernel, but the sheaf cokernel is defined to be a(coker v(f)) where 'coker p' denotes the presheaf cokernel. In particular, one sees that a sequence 0 -~ Y -~ ~ - ~ ~ " -~ 0 of sheaves of O x - m o d u l e s is exact iff o -0
-.
--.
o
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Algebraic K-Theory
is exact as a sequence of presheaves; this is equivalent to the exactness of o ---,
--.
--,
for each x E X. The category of presheaves of O x - m o d u l e s has direct sums, and direct and inverse limits over directed sets. A finite (presheaf) direct sum of sheaves of O x-modules is a sheaf. The inverse limit presheaf of a directed family of sheaves of Ox-modules is in fact a sheaf, but the direct limit in the category of sheaves of Ox-modules is the sheaf associated to the presheaf direct limit. However there is one case where the presheaf and sheaf direct limits coincide: when the topological space X is Noetherian, i.e., satisfies the descending chain condition for closed subsets, that any strictly descending chain of closed subsets of X is finite. (D.8) We mention some other basic operations on presheaves and sheaves. If f : X --+ Y is a continuous map, and ~ is a presheaf on X, then we can define a presheaf f , ~ on Y by f.Jz(U) = ~ ' ( f - i ( U ) ) . We call f.~" the direct image of ~'. If 5~ is a sheaf, so is f . ~ . If O x is a sheaf of rings on X , then f . O x is a sheaf of rings, and for any O x - m o d u l e ~ , the direct image f.~" is an f.Ox-module in a natural way. T h e direct image functor is left
exact. The direct image functor f . from presheaves (or sheaves) of Abelian groups on X to those on Y has a left adjoint f - 1 called the inverse image functor. On presheaves, it is defined (on objects) by (f-l~)(U)=
lim __...+
v~y(u)
~-(V).
This clearly defines a presheaf on X, and the adjointness property Hom (.t'- 1~.,, .T') ~ Horn (.~", .t',.~') is easily verified. The sheaf inverse image is the sheaf associated to the presheaf inverse image; the adjointness property follows from the adjointness at the level of presheaves, and the adjointness of the 'associated sheaf' functor a. If f(x) = y, then for any presheaf ~" on Y, we have an identification of stalks f - l ( ~ ) z -~ ~-~. In particular, f - 1 is an exact functor. If Oy is a sheaf of rings, then so is f - l O y , and f - 1 takes Oy-modules into f - ] Oy-modules, and converts Oy-linear maps into f - 1 O y - l i n e a r ones. In particular, if j : U ~-~ X is the inclusion of an open subset, we have ( j - t ~ ' ) ( V ) = ~'(V) for any open set V C U. We also denote j - i ~ by Iv- The functor j - 1 from sheaves of Abelian groups on X to those on U has a left adjoint j!, called extension by O, where j.w~" is the sheaf associated
Appendix D: Results from Algebraic Geometry to the presheaf
V~-*
299
{ ~-(V) ifVcU 0
i f V C. U
The sheaf j!~" is characterized by the properties t h a t j-lj:.T" ~- .T" and (j!~')x = 0 for x E X - U . Note that there is a natural inclusion j ! ( ~ Iv) --4 9~ for any sheaves j r of Abelian groups. If i : Z ~-. X is the inclusion of a closed subset, let ~ Iz = i - l ~ ". The functor i. gives an equivalence of categories between sheaves of Abelian groups on Z and the full subcategory of sheaves of Abelian groups 5r on X with Jr I x - z - 0. If Ox is a sheaf of rings on X, let Ov = O x Iv. If Jr, G are presheaves of Ox-modules, define a presheaf of Abelian groups T/Omox (~', G) by the assignment U ~-. H o m o . ( ~ - [ u , ~ [u). If 5r, G are sheaves, so is Ham Ox (.T, ~). If O x is a sheaf of commutative rings, Horn ox (jr, G) is a sheaf of O x - m o d u l e s in a natural way. In particular, if O x is commutative, we have a notion of dual; the dual ~'* of a sheaf ~" of O x - m o d u l e s is
~ o m ox (.T', O x ). If O x is a sheaf of rings, not necessarily commutative, let 0~rp be the corresponding sheaf of opposite rings, so t h a t an O~r is a right Ox-module. For any O~cP-module ~-, and any sheaf ~ of Abelian groups, the sheaf ?'lore Zx (.T', ?l) is an O x - m o d u l e in a natural way, via the action ( s . ~o)(f) = p ( s . f ) for sections s E O~P(U) = O x ( U ) , f E Jr(U) and ~, ~ I - t o m z ,, ( . r , ~ ) ( U ) .
Let O x be a sheaf of rings, ~" an O~P-module. T h e funetor H H ~ o m zx ( ~ , ~/) (from the category of Zx-modules to t h a t of Ox-modules) has a left adjoint G H ~ | G. Thus, by definition, there are natural isomorphisms Hom o• (G, Horn zx (~', ~/)) ~ Hom Zx ( ~ |
~, 7"l),
which characterizes ~" | ~ in terms of the usual universal property for bilinear maps of sheaves ~" x ~ --, 7~. One checks that the sheaf associated to the presheaf U ~ Jr(U) | G(U) satisfies this universal property, so that this defines the sheaf ~" | ~. When O x is commutative, if ~ is also an O x - m o d u l e , then we have a commutative diagram Hom Ox (G, 9iota zx (.T, 7-l))
t Hom Ox (~, Tlom ox (Jr, ~/))
-~, Horn zx ( ~ |
g, Tl)
- - ~ Hom o x (Jr |
x G, ~/)
T
where the vertical arrows are each induced by the natural inclusion of the Abelian group of O x - l i n e a r maps into that of Zx-linear ones.
300
Algebraic K-Theory
In a similar fashion, one may define s y m m e t r i c powers, exterior powers, etc. when O x is commutative. Convention: From now on, we will assume O x is a sheaf of commutative rings, unless explicitly mentioned otherwise. Some statements made below may have generalizations to the non-commutative case; we leave these to the interested reader. (D.9) One way to define a sheaf on a space X is through patching: let {Ui}i~l be an open cover of X, and let ~'~ be a sheaf on U~, for each i, such t h a t (i) for any pair of distinct indices i, j there is an isomorphism
(ii) ~oji = ~o~ 1 (iii) for any 3 distinct indices i, j, k we have ~ojk o~oij = ~oik on Ui n Uj N Uk. T h e n there is a sheaf jr, such t h a t there are isomorphisms ~o~ : j r ]u, ~i compatible with the ~oij; further such an j r is unique up to unique isomorphism compatible with the ~oij. One way to construct ~" is to define a presheaf ~'0 as follows: let/d be the collection of open subsets of X which are contained in some U,, and choose a function f :/d ~ I such t h a t V C Ul(v ) for all V ~ / d . Define 9r0(V) = 0 for V r and Jr0(V) = J r / ( v ) ( V ) for V ~ / d . Using the isomorphisms ~o~j we see t h a t there are natural restriction maps making 5r0 a presheaf, together with given isomorphisms 9r0 lu~ ~- JriT h e n j r = a(Jr0) is the desired sheaf obtained by patching the Jri using the isomorphisms ~o~j. We leave it to the reader to cheek the uniqueness assertion. (D.10) An O x - m o d u l e is free of rank n if it is isomorphic to O~r n. An O x - m o d u l e ~ is called locally free (of finite rank) if each x E X has an open neighborhood U such that j r ]u is a free O u = O x Ju-module of finite rank. A locally free O x - m o d u l e of rank 1 is called an invertible O x - m o d u l e . Locally free modules have several good properties. For example, if s is locally free, then the funetors ~" ~ s 1 7 4 ~ , ~ ~-~ ~lom ox (s ~r) are exact. We also have isomorphisms of funetors (in jr) Horn ox (s Jr) "~ s | ~, and Hom Ox (s ~- (eE* | J:)(X). T h e n a t u r a l map s --. (s from s to its double dual is an isomorphism. For any locally free O x - m o d u l e s there is a natural Ox-linear surjeetion C | s ~ O x , which is an isomorphism if ~: is invertible. However, note t h a t in general, locally free O x - m o d u l e s are not projective objects in the category of O x - m o d u l e s . (D.11) Recall t h a t an object I of an Abelian category .4 is injective if the funetor X ~ Hom A(X, I) is exact. For any (possibly non-commutative) sheaf of rings O x , the Abelian category of sheaves of O x - m o d u l e s has
Appendix D: Results from Algebraic Geometry
301
enough injectives, i.e., for any sheaf ~" of Ox-modules, there is a monomorphism 3c -4 2", where iT is an injective Ox-module. To prove this, one notes t h a t if J is an injective sheaf of Abelian groups, then iT = Tlom Zx ( 0 ~ , J ) , which is naturally an O x - m o d u l e , is in fact injective; this follows from the natural isomorphism Horn o• (.~,
~om z• ( O ~p, ,7)) ~ Horn z• (O~-p |215 ~', if) = H o m z(.~, ,7)
for any sheaf ~- of Ox-modules. This reduces us to proving the result when Ox = Z x . One sees easily that if {Ix}~ex is a family of injective (= divisible) Abelian groups indexed by points of X, and Z(U) = rixeu ix, then : / i s an injective sheaf of Abelian groups. Now for any sheaf ~- of Abelian groups, if we choose inclusions 3c~ ~-. Ix into injective Abelian groups, then we obtain an injection of sheaves ~(~') ~-~ iT, where Z is defined by the chosen family { Ix } xex; composing with the natural injection ~" ~-~ ~(~') (since Jr is a sheaf, the natural map is an inclusion), we are done. Thus any sheaf ~- of O x - m o d u l e s has an injective resolution O-,
~
~
~o -,
Zl
-,
...
--, I,
-~ ...
in the category of O x - m o d u l e s , and this is unique up to chain homotopy (by standard arguments using the universal property of an injective object). Hence for any left exact functor F from the category of sheaves of Oxmodules to an Abelian category, we may define its derived functors RiF
by
RiF(~)
_
_
ith cohomology object of the complex F(I.).
If 0 --~ ~" --+ ~- -4 ~ " --+ 0 is an exact sequence of sheaves, we have functorial boundary maps R*F(Jz'') --+ R'+lF(J z') giving a long exact sequence of derived functors (where we identify R~ with F)
0---+ F(.T")---, F(.T')---, F(.T"')~ RiF(.T")~...---, R'F(.T")--, R'F(.T)~ R'F(.T"')---+ R ' + I ( . ~ ') --+--. Any natural transformation between left exact functors induces a unique natural transformation between their derived functors, compatible with boundary maps in the respective long exact sequences. (D.12) I m p o r t a n t examples of left exact functors on sheaves and their derived functors are as follows. (i)
Let f : X -4 Y be a continuous map, Ox a sheaf of (possibly noncommutative) rings on X. Then f. is a left exact functor from Oxmodules to f.Ox-modules, whose derived functors Rif. are called
302
Algebraic K-Theory the higher direct image functors of the map f. In particular, if Y {y} is a point, then f . O x is identifed with the ring R - O x ( X ) , and f . ~ is identified with the R-module Jc(X) of global sections. The sheaves R ~ f . ~ yield R-modules H~(X,.~) called the cohomology groups (really, cohomology R-modules) of ~-.
(ii)
Let g be an Ox-module. Then ~" H Hom Ox (g, 5~), ~" H ~/om Ox (g, ~') are left exact functors. Their i th derived functors are denoted by Ext o x (g , ~') and Ext io• (g, ~ ) , respectively.
We have a natural isomorphism H o m o • ( O x , ~ ) ~ ~ ( X ) = H~ Hence there are natural isomorphisms E x t o x (Ox,J:) "~ H~(X,~). Note t h a t if O x is commutative, and ~; is a locally free O x - m o d u l e , then
s
iOx ( E , 5 ~ ) = 0 f o r a l l i > 0 ,
and there are natural isomorphisms Ext ~gx (~:, ~ ) ~ H ' ( X , s
|
~).
(D.13) Derived functors may also be computed using acyclic resolutions, i.e., if 0 --, ~" --. ~'o is a resolution, and F a left exact functor with RiF(JZj) = 0 for all i > 0, j >_ 0, then the ith cohomology object of the complex F ( ~ ~ is naturally isomorphic to R~F(~). We claim that flasque sheaves of Abelian groups are acyclic for f . for any m a p f : X -4 Y. Indeed, one shows t h a t the following statements hold (see [HI, II, Ex. 1.16). (i) Injective sheaves of Abelian groups are flasque. Indeed, if j 9U ~-, X is an open set, then the map Horn Zx ( Z x , Z) --~ Hom Zx (jtZu, :~), induced by the inclusion of sheaves j!Zu --~ Z x , is surjective for any injective sheaf I , i.e., Px, u : Z ( X ) --~ Z(U) is surjective. Similarly, working with Ox and j~Ou, we see t h a t injective O x - m o d u l e s are flasque for any (possibly non-commutative) sheaf of rings Ox.
(ii) If 0 --~ ~ ' ~ ~ --~ 9r ' ' --* 0 is exact with ~-' flasque, then ~-(U) - , ~ " ' ( U ) is surjective for each open U C X . (iii) If 0 --~ ~ ' --* ~ --* jc,, __. 0 is exact with ~",~" flasque, then ~ " is flasque.
Appendix D: Results from Algebraic Geometry
303
From (i) and (iii), the quotient of an injective sheaf by a flasque subsheaf is flasque. From (ii), given a short exact sequence 0 --, ~-t -4 j r __, ~ __, 0 with ~-~ flasque, we get t h a t for any continuous map f 9X --, Y, the direct image sequence 0 --, f.~'~ --, f.~" - . f.~'~ --, 0 is exact. Hence if ~" is a flasque sheaf of O x - m o d u l e s , 0 --, ~- --, Z~ is an injective resolution by sheaves of Ox-modules, then 0 --, f . j r __, f.Zo is exact for any continuous map f 9X --, Y. Hence R i f . j r = 0 for all i > 0. Since injective O x - m o d u l e s are flasque, we see t h a t the cohomology (or higher direct images) of an O x - m o d u l e ~-, computed with resolutions by injective Ox-modules, equals the cohomology (or higher direct images) of the underlying sheaf of Abelian groups jr. Another application is the following: if f 9X --, Y is the inclusion of a closed subset, then f . is an exact functor from sheaves of Abelian groups on X to those on Y, which sends flasque sheaves on X to flasque sheaves on Y. Hence Rif..7z = 0 for all i > 0, and there are natural isomorphisms H i ( X , ~ ) ~- Hi(Y, f . ~ ) for all i > 0 (an injective resolution of ~ on X yields a flasque resolution of f.~" with the same complex of global sections). (D.14) One important tool in computing sheaf cohomology is Leray's theorem, which relates the cohomology groups defined above to Cech cohomology. We first recall the definition of (~ech cohomology (in a simple context sufficient for our needs). Let L / = (Ui}iel be an open covering of a topological space X , where we fix a well ordering of the index set I, and let ~" be a sheaf of Abelian groups on X. Define groups
=
1-[
n...
n u,,,)
and maps 5P" (?P(/4, jr) __, (?p+ 1(/1, ~ ) by p+l :
j=O
..... .? .....
A
where ij means t h a t the index i i is omitted. Then ((?~ ~~ is a complex, called the Cech complex of ~" with respect to L/, whose cohomology groups are called the Cech cohomology groups of ~- with respect to L/, and are denoted by/:/i(U, ~ ) . There is a natural m a p / : / ' ( U , J r) --. H i ( X , J r) for each i (see [HI, III, (4.4)). Leray's theorem asserts that if HJ(U~,, n . . . A Uz,,,J c) - 0 for all finite intersections of open sets in the covering, and for all j > 0, then these natural maps are isomorphisms (see [U], III, Ex. 4.11).
304
Algebraic K-Theory
Schemes (D.15) The basic building block for schemes is the spectrum of a commutative ring. If R is a commutative ring (with 1), its spectrum Spec R is the set of all prime ideals in R. (Recall t h a t an ideal P C R is prime if R / P has no nontrivial zero-divisors; equivalently, if a, b E R with ab E P, then a E P or b E P.) This set is given the Zariski topology in which the closed sets are of the form V(I) = ( P E Spec R I I C P}, where I is an ideal. The complementary open set Spec R - V(I) is denoted by D(I); if I = (f) is a principal ideal, it is also denoted by D(f). The sets n ( f ) form a basis for the Zariski topology on Spec R. The closed points of Spec R are precisely the maximal ideals. Note that for any ideal I c R, V(I) = V(x/7), where x/7 = {x I xn E I for some n} is the radical of I. But x/7 -- Apev(I)P. Hence V(II) = V(I2) r ~ = V~2. We also have n a V ( I a ) = U ( ~ , ~ I~),
V(I) U V(J) = V(IJ). If k is an algebraically closed field, X an affine variety over k with coordinate ring R, then from the Nullstellensatz, evaluation at points of X determines surjections R -~ k whose kernels are precisely the maximal ideals of R; the subspace topology on the set of maximal ideals from Spec R is precisely the classical Zariski topology on the affine variety X. This also motivates the following definition: if R is a ring, the aj~ne n-space AnR over R is defined to be S p e c R [ X l , . . . ,Xn] where R[X1,... , X , ] is the polynomial ring in n variables over R. Let Rp denote the localization of R at a prime ideal P, i.e., with respect to the multiplicative set R - P. For f 6 R, let R I denote the localization with respect to powers of f; thus R S ~ R [ T ] / ( f T - 1). There is a sheaf of rings OSpec R defined on the topological space Spec R, by
OSpecR(n(I)) = {s E YI Rp I locally on D(I), s equals a quotient of two elements of R}. FED(I) Here, we mean that for each P E D(I), there is (i) a neighborhood D(f) of P in D(I) (i.e., f e I, f r P), and (ii) an element a / f n E R f , such that for each Q E D(f), s and a / f n have the same image in RQ. Because of the 'local' nature of the condition defining the sections s, the above definition yields a sheaf of rings on Spec R, such that (see [H], II,
(2.2)) (i)
the stalk of Ospec R at P is Rp
Appendix D: Results from Algebraic Geometry (ii)
305
there is a natural isomorphism OspecR(D(f)) '~ R/; in particular,
OSpecR(Spec R) ~ R. In (ii) above, one can show that in fact there is an isomorphism of sheaves of rings OSpec R IO(/) '~-~OSper R! on D(f). From now onwards, we regard Spec R as the pair consisting of the above topological space equipped with its sheaf of rings, the structure sheaf
OSpec R. (D.16) A ringed space is a pair (X, Ox) consisting of a topological space X and a sheaf of rings Ox on it, which we call the structure sheaf on X. This is called a locally ringed space if the stalks of Ox are local rings. For example, (Spec R, OSpec R) is a locally ringed space. Given a point x of a locally ringed space (X, Ox), the local ring of X at x is the stalk Ox,= = (Ox)=, and the residue field of this local ring is called the residue field at x, sometimes denoted by k(x). A morphism f 9 (X, Ox) --* (Y, Oy) of locally ringed spaces is a pair, consisting of a continuous map f 9X --, Y, and a map of sheaves of rings f # 9 f - l O y --* O x , where f # is required to satisfy the following property: for any x E X , if y = f(x), then the map on stalks at x induced by f # is a homomorphism of local rings fx# 9 (Oy,y,.M~) ~ (Ox,=,.h/l=) where .M= COx,=, M y C Ov',u are the respective maximal ideals; then we require that f=#(My) C fl4x. If (X, Ox) is a locally ringed space, U C X an open subset, Ou = O x IrJ, then a simple example of a morphism is the inclusion j u " (U, O r ) --~ (X, Ox), where j~ l Ox = Ou and j u# is the identity. We remark t h a t if f 9 R --. S is a homomorphism of rings, then p ~ f - l ( p ) gives a continuous map f 9 S p e c s --~ SpecR, and the ring homomorphism f induces a homomorphism of sheaves of rings f # 9 f - l O S p e c R ---* OSpecS; one checks that ( f , f # ) is in fact a morphism of locally ringed spaces. We recover the ring homomorphism f 9 R --, S by the map induced by f # on global sections. This construction in fact gives us an equivalence of categories between the opposite of the category of commutative rings and the full subcategory of the category of locally ringed spaces consisting of spectra of rings. This generalizes the classical equivalence between the category of affine varieties over an algebraically closed field k and the opposite category of the category of finitely generated k-algebras which are reduced (see (D.21)). The spectrum of a ring, considered as a locally ringed space, is called an affine scheme. (D.17) A scheme is a locally ringed space (X, O x ) such t h a t for some open cover {Ui} of X, we have isomorphisms of locally ringed spaces (Ui, Ox Ivy) ~ (SpecRi, OSpecR,); in particular, we must have Ri
306
Algebraic K-Theory
O x(Ui). A morphism of schemes is defined to be a morphism of locally ringed spaces; thus schemes form a full subcategory of locally ringed spaces. For example, for any scheme (X, O x ) , there is a unique morphism f 9 (X, Ox) --* SpecZ, where Z is the ring of integers. If in a given context, the structure sheaf O x is known, we often denote the scheme (X, Ox) by just X. We have already used this convention in writing Spec R to mean both the topological space and the affine scheme. If X = SpecA, and U C X is open, then (U, Ox [u) is a scheme. Indeed, if U = X - V(I), and I is generated by {f~ }, then U is covered by the open subsets D(fo,) = Spec AI,~ of X. Hence if (X, Ox) is any scheme, U C X an open subset, then (U, Ox [u) is a scheme. This is called an open subscheme of (X, Ox). (D.18) The construction of affine schemes is modeled after that of affine varieties in 'classical' algebraic geometry, through their relationship with their coordinate rings; in a similar manner, one can associate a scheme Proj R to a graded ring R which mimics the relationship between a 'classical' projective variety and its homogeneous coordinate ring. This is done as follows. Let R = ~n>0Rn be a (non-negatively) graded ring. An ideal I c R is called homogeneous if it is generated by homogeneous elements; thus an element of R lies in I if and only if its homogeneous components lie in I. Thus an ideal is homogeneous if and only if it is a graded R-submodule of R. A homogeneous ideal P is prime if and only if for any pair a, b E R of homogeneous elements with ab E P, either a E P or b E P. The ideal R+ = ~n>0Ra is called the 'irrelevant homogeneous ideal'. Define Proj R to be the set of all homogeneous prime ideals P of R such that R+ ~ P. If I is a homogeneous ideal, let V(I) be the subset of Proj R of all prime ideals containing I; give Proj R the topology such that these are the closed subsets. Let n+(I) = Proj R - Y(I). If I = (f), we also denote n+(I) by D+(f), and these open sets give a basis for the topology of Proj R. Finally, we define a sheaf of rings Oproi R on Proj R, as follows. For any homogeneous prime P E Proj R, let S(p) be the multiplicative set of homogeneous elements of R - P ; let R(p) be the subring of S~-A)R consisting of elements a/b where a E R, b E S(p) are homogeneous of the same degree (so that a/b has degree 0). This is seen to be a local ring. Now define
OProjR(D+(I)) = {s E H
R(P) I s is locally equal to an element
PED+(1)
a/b
where a, b e R have the same degree}.
The condition on s means that for each P E D+ (I), there exists a homoge-
Appendix D: Results from Algebraic Geometry
307
neous f E I - P, say of degree d, such that for some homogeneous element a E Rnd for some n, s and a / f n have the same image in R(Q) for each
Q E D+(f). One checks (see [U], II, (2.5)) that Oproj R is a sheaf of rings, whose stalk at P is R(p), and whose ring of sections over D+(f) is R(I), the subring of the Z-graded ring R$ consisting of elements of degree 0. In fact, (D+(I), OProjR ID+(I)) ~ (SpecR(I),OSpecR(1)) as locally ringed spaces. Hence (Proj R, OProj R) is a scheme. (D.19) In particular, if A is any commutative ring, R = A[To,..., Tn] the polynomial ring, graded by defining elements of A to have degree 0, and each variable to have degree 1, then Proj R is called projective nspace over Spec A (or just projective n-space over A); it is denoted by IP~. This terminology is justified to some extent by the fact that if A = k is an algebraically closed field, then the set of closed points of the scheme IP~, with the subspace topology, is naturally identified with the 'classical' projective n-space of hyperplanes in an n + 1-dimensional k-vector space, with its Zariski topology. (D.20) Unlike in the case of affine schemes, the scheme Proj R does not determine the graded ring R. For example, if S {~)nSn (7_.R = (~)nRn is a graded subring such that Sn = t ~ for all sufficiently large n, then Proj R and Proj S are naturally isomorphic (see also [H], II, 5.16.1). Further, if R --, S is a homomorphism between graded rings, we do not necessarily get a corresponding morphism Proj S --, Proj R by taking inverse images of homogeneous prime ideals. The problem is that even if a homogeneous prime P of S does not contain the irrelevant ideal S+, its inverse image in R may contain the irrelevant ideal R+. However, if I = R+S, so t h a t I is a graded ideal in S, then a homogenous prime ideal P of S which does not contain I has an inverse image in R which does not contain R+, and conversely. Using this, one sees that there is a naturally defined morphism of schemes Proj S - V(I) --~ Proj R, which on points is given by taking the inverse images of homogeneous prime ideals (see [n], II, Ex. 2.14). =
Some
Properties
of Schemes
In this section we recall the definitions of certain basic properties of schemes and morphisms. A scheme (X, Ox) is reduced if for each open set U C X, the ring Ox(U) has no non-zero nilpotent elements (i.e., Ox(U) is a reduced ring). For any commutative ring A, let nil A denote the ideal of nilpotent elements of A. Then, nil A is the intersection of all prime ideals of A. If (X, Ox) is an arbitrary scheme, then there is an associated reduced scheme (D.21)
308
Algebraic K-Theory
(X, (OX)red), denoted for short by Xred, where
(Ox)~a(U) O n e checks
that
this
does
=
define
Ox(U)/(nilOx(U)). a scheme
structure
space X, such that if X = Spec A, then Xred
=
on the
topological
Spec (A/nil A).
(D.22) A scheme (X, Ox) is integral if for each open subset U c X, the ring Ox(U) is an integral domain. A scheme (X, Ox) is integral r (X, Ox) is reduced and X is irreducible (i.e., X is not the union of 2 proper closed subsets). If U = Spec A is a non-empty affine open subscheme of an integral scheme (X, Ox), then (0) c A is a prime ideal, so yields a point r/of Spec A. Now {71} is dense in Spec A, hence in X, and so lies in any non-empty affine open subset V = Spec B of X. We see easily that r/also corresponds to the prime ideal (0) of B. The point 7/thus lies in all non-empty open subsets of X , and is called the generic point of X. We may define the function field of an integral scheme (X, Ox) as (the residue field of) the local ring Ox,, at the generic point, which is in fact the quotient field of A for any non-empty affine open subset U = Spec A of X. (D.23) A topological space X is quasi-compact if every covering of X by open sets has a finite subcover. If (X, Ox) = SpecR is an affine scheme, then X is quasi-compact. Indeed, if U~ = X - Y(I~) is an open covering, then N~V(I~) = V(~-~,I~) is empty, so that ~ i I i = R. But then 1 6 ~ i I i , so 1 = a l + .-- + ar for some aj 6 Ii~. Then ~-~j Iij = R, and UjUij = X. (D.24) A topological space X is called Noetherian if it satisfies the descending chain condition for closed subsets. The motivating example for this definition is X = Spec A, where A is a Noetherian ring. A space X is Noetherian if and only if any open subset is quasi-compact. A Noetherian topological space X has a unique decomposition X = U[=lXi for closed subsets Xi such t h a t (i) each X~ is irreducible, i.e., cannot be expressed as the union of 2 proper closed subsets (ii) the decomposition is irredundant, i.e., for each i, we have X~ ~ Uj#~Xj. The Xi are called the irreducible components of the Noetherian space X. If X is irreducible, a generic point of X is a point x 6 X such t h a t the set {x} is dense in X. If (X, Ox) is a scheme, then every non-empty irreducible closed subset of X has a unique generic point. A scheme (X, Ox) is called Noetherian if it has a finite open cover by affine open subschemes Ui = Spec Ai such that each As is a Noetherian ring. One can show (see [HI, II, 3.2) that this implies the condition that for any affine open subscheme U - Spec A of X, the ring A is Noetherian. The
Appendix D: Results from Algebraic Geometry
309
topological space underlying a Noetherian scheme is Noetherian, t h o u g h the converse is false. (D.25)
Recall t h a t a Noetherian local ring (R, A4) of dimension d is called
regular if f14 is generated by d elements; equivalently, f14/f142 is an R/.A4vector space of dimension d. The local ring (R, f14) is called normal if it is an integral domain which is integrally closed in its quotient field; it is called Cohen-Macaulay if it has depth d, i.e., there exist x l , . . . ,Xd E .A4 which are non-zero-divisors in R, such t h a t x~R N (Y]~j#~xjR) = Y~j#~ xzxjR for all 1 < i < d (this is equivalent to the exactness of the Koszul complex of the xi with respect to R). A scheme (X, Ox) is called regular (respectively normal or CohenMacaulay) if every local ring Ox,x is regular (respectively normal, or CohenMacaulay). (D.26) A morphism f 9 (X, Ox) --* (Y, 0~.) is of finite type if Y has a covering by affine open subsets Ui = Spec Ai such t h a t for each i, f - l ( u i ) has a finite covering by affine open subschemes Spec Bij of X, where Bij is a finitely generated Ai algebra. We say t h a t (X, O x ) is of finite type over Y if there is a morphism f " X --, Y of finite type. One can show t h a t f is of finite type r for every affine open subscheme U = Spec A of Y, f - l ( U ) has a finite open covering by the spectra of finitely generated A-algebras. The Hilbert basis theorem implies t h a t if Y is a Noetherian scheme and f 9X ~ Y is a morphism of finite type, then X is a Noetherian scheme. In particular, projective space 1P~ over a Noetherian ring A is a Noetherian scheme. (D.27) A morphism f 9 (X, Ox) -* (Y, Oy) is finite if there exists a covering of Y by affine open subschemes Ui = Spec Ai such t h a t f - l ( U i ) Spec Bi is affine, and Bi is a finite Ai-module, for each i. This is equivalent to requiring t h a t for every affine open subset U S p e c A of Y, f - l ( U ) = S p e c B is affine and B is a finite A-module. If (Y, O~,) is an integral scheme of finite type over a field k (i.e., over Spec k), then we can construct a finite morphism f 9 (X, Ox) -* (Y, Oy) called the normalization of Y, with the following property: for each affine open set U = S p e c A c Y, f - l ( U ) = S p e c B , where B is the integral closure of A in its quotient field, which is the function field of Y. This is a finite morphism, because B is a finite A-module (see [H], I, 3.9 A). A finite morphism f 9X --* Y is closed, i.e., if Z c X is closed, then f ( Z ) c Y is closed. To prove this, one reduces to the case when Y, and hence X, is affine; further one may assume Z -- X. If X - Spec B, Y = Spec A, We may replace Y by the closed subscheme Spec A / I , where I = ker(A --~ B); hence we m a y further assume the ring homomorphism A -4 B is injective. Now we want to prove Spec B --. Spec A is surjective. If P C A is prime,
310
Algebraic K-Theory
Sp = A - P, then S~,IB is a finite Ap-module; so it suffices to prove t h a t when A is local with maximal ideal M , then B has a prime ideal contracting to M. Now B is a non-zero finitely generated A-module; hence M B ~ B by Nakayama's lemma; if M ' is a maximal ideal of B containing M B , then M ' N A is an ideal ~ A (since 1 g~ M ' ) which contains M , hence equals M. (D.28) A morphism f 9 (X, Ox) ---* (Y, OF) is affine if there exists a covering of Y by affine open subschemes U~ = Spec A~ such t h a t f-l(U~) is affine for each i. Thus finite morphisms are affine. A morphism f " X --~ Y is affine if and only if f - l ( U ) is affine for every affine open subscheme U of Y. (D.29) A morphism of schemes f " (X, Ox) -* (Y, Oy) is an immersion if f # 9 f - l O y --, Ox is surjective. In particular, if f is the inclusion of a closed subset of Y, we call (X, Ox) a closed subscheme of (Y, Oy). The morphism f is called a closed immersion if it is an immersion which is an isomorphism of (X, Ox) with a closed subscheme of (Y, Oy). If X = Spec R, and I is an ideal of R, then there is a natural bijection between the prime ideals of R / I and the set V(I) of prime ideals of R containing I. This bijection in fact makes Spec R / I a closed subscheme of Spec R. One can show ([H], II, 5.10) t h a t every closed subscheme of Spec R is obtained this way. If R is a graded ring, X : Proj R, and I c R is a homogeneous ideal, then again the bijection Proj R / I ~ V(I) makes Proj R / I a closed subscheme of Proj R. If R = @i>_0R~ where R1 is a finite R0-module, and R is generated by R1 as an Ro-algebra, then any closed subscheme of Proj R is of this form ([U], II, 5.16). (D.30) T h e KruU dimension dim X of a topological space X is the suprem u m of all integers n such that there exists a chain Zo C Z1 C ... C Zn of distinct, irreducible closed subsets of X. The Krull dimension of a scheme is t h a t of its underlying topological space. This notion of dimension is called Krull dimension because for X = Spec A, this agrees with the Krull dimension (as defined in commutative algebra) of A. The codimension codim x Z of an irreducible closed subset Z c X is the supremum of all integers n such t h a t there is a chain Z = Z0 C Z1 C 9.- C Z~ of distinct irreducible closed subsets of X. Note t h a t it is false in general t h a t (i) for a n o n - e m p t y open subset U C X of an irreducible space X, we have dim X = dim U, and (ii) for an irreducible closed subset Z C X , we have dim Z + codim x Z = dim X. However these statements are true if X is the topological space underlying an irreducible scheme of finite type over a field (see [H], II, 3.2.8 and Ex. 3.20). In this case, the dimension function has other good properties; for example, if f " X -~ Y is any morphism between irreducible schemes
Appendix D: Results from Algebraic Geometry
311
of finite type over a field, with dense image, then the image contains an open dense subset of Y, and dim ] - I (y) = dim X - dim Y for all y in some (perhaps smaller) open dense subset of Y (see [H], II, Ex. 3.22). Hence if dim X < dim Y, a morphism f " X --, Y cannot have dense image, i.e., the closure of its image is a proper subvariety of Y. This is a very useful fact in practice when making general position arguments (see l e m m a (D.69) below for an illustration). If x E X is a point, its codimension is defined to be the Krull dimension of Ox,x. If X is a scheme, we will use sometimes use X p to denote the set of points of X of codimension p. This convention is used in C h a p t e r 5 of the main text; see (5.20). (D.31) T h e category of schemes has fibered products. Thus if f 9X --. S, g 9Y ~ S are morphisms of schemes, there exists a scheme X x s Y together with morphisms p " X x s Y -~ X, q" X x s Y ~ Y such t h a t the d i a g r a m
XxsY
p~
X
q
Y
~I
S
lg
commutes, and such t h a t for any scheme Z which fits into a c o m m u t a t i v e diagram t
Z
--. q
y
x
~
s
p'~
lg
there is a unique morphism of schemes h 9Z --, X • s Y such t h a t p' = p o h, q' = q o h. This is proved in [H] (II, 3.3), by reducing to the case when X = S p e c A , Y = Spec B, S -- S p e c C are all affine; in this case X x s Y ~ Spec (A | B). This leads to two i m p o r t a n t notions. First, if f " X --~ Y is a m o r p h i s m of schemes, the fiber X v (also called the scheme theoretic fiber) of f over a point y E Y is defined to be the fiber product X X v Speck(y). Here k(y) is the residue field of y, and the morphism f " S p e c k ( y ) --~ (Y, O v ) is the inclusion of {y} in Y, together with the map of sheaves f - i o y = Ov, y-+-,k(y) (where we identify sheaves on (y} with Abelian groups). T h e topological space underlying Xy is the fiber of the continuous m a p f 9 X --, Y over y with its subspace topology; the scheme structure on Xu includes more information (for example, even if X, Y are integral, Xu m a y not be reduced, corresponding to the geometric ideas of 'multiple fibers' or 'singularities' of the morphism f). Another i m p o r t a n t notion is t h a t of base change. If f 9X --~ S is a morphism, we regard X a.s a scheme over S; if f - X --~ S, g " Y --~ S are
312
Algebraic K-Theory
schemes over S, an S - m o r p h i s m from X to Y is a m o r p h i s m h : X --. Y such t h a t g o h -- f . For example, one m a y consider the c a t e g o r y of schemes over a field k (i.e., over Speck). One writes ' S - s c h e m e ' instead 'scheme over S'; in t h e context of S-schemes, the m o r p h i s m to S is referred to as the structure morphism. Given any m o r p h i s m S t --, S, we obtain a scheme X t = S t • s X over S t. This gives a functor from schemes over S to schemes over S t, called the base change corresponding to S t --~ S. For example, one m a y take k to be a field, S = Spec k, K an e x t e n s i o n field, S t = Spec K . A m o r p h i s m f : X ~ S is closed if for any closed s u b s e t Z c X , f ( Z ) c S is closed; f is called universally closed if for a n y base change S t --~ S, the corresponding m o r p h i s m f t : X t = S t • s X --~ S t is closed. In a similar way, if P is a p r o p e r t y of m o r p h i s m s of schemes, one says a m o r p h i s m is 'universally 7~' if 7~ holds for every m o r p h i s m o b t a i n e d by an a r b i t r a r y base change. (D.32) T h e notion of fiber p r o d u c t s allows us to define t h e 'correct' analogues for schemes of the notions of Hausdorff spaces and p r o p e r m a p s of topological spaces; t h e definitions involve t h e diagonal. If f : X --~ S is a m o r p h i s m , there is a unique induced m o r p h i s m A from X to t h e fiber p r o d u c t X • s X , called the diagonal m o r p h i s m , such t h a t t h e d i a g r a m X
X
xsX
f X
~
S
commutes. Here l x denotes the identity m o r p h i s m on X . In particular, one sees at once t h a t A is an immersion. (D.33) A m o r p h i s m f : X ~ S is defined to be separated if A is a closed immersion. A scheme X is separated if X --~ Spec Z is separated. Since A is anyway an immersion, we see t h a t f is s e p a r a t e d if and only if t h e image of A is a closed subset of X • s X . If X is a scheme of finite t y p e
Appendix D- Results from Algebraic Geometry
313
over C, the field of complex numbers, then the set of closed points of X is naturally identified with the set of points of a complex analytic space (this boils down to the assertion that a polynomial with complex coefficients is an analytic function). In particular, we can associate a topological space to this set of closed points; this space is Hausdorff precisely when X ~ Spec C is separated. If f 9 Spec B ~ Spec A is any morphism of affine schemes, then f is separated. In fact A is the closed immersion Spec B --~ Spec ( B | which corresponds to the surjection of rings BC~A B ~ B given by bt •b2 H bib2. One good property of domains of separated morphisms is the following: if f 9X - , Spec A is separated, for some ring A, then for any pair of affine open subsets U, V c X , their intersection U M V is also affine. Indeed, U N V ~ (U XSpecA V)N AX, where the intersection on the right is as subschemes of X XSpecA X, and A x is the diagonal subscheme. Now U, V are affine, so U • Spec A V is affine; if f is separated, A x is a closed subscheme, so t h a t U M V is isomorphic to a closed subscheme of the affine scheme U x Spec A V; hence U M V is affine. (D.34) A morphism f 9X ~ S is proper if f is separated, of finite type, and universally closed. (D.35) It may not be so easy to check directly, using the definition, t h a t a morphism is proper, or even separated. However, it follows easily from the definitions t h a t separated and proper morphisms are each stable under composition and arbitrary base change; further, an open immersion is separated, and a closed immersion is proper. Also, separatedness and properness are local on the base, i.e., to check either condition for f - X S, it suffices to check it for each of the morphisms f - 1 (U i) --~ U~ for some open cover U~ o r S . I f X is Noetherian, and f " X---, Y, g " Y ~ Z are morphisms, such t h a t g o f 9X ~ Z is either separated or proper, then f has the same property. Finally, we have shown (see (D.27)) that a finite morphism is closed; since it is affine, it is separated; since any base change of a finite morphism is finite, any finite morphism is proper. One can show t h a t (i) a proper morphism between affine schemes is finite; (ii) a proper morphism with finite fibers is finite (see [HI, III, Ex. 11.2). For morphisms f 9 X ---, Y with X Noetherian, one has the valuative criteria for separatedness and properness. These are as follows. Let R be a valuation ring with quotient field K . Thus R is a local integral domain with quotient field K such t h a t for any non-zero x E K , either x E R or x -1 E R. There is then a homomorphism (called a valuation) v" K* ~ G to a totally ordered Abelian group G such t h a t R - (0} consists of the elements x E K* such that v(x) > OG, where 0 c E G is the identity. Further, one has t h a t v(x + y) >_ min(v(x), v(y)) for all x, y E K* (D.36)
314
Algebraic K-Theory
with x + y ~ O. Familiar examples of valuation rings are discrete valuation rings; another type of valuation ring, contained in the quotient field of a ring of formal power series, is described in the course of the proof of the Mercurjev-Suslin theorem in Chapter 8 (see the remark after lemma (8.20)). The criterion for separatedness is as follows. Let f : X --, S be a morphism with X Noetherian. Then f is separated r for any valuation ring R, with a morphism h : S p e c R --, S, and for any morphism g : Spec K --~ X such that the diagram Spec K
--, g
X
Spec R
h
S
commutes, there exists at most one extension of g to a morphism ~ 9 Spec R --, X such that ~ lifts g. Similarly, if X is Noetherian and f is of finite type, then f is proper r in the above situation, there exists a unique lift ~ of g as above. The proofs of the above two valuative criteria are given in [H], II, Theorem 4.3 and Theorem 4.8. (D.37) We use the valuative criterion to show t h a t f 9I1~ ~ Spec A is proper, for any ring A. If this is true for A - Z, the ring of integers, then it is true for any A, since properness is preserved under base change. So we may assume f " ll)~ -4 Spec Z. Since f is clearly of finite type, ll~ is Noetherian, and it suffices to show t h a t if R is a valuation ring with quotient field K, then any morphism g 9 Spec K --~ IP~ extends uniquely to a morphism ~" Spec R --. ll~. (Note t h a t there are unique morphisms Spec K --~ Spec Z and Spec R --~ Spec Z.) Let x E II~ be the image of the unique point of Spec K. Then the morphism g determines an injective h o m o m o r p h i s m k ( x ) ~ K , where k ( x ) is the residue field at x. We may explicitly realize this as follows. ll~ = Proj Z [ X 0 , . . . ,Xn] is covered by the affine open subschemes Ui = D + ( X i ) = S p e c Z [ X o / X i , X 1 / X ~ , . . . , X n / X ~ ] ; if x E Ui, then the morphism Spec K -~ I1)~ factors through a m o r p h i s m Spec K --~ U,, i.e., corresponds to a homomorphism p : Z [ X o / X ~ , . . . , X n / X ~ ] ~ K . The point x E Ui is the prime ideal which is the kernel of this homomorphism. Let v : K* --. G be a valuation corresponding to the valuation ring R c K , where G is a totally ordered Abelian group. Extend v to a m a p p i n g v : K --, G (A {co} where G td {co} is a totally ordered set, such t h a t co is an element larger than any element of G; we define v(0) = co. Choose an index j such that v(p(Xj IX,)) = min(v(p(Xo/X,)), v(p(X1/X,)), . . . , v(p(Xn/X,))}.
Appendix D: Results from Algebraic Geometry
315
Since p(X~/Xi) = 1, this minimal value of v is an element of G. T h e n p(Xj/X~) ~ O, and so p extends to a homomorphism
Z [ X o / X i , . . . , X n / X , , ( X j / X i ) - ' I -* K, which we also denote by p. Now Xl/X.i = ( X I / X , ) ( X j / X , ) - l , and v(p(x,/xj))
=
,,(p(x,/x,))
-
v(p(x~/x,))
>
o.
Hence p ( X l / X j ) E R for all l, and we have a commutative diagram of rings Z[Xo/X~
P
~o~ . ,x,,/xjl
1
n
1
Z[Xo/X~,. . . ,Xn/X~, (X~/X~)-ll
A
K
which yields a c o m m u t a t i v e diagram of schemes and morphisms Spec R
t
SpecK
--,
Uj
--,
UiNUj
t
In particular, the morphism Spec K --, IP~ extends to a morphism Spec R --, IP~. One verifies easily t h a t this extension is unique, and in particular, does not depend on the choice of the index j. In particular, if k is an algebraically closed field, IP~ --, Spec k is proper. Hence the projection ~r 9 IP~ • k A~ --, A~' is a closed map. In concrete terms, this means the following. Let X 0 , . . . , Xn be variables corresponding to homogeneous coordinates on IP~ (i.e., IP~ = Proj k [ X o , . . . , X , ] ) and let Y t , . . . , Ym be the variables corresponding to coordinates on A~ (i.e., A~ = Spec k [ Y l , . . . , Ym]). A closed set W in lP~ • A~n is the zero locus of a finite set of polynomials F~ (Xo,
. . . , X,,;
Y~ , . . . , Y,,,),
. . . , F~(Xo,
. . . , Xn;
Y~ , . . . , Y,,,),
where the Fj are homogeneous in the Xv. To say t h a t its image lr(W) in A~ is closed is to say t h a t ~r(W) consists of the zeroes of a set
al(Yl,...
, Y,,),
. . . , G~(Y~,
. . . , Ym)
of polynomials in the Yj alone; if I c k[Xo,... ,X,.,,Y1,... ,Ym] is the ideal generated by the Fi, we take Gj to be generators for the intersection of I with the subring k [ Y l , . . . , ym], i.e., the Gj are obtained from the Fi by 'eliminating the variables X~'. The assertion, t h a t the zero locus of these Gj describes the image Ir(W), is the content of 'elimination theory' in classical algebraic geometry, and may be proved directly without appealing to schemes, valuation rings, etc. (for example, for the case when
316
Algebraic K-Theory
k = C, see D. Mumford, Algebraic Geometry I: Complex Projective Varieties, Grundlehren Math. 221, Springer-Verlag (1976), Theorem (2.23) and
(2.25)).
(D.38) Let S be a scheme. Define projective n-space over S, denoted ]P~, to be the fiber product IP~ x SpecZ S. For S -- Spec A, this is consistent with our earlier definition of lP~. Since properness is preserved under base change, t h e n a t u r a l m a p p " IP~ --~ S is proper. A m o r p h i s m f 9X - , S is called projective if there is a closed immersion (see (D.29)) i" X --, IP~ such t h a t f = poi, where p" IP~ --, S is the n a t u r a l map. We say t h a t X is projective over S if there is a projective m o r p h i s m f 9 X --. S. A scheme X is said to be quasi-projective over S if it is isomorphic to an open subscheme of a projective S-scheme X . Since a closed immersion is proper, and a composition of proper morphism is proper, we see t h a t any projective m o r p h i s m is proper. T h e converse is false, even if S = Spec C, where C is the field of complex numbers; however, if f - X --, S is proper, and S is Noetherian, then there is a projective morphism g" X ' ~ S and a m o r p h i s m h " X ~ --, X such t h a t (i) g = f o h (ii) there is a dense open subset U C X such t h a t h - i (U) --~ U is an isomorphism. This assertion is known as Chow's lemma; in practice, it reduces the proofs of many assertions a b o u t proper morphisms to the special case of projective morphisms. (Note t h a t the morphism h" X ~ --, X is also projective, since the closed immersion i 9X ~ --, IP~ used to factorize g induces a closed immersion i ~" X ' -4 ItD~: ~- I?~ • s X which factorizes h.) T h e structure morphism X --, S of a quasi-projective S-scheme X is separated and of finite type. (D.39) We will use the term variety over a field k to denote a separated k-scheme of finite type. A quasi-projective variety over k is a s e p a r a t e d scheme of finite type over k which is quasi-projective over k. This includes the notions of affine and projective varieties over k. (D.40) A morphism f " X --* Y is fiat if for each x E X , if y = f ( x ) , t h e n the induced map of rings Oy, y ~ Ox,x is flat (i.e., the functor - | Ox,= is exact). F l a t morphisms of finite type between Noetherian schemes are open. This can be deduced using the following assertion from c o m m u t a t i v e algebra: if f 9R ~ S is a flat homomorphism between Noetherian rings, t h e n the going down theorem holds, i.e., given a pair of prime ideals P C p , of R and a prime Q' of S contracting to P ' (i.e., with f - l ( Q , ) = p,), t h e r e exists a prime ideal Q c Q' of S which contracts to P. To see this, by localising at P ' and Q', we may assume R, S are local with maximal ideals P~ and Q' respectively, and Q~ contracts to P~; now R I P ~ R p / P R p ,
Appendix D: Results from Algebraic Geometry
317
so by flatness, S I P S ~ S | (Rp/PRp), and S I P S ~.~.S/Q' ~: O; hence P(S | Rp) is not the unit ideal. If Q1 is any maximal ideal of S | Rp containing P(S | Rp), then under the homomorphism R --, S | Rp, we see that Q1 contracts to P (the contraction contains P, but cannot be larger since Q1 is not the unit ideal). Take Q to be the contraction of Q1 to S. In particular, if f 9 (R, ~/[R) --* (S, J~4s) is a flat local homomorphism between Noetherian local rings, then dim S _> dim R. If ~/A/IRS = M s , then dim R = dim S (from dimension theory in commutative algebra, dim R = d r there exist x 1,... ,xd E M R with v / ~ R f i = MR, and d is the smallest such number; since also x / ~ Sfi M s , we have dim S <_ dim R). This means that if f 9X --~ Y is a fiat morphism, and y E yn is a point of codimension p, and x E X is the generic point of an irreducible component of the fiber Xu, then x E X p. This means that f-1 preserves codirnension of closed subschemes; this is important for K-theory. We give a criterion for a morphism to be fiat: if f 9 X - , Y is a morphism between irreducible varieties over a field k (i.e., X, Y are irreducible separated k-schemes of finite type) such that Y is regular, X is Cohen-Macaulay (for example, regular) and all fibers of f have dimension dim X - dim Y, then f is fiat (see [H], IiI, Ex. 10.9). For example, any finite surjective morphism between integral regular schemes is fiat. (9.41) If (f, f # ) " (X, Ox) ~ (Y, Or,) is a morphism of ringed spaces, and ~" is any sheaf of Ox-modules, then the natural homomorphism Or, --, f, Ox (induced by f # ) makes the direct image f . ~ into an Or,-module. Then f . , considered as a functor from Ox-modules to Oy-modules, has a left adjoint f*, given by f* G = Ox | ov f - 1~. We call f* G the module theoretic inverse image of G. Note that if f is a fiat morphism of schemes, then f* is an exact functor. For any Ox-module ~ and any locally free Or'module •, we have a natural isomorphism f.(Jr| x f ' E ) ~- f . ( j z ) | E, called the projection formula. Coherent
and
Quasi-coherent
Sheaves
(D.42) For any ringed space (X, Ox), we have defined the notion of a sheaf of Ox-modules. When X = Spec R, there are certain distinguished Ox-modules which correspond naturally to R-modules, in the sense that (i) the category of these modules is a full Abelian subcategory of the category of all Ox-modules, and (ii) this full subcategory is naturally equivalent to the category of R-modules, in a fashion analogous to the equivalence of the category of affine schemes with the opposite of the category of rings.
318
Algebraic K-Theory
First, to any R-module M, we associate an O x - m o d u l e M as follows. For any P E X, let MR denote the localization of M at the multiplicagive set R - P, so that MR ~ M | Rp. For any open set D(I) c X , let
"M(D(I)) = {s E H Me [locally on D(I), s equals a quotient PED(I)
m / a with m e M and a E R}. Here, we mean that for each P E D(I), there is (i) a neighborhood D(f) of P in D(I) (i.e., f e I, f r P), and (ii) an element m / f n e M f, such t h a t for each Q ~_ D(f), s and m / f '~ have the same image in MQ. Because of the 'local' nature of the condition defining the sections s, the above definition yields a sheaf of O x - m o d u l e s on X = Spec R, such that (see [H], II, (5.1)) (i) (ii) In of M of
the stalk of M at P is Mp there is a natural isomorphism M ( D ( f ) ) ~ MI; in particular, M(Spec R) ~ M.
(ii) above, one can show that in fact there is an isomorphism of sheaves OSpec R1-modules M [O(f)ff My..It is clear from the construction t h a t ~-~ M is functorial in M. We may also use the notation M - i n s t e a d M. It is shown in In] (II, (5.2)) that:
(i)
for any ring R, the functor M ~-. M is exact and fully faithful
(ii)
if f 9 Spec S --~ Spec R is a morphism, corresponding to a homomorphism ~0 : ~ S, then the direct image functor f . and module theoretic inverse image functor f* have the following descriptions: for any Rmodule M, we have a natural isomorphism f * M ~- (S | M ) , and for any S-module N, the direct image f . N is NR, where NR is N regarded as an R-module via ~o
(iii)
for any family of R-modules Mi, we have ~ M i ~ ~ i M i
(iv)
for any pair of R-modules M and N, we have M | (M | Y)".
N
It is easy to see from the definitions t h a t the functor M ~-. M from Rmodules to OSpecR-modules is left adjoint to the global section functor jr ~_, F(Spec R, jr).
Appendix D: Results from Algebraic Geometry
319
(D.43) A sheaf j r of O x - m o d u l e s on a scheme X is called quasi-coherent if there exists an aftine open cover Ui = Spec Ri of X and R~-modules Mi such that there are isomorphisms of Ou,-modules ~r Iv, -~ M~. Note t h a t Ox is quasi-coherent; more generally, any locally free O x module is quasi-coherent. If j : U --. X is the inclusion of an open subscheme of X, then in general j!Ou is an Ox-module (in fact a sheaf of ideals) which is not quasi-coherent. One can show t h a t quasi-coherent sheaves have the following properties (see the discussion below, and [HI, II, (5.4), (5.7), (5.8), III, (8.5)). (i) Let X be a scheme. An O x - m o d u l e }- is quasi-coherent if and only if for each afline open subscheme U = Spec R of X, we have an isomorphism of On-modules jr Iv ~ (~-(U))". (ii) The kernel, cokernel and image of any Ox-linear map between quasicoherent O x - m o d u l e s is quasi-coherent; any extension of quasicoherent O x - m o d u l e s is quasi-coherent. Any direct sum of quasicoherent O x - m o d u l e s is quasi-coherent; if ~', ~; are quasi-coherent, then j r | ~ is quasi-coherent. (iii) For any morphism f : X ~ Y, if !7 is a quasi-coherent Oy-module, then f*~; is a quasi-coherent Ox-module. If X is Noetherian, or f is quasi-compact (inverse image of any aftine open set is quasi-compact) and separated, then for any quasi-coherent O x - m o d u l e jr, the direct image f . ~ is quasi-coherent. (iv) If X is Noetherian, then for any morphism f : X -~ Y and any quasi-coherent O x - m o d u l e ~', the higher direct images Rif.J r are quasi-coherent, and for any atiine open subscheme U of Y, we have RZf.Jz luu Hi(I-I(U),.~ I$-~(u))-- For any quasi-coherent Oxmodule ~ and any locally free Oy-module E, we have the projection
formula
f'c)
c.
The assertion (ii) follows from the case when X is affine; t h a t an extension of quasi-coherent sheaves is quasi-coherent, reduces to the assertion t h a t if 0 ~ jr, __, j r __~ ~',' __, 0 is exact with ~", jr,, quasi-coherent, and X is affine, then 0 --, ~ " ( X ) - , Jr(X) - , ~"'(X) - , 0 is exact ([H], II, (5.6)). In (iii), the point is t h a t the inverse image of any affine open subset of Y is a finite union of affine open subsets of X, such that the pairwise intersections of any two of these open subsets of X is again a finite union of open affines. Hence there is an exact sequence of the form
320
Algebraic K-Theory
where {Uijk}k is an affine open cover of Ui N Uj. The second and third term are quasi-coherent from the case when b o t h X and Y are affine, since -" commutes with direct sums; by (ii), f . ~ " is also quasi-coherent. (iv) follows from (iii), that is the case i - 0, by induction on i, using the fact t h a t any quasi-coherent O x - m o d u l e on an affine scheme X = Spec R is a submodule of a flasque quasi-coherent O x - m o d u l e (namely one of the form I where I is an injective R-module; see [H], III, (3.4)); now one uses the fact t h a t Rif. and H i both vanish for a flasque sheaf, and the long exact sequences for Rif. and for H i. For the projection formula, we use the fact t h a t tensoring with a locally free sheaf is exact, and the projection formula for the case i = 0. In particular, if i 9Y ~ X is the inclusion of a closed subscheme of X (see (D.29)), let Zy = ker(Ox .~~.i.Oy) be the sheaf of ideals of Y in X. Now i.Oy is quasi-coherent on X, since i is quasi-compact and separated, and hence 2"y is quasi-coherent. Conversely, one sees ([H], II, (5.9)) t h a t any quasi-coherent sheaf of ideals in O x determines a subscheme of X . In particular, the closed subschemes of an affine scheme Spec R are precisely the schemes Spec R/I for all ideals I of R. Note t h a t it is not true that for a r b i t r a r y quasi-coherent sheaves 9r, ~, the O x - m o d u l e 7-lOmox(~, ~) is quasi-coherent, since for modules M, N over a ring R and a multiplicative set S, it is in general not true t h a t S - 1 H o m n(M,N) ~- Horn s-~a(S-1M, S-1N). However this is true if M is a finitely presented R-module. This p a r t l y motivates the following notion. (D.44)
A sheaf ~" of O x - m o d u l e s is coherent if
(i)
for any x E X, there exists an open neighborhood U of x and a surjection O~ ~ - , j r Iu, for some positive integer r (i.e., 9r is locally finitely generated), and
(ii)
for any open U C X and any h o m o m o r p h i s m u " O ~ r --, j r Iv, the kernel of u is locally finitely generated.
Note t h a t a coherent sheaf is locally a cokernel of a map O~ s -~ O~ ~, and is hence quasi-coherent. This implies t h a t if ~ is coherent and G is quasi-coherent, then 7"lore Ox (~', G) is quasi-coherent, and it is coherent if G is coherent. If X is Noetherian, then j r is a coherent O x - m o d u l e if and only if for each affine open set U - Spec R of X , we have j r [t]~ ~'(U), where j r ( U ) is a finitely generated R-module (see [HI, (5.4)); it is enough to check this condition for all U belonging to some affine open cover of X. One sees that if 0 - , ~" --, 5r --, ~'~' --, 0 is exact, and any two of the three Ox-modules is coherent, so is the third. If X is Noetherian, the kernel, cokernel and image of any m o r p h i s m between coherent sheaves is
Appendix D: Results from Algebraic Geometry
321
coherent. If f : X --, Y is a morphism between Noetherian schemes, and {7 is coherent on Y, then f*{7 is coherent on X. It is not true t h a t the higher direct images (or even the direct image itself) of a coherent sheaf under a morphism between Noetherian schemes are coherent; however, this is true for proper morphisms (Grothendieck, EGA III, 3.2.1). For projective morphisms, this is shown in [H], III, (8.S); we discuss this further below (see Theorem (D.62)). (D.45) An i m p o r t a n t notion for K-theory is that of the support of a coherent O x - m o d u l e ~ on a Noetherian scheme X. If X = Spec A is affine, and ~" - M , then M is a finite A-module. The s u p p o r t of M is Spec A / a n n M, where ann M = {a E A l a M = 0} is the annihilator of M. Depending on the context, we may regard the support as a subscheme or a subset of X. In general, the support of ~- is defined to be the subscheme Z of X such t h a t for any affine open subscheme U = Spec A of X, the scheme U n Z is the support of ~ It]= ~ ( U ) . Equivalently, we can define the sheaf of annihilators of jr, which is a coherent sheaf of ideals in O x ; then Z is the corresponding subscheme. In some contexts, one regards the s u p p o r t of ~" as the closed subset of X underlying the scheme Z. We can also define the notion of an associated point of a coherent sheaf on a Noetherian scheme X; this is a point x E X such t h a t the maximal ideal of Ox,x is an associated prime (in the sense of commutative algebra) of Jrx. If U = Spec A is an affine open subscheme containing x, and ~" It]= M , and x corresponds to a prime ideal P, then x is an associated point of j r if and only if there is an m E M whose annihilator in A is P, i.e., P is an associated prime of M in the sense of commutative algebra. We remark t h a t if f is a coherent O x - m o d u l e on a Noetherian scheme X, then for any r > 0, the set {x E X I dimk(~)~'~ |
k(x) > r}
is closed. This follows easily from the case when X is affine. In particular, if X is integral, with generic point r/, the rank of ~" is the dimension of ~-u, which is a finite dimensional vector space over the function field of X; if ~" is of rank r, then {x e X I d i m k ( ~ ) ~ |
k(x) = r}
is open, and is the largest subset of X on which 9r restricts to a locally free sheaf of rank r (we make use of the easy lemma that a finitely generated module of rank r over a local integral domain is free r tensored with the residue field, its dimension is r; this follows at once from N a k a y a m a ' s lemma).
322
Algebraic K-Theory
(D.46) We now describe, the correspondence between graded modules over a finitely generated graded ring R and quasi-coherent and coherent sheaves on Proj R. If M is a graded R-module, we associate to it the sheaf M on Proj R, defined as follows. For any homogeneous prime P E Proj R, recall that S(p) is the multiplicative set of homogeneous elements of R - P; let M(p) be the subgroup of S(p)x M consisting of elements (1/b)m where m E M, b ~_ S(p) are homogeneous of the same degree (so that (1/b)m has degree 0). This is a module over the local ring R(p). Now define -
I~(D+(I)) = (s e 11 M(P) I s is locally equal to PED+(I)
an element (1/b)m where m E M, b e R have the same degree}. The condition on s means that for each P E D+(I), there exists a homogeneous f E I - P, say of degree d, such that for some homogeneous element m E Mnd for some n, the elements s and ( 1 / f n ) m have the same image in M(Q) for each Q E D+(f). One checks (see [HI, II, (5.11)) that (i) (ii)
the stalk of M at P is M(p) for any homogeneous element f E R, if M(S) is the R(y)-submodule of elements of degree 0 in the localization MS, then M ID+(I)~
M(I ) (the sheaf on the right is the quasi-coherent sheaf on D+(f) = Spec R(I ) associated to the R(i)-module M(I)) Off) M is a quasi-coherent Op~oj R-module; if S is Noetherian and M is finitely generated, then M is coherent. (iv)
there is a natural homomorphism M0 --~ F(Proj R, M).
Here M0 is the group of elements of degree 0 in M. For any graded R-module M, we can define a new graded R-module M(n) by setting M(n)d = Mn+u, with the evident R-module structure. In particular, let R(n) be the graded R-module given by shifting the grading on R by n, i.e., R(n)d = Rn+d. Let OProj R(n) = R(n) for each n. For any Oproj n-module jr, define ~'(n) = ~ | R OProj n(n). One can show that if R is generated over R0 by R1, then
(i) OProjR(1)
is an invertible Oproj R-module (locally free sheaf of rank 1), and OProjR(n) -- OProjR(1) | (if n < 0, the right side of this isomorphism is defined to be (Oproj R (1) | n). );
(ii)
for any graded R-module M, we have M(n) = M(n);
Appendix D: Results from Algebraic Geometry (iii)
323
there is a natural homomorphism Mn --* F(Proj R, M ( n ) ) .
The operation M ~ M ( n ) on graded modules (or ~ H ~ ( n ) on sheaves) is called twisting. It allows one to associate a graded module to any quasicoherent OProjn-module 9r , by the formula F.(~') = S n e z F ( P r o j R , ~ ( n ) ) . This is a graded R-module in a natural way, since elements of P~ give global sections of Oproj R(n). In particular, we have a graded R-algebra F . ( R ) . One has a natural isomorphism R --. F.(R), which is an isomorphism (see [H], II, (5.13)) if R = A[Xo,..., X~] is a polynomial ring (with the usual grading), i.e., if Proj R = ll~. One can show ([HI, II, (5.15)) that under suitable assumptions, there is a natural isomorphism F. (~-)- -~ ~- for any quasi-coherent sheaf on Proj R. This follows from lemma (D.48) below, which also has other applications. (D.47) Let X be a scheme, and s an invertible Ox-module. Let f E F(X, E), and let X S be the set of points in X where the image fx of f in the stalk s does not lie in A/l~s where A4x C Ox,x is the maximal ideal. We may express the condition that fx r J~4xEx by saying t h a t "f does not vanish at x", since the image of fx in s = Ex | k(x) is the 'value' of the section f in the fiber at x. Now f determines an Ox-linear map s _. O x , whose image is a quasi-coherent sheaf of ideals on X, hence defines a closed subscheme; X I is the complement of this closed subscheme. L e m m a D . 4 8 Let X , s f E F ( X , s quasi-coherent sheaf on X .
and X ! be as above. Let j z be a
(i) Assume X is quasi-compact, and s E F ( X , ~ ) such that s ~-, 0 E ~ ( X s ) . Then for some n > O, we have s | f'~ = 0 E F(X,~" | E| (ii) Assume further that X has a finite open cover by open a ~ n e s Ut such that s lu~ is free, and Ui M Ui is quasi-compact for each i, j. Given a section t E :Tz(Xs), there exists n > 0 such that t | f " is the restriction to X ! of a global section of J: | s174
Proof. (i) First choose a finite cover (Ui} of X by affine open sets, such that s Iv,~ Or, for each i. Write ~" Ivy= Mi for some Ai = Ox(Ut)-module Mi; then the restrictions of s yield elements st E Mi. Since s Iv, is free for each i, the section f yields an element ft E At, and X I M Ut = Spec(At)/,. We also have jc | s174 ~ M~, and 3z | s174 M XS) = (M~)I,. Now s vanishes on XS; hence st E Mt vanishes in the localization (Mr)s,, for each i. Hence I n ' st = 0 in Mt for each i. Hence if n = max nt, then s | f " restricts to 0 in each Mr, hence is 0, as a section of 0c | s174
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Algebraic K-Theory
(ii) Let 5~ Jv~= Mi, and let f Ivy= f~ e F(U~, Ov,). Let t~ be the restriction of t to X I MUi. Then ti E (Mi)I, for each i. Hence there is an nl > 0 such t h a t ti = t~f~ ~ -n~ for each i, for some t~~ E M~. T h e n t ti can be regarded as a section of $-| s174 on Ui, for each i. Now t~ - tj is a section of this sheaf on U~ O Uj which vanishes on X I O U~ M Uj. Hence by (i), there exists n2 > 0 such that (t i - t~) | f~2 = 0 in 9v | s174 ~ n Uj), for all i , j . Hence the sections t~ | f~2 of ~ ' | s174 agree on the intersections Ui M Uj, and patch up to a global section of ~ | s174 with n = n l + n2. Note t h a t the quasi compactness and finiteness assumptions of the lemma are valid if X is Noetherian. If X = Proj R, where R is generated by a finite subset of R1 as an algebra over R0, the lemma easily implies t h a t for any homogeneous element f of degree 1 in R, which we may regard as a section of/2 = Oeroj n(1), we have a natural isomorphism F.(JC)(D+ ( f ) ) 9r(D+ (f)), for any quasi-coherent sheaf ~" (the finite generation assumption on R implies that the hypotheses of the lemma are valid; note t h a t for any homogeneous element f of degree 1 in R, we have s ID+(f)~ OD+(I)). In particular, for any ring A, any closed subscheme X C IP~ is defined by a graded ideal I c A [ X o , . . . , Xn]. Indeed, if :Z is the ideal sheaf of X, and I = F.(Z), then I C F.(OProj n) = R defines the subscheme of Proj R with ideal sheaf I - :Z. Thus X is projective over a ring A if and only if X = Proj R where R is a graded A-algebra generated over Ro = A by a finite set of elements of degree 1. T h e o r e m D . 4 9 (Serre) Let X be a projective scheme over a Noetherian ring A, and let J: be a coherent Ox-module. Then there exists an integer no such that Jr(n) is generated by global sections for all n :> no. Proof. Let i" X ~ IP~ be an embedding as a closed subscheme. T h e n i . ~ is coherent on IP~. So we reduce to the case when X --- IP~. Let F~ = Proj R, where R = A [ X o , . . . ,Xn] is the homogeneous coordinate ring of P~. Let U~ - D+ (X~), so that the U~ are a finite covering of IP~ by affine open sets. Let ~'(U~) = Mi, so t h a t Mi is a finite R(x,)-module for each i. Choose finite sets of generators mij of M~. T h e n there exists an no such that for each n _> no, we have (by the above lemma) t h a t mij | X ~ extends to a global section of ~'(n) for each i, j; now one sees at once t h a t these sections globally generate .~. C o r o l l a r y D . 5 0 Let X , :F be as above. Then :7: is a quotient of O x ( - n ) r for any sufficiently large n > O, and a suitable m > 0 (depending only on J=). T h e applicability of the above theorem and corollary are extended by the following lemma, which is also useful elsewhere.
Appendix D: Results from Algebraic Geometry
325
L e m m a D . 5 1 Let X be a Noelherian scheme, U C X an open subscheme, and let ]z be a coherent Or-module. Then there exists a coherent O x module -~ such that -~ Itl ~- .7:.. Proof. First suppose X = Spec A is affine. If j : U --. X is the inclusion, then j.~" is quasi-coherent, hence of the form M, and M Iv= .T'. If f l , . . . , fr E A such t h a t tA~D(fi) = U, we are given t h a t MI, = .T'(D(f~)) is a finite Aft-module for each i. Choose a finite subset S of M containing generators for each of the modules Mr,, and let N be the A-submodule of M generated by S. Then N is coherent, and N ~ M, so t h a t N [v_.__~ M Iv for any open set V C X. In particular, N [D(I~)= NI, ~-~ MI~ , while NI,
.~; M I,. Hence NI, = Mr,, and so g [u= ~ . In general, work by induction on the number of affine open subsets of X needed to cover X - U. If there exists an affine open subset V with U tJ V = X, then ~" Ivny extends to a coherent sheaf G on V; now j r and patch up to give a coherent sheaf on X (see (D.9)). In the inductive step, suppose U t.J U1 tA . . . tA Un - X; by induction, ~- has a coherent extension ~7 to U tJ U1 tA.-. tAUn- 1; by the first case, G now has a coherent extension to X. C o r o l l a r y D . 5 2 Suppose X is a quasi-projective scheme over a Noetherian ring A, and 3: is a coherent sheaf on X . Let f 9 X ~ IP~ be an immersion onto a locally closed subset of ~ , and let s = f* (Pp, (1). Then there exists no such that J~ | E| is generated by global sections for all n >_ no. In particular, ~ is a quotient of a locally free O x - m o d u l e of finite T'a fl,k.
(D.53) There is another situation in which any coherent sheaf is a quotient of a direct sum of invertible sheaves. Recall (see (D.25)) t h a t a Noetherian scheme X is called regular if each of the local rings Ox,x are regular local rings (i.e., fl4x/ f 1 4x2 is a k(x)-vector space of dimension equal to the Krull dimension of O x,x). If x E X l is any point of codimension 1 in a regular scheme X, then Ox,x is a regular local ring of dimension 1, i.e., a discrete valuation ring. If x E X 1, its closure D C X (with the reduced structure) is a subscheme of X, such that for any y E D, the stalk of the ideal sheaf 2"o,y c O i , u is a prime ideal of height 1 (since O x , u / I D , u ~-- OD,u which is an integral domain, :~D,u is a prime ideal; further the 1-dimensional local ring Ox,x is the localization of Ox,u at ZD,u). Since Ox,u is regular, hence is a unique factorization domain, any prime ideal of height 1 is a principal ideal. Hence ZrD is a coherent O x - m o d u l e , all of whose stalks are free modules of rank 1, i.e., ZD is an invertible O x module. Define O x ( D ) = 7-lOmox(ID, OX) for any irreducible D. T h e
326
Algebraic K-Theory
inclusion ZD C O x gives a global section s of O x ( D ) , hence an O x - l i n e a r m a p O x --* O x ( D ) which is an isomorphism on X - D, and restricts to 0 on D.
Let X be a Noetherian regular scheme. A divisor on X is an element of the free Abelian group on X 1. A divisor is called effective if it is a positive linear combination of elements of X 19 If D = ~--]~i=1 r niD~ is any effective divisor, we may associate to it the invertible sheaf O x ( D ) = O x ( D 1 ) | | 9. . | | If si is the canonical global section of O x ( D i ) described above, for each i, then s~ nl | | s~rn'' is a canonically defined global section of O x ( D ) , which gives an isomorphism O x --* O x ( D ) on X - U i D ~ , and restricts to 0 on UiDi. This inclusion O x --~ O x ( D ) also identifies the coherent sheaf ? - l O m o x ( O x ( D ) , O x ) with a sheaf of ideals ZD C O x , which defines a canonical structure on D of a closed subscheme of X. Any irreducible closed proper subset of X is contained in an irreducible divisor. We deduce t h a t if X is a Noetherian regular scheme, then there is a basis of open sets in X consisting of sets Xs = {x [ s 9 O x --~ E restricts to 0 on precisely the complement of Xs }, for some section s of an invertible O x - m o d u l e E. T h e following is a particular case of a result of Kleiman. T h e o r e m D . 5 4 Let X be a Noetherian regular scheme. Then any coherent Ox-module J~ is a quotient of a direct sum of invertible Ox-modules. Proof 9 Suppose given an invertible O x - m o d u l e E and a global section s. Let X8 be the corresponding open subset of X . Let a E j r ( X s ) . T h e n by l e m m a (D.48), we see t h a t s | | a is the restriction to X s of a global section of j r | E| This yields a morphism (E*) | --~ ~-, such t h a t the image of the restriction to Xs contains a. Since X is Noetherian, hence quasi-compact, and j r is coherent, we can find a finite n u m b e r of such open sets Xs, (corresponding to invertible sheaves Ei and si E s and sections ai E yr(Xs~), such t h a t these sections generate the stalks of ~" at any point of X; now the corresponding map (gi(E~)| __. ~- is surjective. C o r o l l a r y D . 5 5 Let X be a Noetherian regular scheme of Krull dimension <_ n. Then for any coherent Ox-module J~, there exists an Ox-linear resolution o - - , E~ --, E ~ _ ~ --, . . . --, Eo --, J : - - ,
o
where the Ci are locally free of finite rank. Proof. From the above lemma, there exists such a resolution where the Ci are locally free for i < n, and s is coherent. To show t h a t C~ is locally free, it suffices to show t h a t for each x E X , the stalk (s is a free Ox,xmodule; here the stalks (s are free for i < n. Since Ox,z is a regular
Appendix D" Results from Algebraic Geometry
327
local ring of Krull dimension _< n, the finite Ox,z-module ~'x has projective dimension _< n. This means t h a t for any resolution 0---~ M ~ Fn-1 ---~ 9 ---~ Fo --~ .Tx --~0 where the Fi are free Ox,z-modules, the module M is also free. (D.56) On any Noetherian scheme X, we say that a coherent (Px-module ~- is of finite homological dimension if it has a finite resolution by locally free O x - m o d u l e s of finite rank. If ~" has finite homological dimension, then for each x E X , the stalk ~'~ has finite projective dimension over Ox,x. If X is a scheme on which any coherent sheaf is a quotient of a locally free sheaf of finite rank (for example, a scheme which is quasi-projective over a Noetherian ring, or a locally closed subscheme of a regular scheme), then ~" has finite homological dimension precisely when its stalks have finite projective dimension. (D.57) An invertible sheaf/2 on a Noetherian scheme X is ample if for any coherent sheaf ~', there exists an integer no such t h a t ~" | s174 is generated by its global sections, for any n >_ no. One exampl e of an ample invertible sheaf is the following: if X is quasiprojective over a Noetherian ring A, and f 9X - , IP~ is a locally closed immersion in IP~, then f*Op~(1) is an ample invertible sheaf on X , by corollary (D.52). If X is of finite type over a Noetherian ring A, then for any ample invertible sheaf s on X, some positive tensor power s174 is of this form ([H], II, Theorem 7.6). Clearly, if there is an ample invertible sheaf on X , then any coherent O x - m o d u l e is a quotient of a locally free O x - m o d u l e of finite rank. (D.58) T h e notions of quasi-coherent and coherent sheaves allow one to 'relativize' the notions of the spectrum of a ring, and of Proj of a graded ring, as follows. If f " Z --~ X is an aifine morphism (i.e., f-l(U) is affine for any affine open set U) then S = f.Oz is a quasi-coherent sheaf of Ox-algebras, such t h a t for any affine open set U c X , we have f-l(U) ~S p e c S ( U ) . Conversely, given any quasi-coherent O x - a l g e b r a S, we m a y define an X - s c h e m e Z = S p e c s by taking patching together the affine schemes Zu -- S p e c S ( U ) for various afiine open subsets U of X (see [H], II, Ex. 5.17). If f 9Z = S p e c S ~ X is the natural map, then f . induces an equivalence of categories from the category of quasi-coherent O z - m o d u l e s to the category of ,.q-modules which are quasi-coherent as O x - m o d u l e s . Similarly, suppose S = ~n>0Sn is a quasi-coherent Ox-algebra, where (i) S0 = Ox (ii) ~d is a coherent O x - m o d u l e for each d (iii) S is generated by ~1 as an O x - a l g e b r a . T h e n we may define (see [H], II, w an X - s c h e m e P r o j S, with a structure morphism f 9 P r o j S ~ X, as follows: if U C X
328
Algebraic K-Theory
is affine, t h e n we may form the Ox(U)-scheme P r o j S ( U ) ; these schemes glue together in a natural way to give the scheme P r o j s T h e invertible sheaves O(1) on the various local schemes P r o j s also glue to give an invertible sheaf Oproj s (1) on P r o j s (D.59) Two applications of the above relativization of Spec and Proj are the following. A geometric vector bundle (in the terminology of Hartshorne's book; see II, Ex. 5.18) of rank n on a scheme X is a scheme V with a morphism ~r 9V --* X, such t h a t for some open covering {Ui} of X, we are given isomorphisms of Ui-schemes ~i " 7r-l(Ui) ~-; A~,, such t h a t the induced 'glueing' automorphisms qoj oqo~-1 of A~,nvj are linear along the fibers, i.e., are given by an element of GLn(Ox(Ui N Uj)). Two such 'local triviality' d a t a {U i, ~o~} and {Vj, Cj} define the same s t r u c t u r e of a vector bundle on V if their union defines a structure of a vector bundle. A morphism of vector bundles on X may be defined to be a m o r p h i s m of X-schemes which is linear on the fibers, in the obvious sense defined using the local trivializations. If lr 9V ~ X is a vector bundle, then the sheaf of sections of 7r, which is a priori a sheaf of sets, has a natural structure as a locally free O x module of rank n; locally, this corresponds to the natural identification of the sheaf of sections of A~] ~ U with O ~ n. Conversely, given a locally free O x - m o d u l e E of rank n, we may associate to it a vector bundle V(E) = S p e c ( S y m (s where Sym (E) is the s y m m e t r i c algebra of E over Ox. Locally, if we choose an isomorphism E [u O ~ n , the symmetric algebra is identified with a polynomial algebra in n-variables, and we get an isomorphism Y(S) [u ~ A~; one checks easily t h a t these local isomorphisms determine a structure of a vector bundle on V(E). Finally, one can show t h a t the sheaf of sections of V(s is naturally isomorphic to E*, the dual locally free sheaf. These construction provide an anti-equivalence between the categories of locally free sheaves (of constant rank) on X and vector bundles on X. Thus one very often uses the t e r m 'vector bundle' to mean 'locally free sheaf'. This convention is followed in the main text; the only place where 'vector bundle' is used to mean a geometric vector bundle is in Proposition (5.17). In a similar fashion, if g is a locally free sheaf of rank r on X , we may define the projective bundle associated to E to be Px(E) -- P(E) = P r o j (Sym (E)). If lr 9 IP(E) --~ X is the natural m a p (the structure morphism) then there are natural isomorphisms S y m n ( s -~ lr.Oe(e)(n) for n _> 0, which are obtained by applying 7r. to natural surjections 7r*Symn(E) --, Oe(e)(n); and 7r.Op(e)(n) = 0 for n < O. The X - s c h e m e IP(s has the following universal property: for any X-scheme Y, with structure morphism f - Y --. X , the X-morphisms g - Y -~ P(E) are in bijection with isomor-
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329
phism classes of surjections f*s ---, E. for invertible sheaves s (two such surjections are isomorphic if they have the same kernel).
Cohomology and Direct Images of Quasi-coherent and Coherent Sheaves (D.60) As observed earlier (see (D.43)(iv)), if f : X -~ Y is a morphism between Noetherian schemes, then the higher direct images Rif..T" of a quasi-coherent Ox-module ~- are quasi-coherent, such that for any affine open subscheme U - SpecA of Y, we have Rif..T" Iv= H i ( f - l ( U ) ,
Iz-,(u)):
On a Noetherian affine scheme X - Spec A, one has H i ( X , 3 c) = 0 for any quasi-coherent sheaf ~'. This is proved as follows (see [HI, III, w First we claim that if I is an injective A-module, then I is flasque. One uses the Artin-Rees lemma to see that for f E A, the natural localization map I -~ I I is surjective (i.e., I is f-divisible); using this and Noetherian induction, one deduces that I is flasque. Now for any quasi-coherent sheaf ~', writing 9w = M, let
O---~ M - - , Io--, II ---~... be a resolution by injective A-modules. Then the corresponding resolution of Ox-modules
O~M~Io~I1
~...
is a flasque resolution, so that it may be used to compute cohomology; taking global sections, we recover the given injective resolution, so that Hi(X,.T ") = 0 as claimed. (D.61) From Leray's theorem, we deduce that if X is separated over a Noetherian ring A, then for any quasi-coherent sheaf 3c, there is a natural isomorphism of H i ( X , .T') with the (~ech cohomology group/:/*(U, ~ ) where /g = {Ui} is a covering of X by affine open subschemes. This is because the separatedness ensures that all finite intersections of the Ui are affine schemes, on which quasi-coherent sheaves have vanishing H i (for i > 0). A direct proof of this isomorphism which does not appeal to Leray's theorem is given in [H] (III, 4.5). The computation using Cech cohomology has the following consequence. Let f : X --~ Y be a separated morphism of finite type between Noetherian schemes, such t h a t d is the maximum of the dimensions of the fibers of f; then for any quasi-coherent Ox-module ~', we have Rif..T" = 0 for i > d. This is local on the base; it suffices to show that for a suitable affine open covering (U i} of Y, the scheme f - l ( U i ) has an affine open covering
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Algebraic K-Theory
such that all d + 2-fold intersections are empty. The existence of such coverings (for 'sufficiently small Ui') can be deduced from the assumption on dimensions of the fibers. A basic theorem of Grothendieck is the following. T h e o r e m D . 6 2 Let f " X --, Y be a proper morphism between Noetherian
schemes, and let ~ be a coherent Ox-module. Then the Oy-modules Ri f . ~ are coherent Or-modules. We will outline a proof of this theorem in the case when f is a projective morphism, i.e., when there is a factorization f = 7roi for a closed immersion i 9X --* IP~, where 7r 9 IP~. ~ Y is the structure morphism (see EGA III, 3.2.1 for the general case). We reduce immediately to the case when X = IP~, since i. is exact and preserves coherence and 'flasqueness'. Further, the theorem is local on the base, so we reduce to the case when Y = Spec A for a Noetherian ring A. In this case, the theorem amounts to the statement that H~(P~, ~') is a finite A-module for each n. We first state a result giving a computation of the cohomology groups of the invertible sheaves Op~ (n) (see [H], III, 5.1). Recall that the tensor product makes @n>oOe~ (n) into a sheaf of graded Or~-algebras , making @n>0H~
Ov~ (n)) into a graded A-algebra.
(a) The graded A-algebra ~n>oHO(pNA, OpN A(n)) is naturally isomorphic to a polynomial algebra A [ X o , . . . ,XN]; thus
T h e o r e m D.63
H~
We also have H~
N, Or~ (n)) ~ A~(N+").
Or~(n)) = 0 for n < O.
(b) H'(IPAN, Or~ (n)) = 0 for 0 < i < N for all n E Z.
(c) H l V ( ~ , O p ~ ( - N - 1)) is a free A-module of rank 1, and the natural pairings H~
Op~(n))|
Or~(-N-l-n))
---, H N ( P ~ , Or~ ( - N - l ) )
~ A
are perfect pairings between free A-modules. The idea of the proof is to compute cohomology as Cech cohomology of the graded quasi-coherent sheaf S = ~ , e z O r ~ ( n ) using the standard affine open covering of IPA N by N + 1 affine open subsets D+(X~). The (~ech complex of S has an explicit description in terms of localized modules over the polynomial ring S = A[Xo,... , XN],
~P(ll, S) = ~o<_~,,<...<~,,<_NSx,,,X,, ...X,,, ,
Appendix D: Results from Algebraic Geometry
331
with differentials given by alternating sums of natural localization maps. From this explicit description, one may compute the cohomology directly by calculations with polynomials, or more elegantly as in [HI. (D.64) Now we prove t h a t H~(PAN,~-) is a finitely generated A-module, for any coherent sheaf 3c on PA ~. As seen earlier (see (D.50)), we may write as a quotient of Op~ ( - m ) Cr for some sufficiently large m, r, giving an exact sequence of coherent Opt-modules 0 --, {~ --, Oe~ ( - m ) Cr --, ~- --, 0 From the induced long exact sequence of cohomology groups, we see t h a t H~(I~A,~-) is finitely generated r Hi+I(IP~, G) is finitely generated. Since H~(IPA~,~) = 0 for any coherent ~ for i > N (using the (~ech complex for the standard open covering, for example), we see by descending induction on i that Hi(PAN, 3c) is finitely generated for all coherent sheaves ~-. (D.65) A similar argument shows that for any coherent sheaf ~', there is an integer no such t h a t for n _> no, we have Hi(IP~r, ~-(n)) = 0 for all n > no, for all i > 0. Indeed, the above exact sequence shows that for n > m - N, and any i > 0, there is an isomorphism g i ( l P ~ , ~ - ( n ) ) Hi+I(IP~, {~(n)). Again the result follows by descending induction. (D.66) We deduce from the above results that for any locally free sheaf s of rank N + 1 on a Noetherian scheme X, if 7r 9IP(~:) --, X is the structure morphism, then (i) there are natural isomorphisms Sym"(s and lr.Op(s)(n) = 0 for n < 0; (ii)
Rir.Op(~)(n)
= 0 for all n E Z and 0 < i < N;
(iii) there is a natural isomorphism pairings
r.Op(E)(n)|
-~ ~r.Op(e)(n), for n > 0,
RNlr.Op(e)(-N-
1) - .
Ox,
and the
RN~.Op(e)(-N-I-n) ~ RN~r.Op(8)(-N-1) ~ Ox
are perfect pairings between locally free Ox-modules; (iv) for any coherent Op(s)-module ~-, the sheaves Ri~r..T'(n) are coherent for any i > 0 and any n E Z, and for some no > 0, they vanish for all i > 0 and n > no. Some
Miscellaneous
Topics
In this section we prove two lemmas about varieties, i.e., schemes of finite type over a field, which are needed in the text. We also discuss the construction of quasi-coherent sheaves using faithfully flat descent.
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Algebraic K-Theory
(D.67) Let B be an A-algebra. If I = ker(B | B "~ B), w h e r e / z is the multiplication map, then we define f~S/A -- I~ I2 to be the module of Kiihler differentials of B relative to A. T h e r e is an A-linear derivation d 9B --~ f~B/A (i.e., a homomorphism of A-modules satisfying d(blb2) = bl db2 + b2 dbl ), given by the formula db = b | 1 - 1 | b. One verifies t h a t this is universal for A-linear derivations from B to B-modules. If L / K is a finitely generated extension of fields, then ~'~L/K is a finite dimensional L-vector space of dimension at least equal to the transcendence degree of L over K; equality holds r L is separably generated over K . The formation of f~B/A commutes with localization; hence if A --. B is an inclusion of integral domains which are finitely generated k-algebras, and if dim B - dim A - n, then 12B/A is of rank > n, with equality if and only if the quotient field of B is separably generated over t h a t of A. Thus for any prime ideal 7~ of B with residue field k ( P ) , we have t h a t dim l2B~,/A | k('P) > n, and the set of primes for which equality holds is an open subset (perhaps empty) of Spec B. T h e r e are two standard exact sequences which are useful in working with K/ihler differentials (see Matsumura, Commutative Algebra, Benjamin (1970), Th. 57 and Th. 58). First, if B is an A-algebra, and C is a B-algebra (and hence also an A-algebra), we have an exact sequence C |
~'~B/A -~ ~'~C/A --'~ ~'~C/B -"* 0
where the maps are induced by the universal p r o p e r t y of Kiihler differentials. Next, if B is an A-algebra, and I C B is an ideal, then there is an exact sequence
1/12
6
~ f~B/A |
B / I ---* f~(B/I)/A ---* 0
where 6 ( f (mod i2)) = df | 1 e ~B/A | B / I . Now we discuss the notion of smoothness. If k is a field, and A is an integral domain which is a finitely generated k-algebra of dimension d, we define A to be smooth over k if its module of K/ihler differentials f~A/k is a projective A-module of rank d. From the second exact sequence above, one can show t h a t this is equivalent to the following: if A = k [ X x , . . . , X n ] / ( f l , . . . , fr), then the minors of size n - d + 1 of the Jacobian matrix [ox"j] ~ vanish in A (i.e. , lie in the ideal ( f l , . . . , fr)), and the minors of size n - d generate the unit ideal in A. If A is smooth over k, then A is regular (see Matsumura, Commutative Algebra, Sec. 29). More generally, if S is a k-scheme of finite type, and f : X -4 S is of finite type, we say t h a t f is smooth of relative dimension d if (i) f is flat (see (D.40) (ii) if X ' C X, S' c S are irreducible components, with
Appendix D: Results from Algebraic Geometry
f ( X ' ) c S', then dim X ' - d i m
333
S' = d (iii) for each x E X, we have
dimk(x)(fl~/s)x |
k(x) = d.
Here nlx/s is the sheaf of relative K~ihler differentials (see [H], II, w it is a quasi-coherent sheaf such that if U = Spec B C X and V - Spec A C: S are affine open subsets such that f ( V ) C V, then ~lx/s I v ~ ~B/A-. If f : X --0 S is a morphism between integral smooth k-schemes of finite type, and d = dim X - d i m Y, then f i s s m o o t h of relative dimension d if and only if ~lx/Y is locally free of rank d on X (see [H], III, 10.4 and II, Ex. 8.1). For any morphism f : X ~ S, there is a largest open subscheme U C X (possibly empty) such that f Iv is smooth, namely that where 121x/y restricts to a locally free sheaf of rank d. We will use this to prove a special case of Bertini's theorem. L e m m a D . 6 8 Let A be a smooth finitely generated k-algebra of dimension n, and let T C Spec A be a finite subset. Then there exist x x,... , xn_l E A and an element f E A such that f ~ NreTP, and the morphism Spec A f
A~-1 is smooth of relative dimension 1. If k is infinite, and A is generated by elements y l , . . . , Ym, then the xi may be chosen to be 'general' k-linear combinations of the yj. (Here 'general' means 'for all elements in a nonempty Zariski open subset of km~'.) Proof. Since the subset of X where a morphism to An-1 is smooth is open, it suffices to consider the case when T consists of closed points. Since A is smooth over k, for any x e T we have dimk(x)(~2~/k)x | k(x) = n. Let R be the semilocal ring of A at the finite set of points of T (the intersection of the local rings of A at the points). Let J be the Jacobson radical of R, and let -R = R / J 2. Then 12R/k | R / J ~ gt-~/k | R / J . From the Chinese remainder theorem, R ~ 1-IxeTRx, and fl~/k ~ 1-Ix f l ~ / k " The elements
{drx [Xr e R} generate f l ~ / k ; so we can find ( r l ) ~ , . . . , (rn)x in Rx which map to a basis of fl~x/k | k(x) ~- k(x) r By the Chinese Remainder theorem, we can find r l , . . . , rn e R lifting the various (ri)x, and can then lift the rj to elements xj E A (since R / J 2 ~- A / ( J 2 N A)). Let X --, A~-I be the morphism determined by x 1,... ,xn-1. We check easily that for x e T, (~21x/A._,)x | k(x) has dimension 1, since (by the second exact sequence for Kiihler differentials) it is the quotient of (fl~/k)x | k(x) by the subspace spanned by the images of dxi. Now X -4 A~-1 is smooth in a non-empty open neighborhood of T, hence on Spec A! for some f which is a unit at points of T. If k is infinite, and the yj generate A, then the images of the dyj generate ~A/k. Hence in the above argument, we may choose the ri to be k-linear
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Algebraic K-Theory
combinations ~ = ~-~ij aij9-~ of the images of the yj; the desired properties of the ri are equivalent to the non-vanishing of certain determinants which are non-zero polynomials in the aij. We also prove a special case of the Noether normalization lemma. Lemma
D . 6 9 L e t A be a k - a l g e b r a o f d i m e n s i o n n g e n e r a t e d by Yl , . . . , Ym,
w h e r e k is a n i n f i n i t e field. T h e n f o r ' g e n e r a l ' k - l i n e a r c o m b i n a t i o n s x l , . . . x n o f the y j , the m o r p h i s m f 9X
,
= Spec A --, A~ w i t h c o o r d i n a t e s x i is
finite.
Proof.
Let X - Spec A. The generators yj determine a closed embedding
We m a y regard A ~ as the open subscheme D + ( Y o ) of IP~n, which has homogeneous coordinates Y o , . . . , Y m , and yi = Y i / Y o . Then H = IP~n A ~ ~ IP~ -1 is the hyperplane defined by Y0 = 0. Let X C IP~ be the closure of X in p~n. Let Y = X N H . Now any proper subset of an irreducible Noetherian space has strictly smaller dimension. Hence Y C H -~ pro-1 is a projective subscheme of dimension < n - 1, since each irreducible component Xi of X has dimension < n, and Y = t A i ( X i - Xi) where dim Xi - Xi < dim Xi = dim Xi <_ n. If x 1 , . . . , x n are homogenous linear polynomials in the yj, let X 1 , . . . , X n be the corresponding homogeneous linear polynomials in the Yj (thus X j = x j Y o ) . Let L C H be the closed subscheme (isomorphic to f f ~ m - n - l ) defined by Yo = X 1 . . . . = X n = 0. The inclusion of graded rings k[Yo, X l , . . . , Xn] r k[Yo, . . . , Ym] induces a morphism (Proj k [ Y 0 , . . . , Ym]) - Y ( ( Y o , X x , . . . , X n ) ) ---* Proj k[Yo, X ~ , . . . , X n ] , t h a t is to say a morphism PL " IP'~ - L --~ IP~ (this is called the p r o j e c t i o n f r o m t h e l i n e a r subspace L ) . Further, if A~ = I P ~ - {Y0 = 0}, then L -* IP~ restricts to the PL- I ( A ~ ) = A~. The induced morphism X morphism f 9X ~ A~ determined by the xi. If L N X = r then we get an induced morphism PL " X --. IP~, which is hence a projective morphism. Hence X N pL 1(A~) - , A~ is projective, i.e., f 9X -4 A~ is projective. In particular f . O x is a coherent OA~-module, i.e., A is a finite module over the polynomial ring k [ x l , . . . , x n ] . So we are reduced to proving t h a t for 'general' linear combinations xi of the yj, the resulting linear space L is disjoint from X, t h a t is to say, from Y. This is done by a dimension count. We may regard the linear spaces L as parametrized by the k-rational points of Z = (A~' - {0})n; consider the incidence scheme F c H x k Z ~ p ~ - I defined by the vanishing of the polynomials ~--:~jm=ltijYj, 1 < i < n, where the tij, 1 _< j _< m are
Appendix D: Results from Algebraic Geometry
335
the coordinate functions on the ith factor of A m - {0} in Z. One finds easily that under the projection F ~ H, the fiber of F over any point x of m--I H is isomorphic to (Ak(x) -- {0}) n, since the condition t h a t a hyperplane pass through a point is a linear relation between the coefficients of its defining equation. Hence the inverse image F v of Y in F has dimension dim Y + n ( m - 1) < n - 1 + n ( m - 1) < m n - dim Z. Hence (see (D.30)) the closure of the image of F y in Z is a proper subscheme; for any k-rational point in the complement (i.e., for a non-empty Zariski open set in k m'') the corresponding linear space L is disjoint from Y. (D.70) We end this section with a brief discussion of the theory of faithfully flat descent, used in the main text in the discussion of SeveriBrauer schemes. A morphism f : U ~ X is faithfully flat if it is flat and surjective; this is equivalent to requiring that any diagram of O y - m o d u l e s .T" ---, .7" --, .7"' is an exact sequence if and only if f*Sr' -4 f*Sr --, f * ~ " is exact. One example of a faithfully flat morphism is as follows: if {U i} is a covering of X by open subschemes, and U = LI ui is the disjoint union of the Ui with the natural scheme structure, then the natural morphism U --, X is faithfully flat. The reader has seen several constructions (including the definition of a scheme) where local objects were defined together with patching d a t a of some kind. For example, we have seen (see (D.9)) that one may define a sheaf on X by giving sheaves ~'i on Ui, and isomorphisms
such that ~ojk o ~oij = ~zk in Hom (Jri lu, r~vjmuk, .T'k Iv, nv~mvk ). The 3ri determine a sheaf ~" on U. If pl, p2 are the two projections U x x U --, U, the ~ij a m o u n t to an isomorphism ~0 9 p ~ F ---, p~.T', satisfying a 'cocycle condition' p~3tp o p~2tp -- Pl3tpon * U • U • U. More generally, suppose f 9U ~ X is a faithfully flat morphism. One aspect of the theory of descent gives an equivalence of categories between quasi-coherent O x - m o d u l e s , and quasi-coherent Ou-modules with d e s c e n t data, i.e., O u - m o d u l e s ~" with isomorphisms ~o 9p ~ - , p ~ " satisfying the cocycle condition. To prove this, one reduces at once to the case when X is affine; replacing U by an affine open cover, we reduce further to the case when U is affine. Now we must show t h a t if B is a faithfully flat A-algebra, then the category of A-modules is equivalent to the category of B-modules M with additional data, consisting of an isomorphism of B | B-modules ~o 9 M | B -, B | M such t h a t (1 | ~o) o (~o | 1) - M |
B |
B ~ B |
B |
M
336
Algebraic K-Theory
equals ~, given by tensoring ~o with B | B | B, regarded as a B | Balgebra via the first and last factors. It is easy to see t h a t if N is an A-module, and M = B | N, t h e n ~N : ( B | N) | B ~ B | (B | N ) given by ~o((bl | n) | 52) = bl | (b2 | n) satisfies the cocycle condition. We claim t h a t N = {x E B | N [ ~pg(X | 1) = 1 | x}, which a m o u n t s to saying t h a t the sequence of A-modules O ---, N
a , B |
N
B, B |
B |
N
...(*)
is exact, where a ( n ) = 1 | n, and ~ ( x ) = 1 | x - ~ON(X | 1), i.e., f~(b | n) = 1 | 1 7 4 1 7 4
1|
Since B is faithfully flat over A, it suffices to check the exactness after tensoring (over A) with B. The tensored sequence is 0 --~ B |
1|
N
B |
B |
N
1|
B |
B |
B |
N
B u t 1 | a is split by the map a ' : b | b' | n ~ bb' | n. Similarly there is a map /~' : B | B | B | N --. B | B | N, bl | b2 | b3 | n H bib2 | b3 | n. One sees at once t h a t / 3 o B' + a ' o c~ is the identity on B | B | N , so t h a t the sequence has no homology at this term. In fact one can show by a similar a r g u m e n t t h a t the above 3 t e r m sequence extends to a resolution of A whose t e r m s are the A-algebras B | Now suppose given a B-module M w i t h descent data, i.e., a suitable ~. Let N = {x E M [ x | 1 = ~a(1 | x)}. T h e n N is an A-module, and there is a canonical B-linear map f : B | N --. M . We claim f is an isomorphism, and t h e m a p ~o corresponds to ~ON. B y construction, the sequence 0----~ N ---* M
~-~ B |
M
is exact, where 7(x) = 1 | x - ~o(x | 1). Since B is faithfully flat over A, the tensored sequence 0 ~ N|
B --, M |
B ~_~l B |
M |
is also exact. We see that the composite
N|
B --, M |
is given by n | b ~-~ 1 | bn.
B
~, B |
M
B
Appendix D: Results from Algebraic Geometry
337
We claim t h a t there is a commutative diagram M |
B
B |
M
~o1
~._@@1 B -~" ,
|
M | I (l|
B
B |
M
B |
where #(b | m) = 1 | b | m - b | 1 | m. Indeed,
(i
,,,:,) o
i)(m = 1|
b) = (i
,p)(1
,n
| b) - ~ ( m | 1 7 4
b) - (1
i)
b)
1) = #o~o(m |
where the second equality is because ~o satisfies the cocycle condition. Since the vertical arrows in the diagrams are isomorphisms, ~o induces an isomorphism ker(7 | 1) -4 ker(#). B u t the sequence 0 - - , M -~ B |
M
~; B |
B|
M
is exact, since B | B is a faithfully flat B-algebra; this is j u s t the sequence corresponding to (,) in this situation. Hence ~p induces an isomorphism N| B --, M , which we c o m p u t e d earlier to be the map given by n| ~-~ bn. This means the canonical m a p f " B | N ~ M is an isomorphism. This is applied in the main text as follows. We say t h a t f 9X --~ S is a S e v e r i - B r a u e r scheme over S if there is a faithfully flat m o r p h i s m ~r 9 S' --, S such t h a t X ' - X • S ' ~ S' is isomorphic to lP~, for some r > 0. For example, we could be considering varieties over a field which become isomorphic to projective space over the algebraic closure of the field. T h e n after replacing S' by a further faithfully flat extension, if needed, we m a y assume t h a t the n a t u r a l automorphism of X ' • X ' obtained by switching the factors is induced by an element ~0 of GL r + l ( F ( O ~ ) ) . Hence any G L r + l - e q u i v a r i a n t quasi-coherent sheaf ~0 on lP~. will determine a quasi-coherent sheaf ~-' on X ' , together with an isomorphism ~o* 9 p ~ - ' -~ p ~ - ' . This isomorphism will in general only satisfy the cocycle condition up to composing with a scalar automorphism; however if ~-0 is a P G L r + I equivariant sheaf, then jr, satisfies the cocycle condition, and yields a quasicoherent sheaf j r on X; we say j r is the sheaf on X determined by ~-0 via descent theory. As another illustration of these ideas, suppose k is a field, L a finite Galois extension, G = Gal ( L / k ) . Then B = L is a faithfully flat A = kalgebra, and by the normal basis theorem, B | B is n a t u r a l l y isomorphic to a direct product of copies Lg of L indexed by elements g E G. A coherent sheaf on Spec B is just a finite dimensional L-vector space V; if dim V - n and we choose a basis for V, an isomorphism ~o 9V | B B | V of B | B - m o d u l e s amounts to a function G --, G L n ( L ) , and
338
Algebraic K-Theory
the cocycle condition means that this is a cocycle in the sense of group cohomology, for the natural action of G on GL,~(L) (see Serre, Local Fields, for example). The assertion that V 'descends' to a k-vector space Vo such that B | V0 = L | V0 ~ V, under which the 'obvious' isomorphism ~0 : B | (B | V0) --* (B | V0) | B corresponds to ~, just means t h a t the above cocycle for group cohomology is a coboundary. Hence in this special situation, descent theory reduces to Hilbert's Theorem 90, and its generalization to GLn, i.e., that Hi(L/k, G L n ( L ) ) = 0. For a further discussion of descent theory, see Milne, Etale Cohomology, I, w and the more detailed references given there.
Bibliography H. Bass: Algebraic K-Theory, Benjamin, New York (1968). A. A. Beilinson: Coherent sheaves on IPn and problems of linear algebra, J. Funct. Anal. Appl. 12 (1978) 214-215 (English translation). S. Bloch, A. Ogus: Gersten's Conjecture and the Homology of Schemes, Ann. Sci. E.N.S. 7 (1974) 181-201. S. Bloch: Lectures on Algebraic Cycles, Duke Univ. Math. Ser. IV, Durham
(1976).
A. Borel: Stable real cohomology of arithmetic groups, Ann. Sci. E.N.S. 7 (1974) 235-272. H. Cartan, S. Eilenberg: Homological Algebra, Princeton Math. Ser. 19, Princeton (1956). A. Dold, R. Thom: Quasifaserungen und unendlische symmetrische Produkte, Ann. Math. 67 (1958) 239-281. S. Dutta, M. Hochster, J. E. McLaughlin: Modules of finite projective dimension with negative intersection multiplicity, Invent. Math. 79 (1985) 253-291. W. Fulton: Intersection Theory, Ergeb. Math. 3 (2), Springer-Verlag, New York (1984). P. Gabriel: Des cat6gories ab61iennes, Bull. Math. Soc. France 90 (1962) 323-448. P. Gabriel, M. Zisman: Calculus of b-~ctions and Homotopy Theory, Ergeb. Math. 35, Springer-Verlag, New York (1967). H. Gillet: Riemann-Roch Theorems for Higher Algebraic K-Theory, Adv. Math. 40 (1981) 203-289. D. Grayson: Higher Algebraic K-Theory H (after Daniel QuiUen), Lect. Notes in Math. 551, Springer-Verlag, New York (1976). D. Grayson: Localization for fiat modules in algebraic K-theory, J. Alg. 61 (1979) 463-496. P. A. Griffiths, J. W. Morgan: Rational Homotopy Theory and Differential Forms, Prog. Math. 16, Birkh/iuser, Boston (1981). R. Hartshorne: Algebraic Geometry, Grad. Texts in Math. 52, Springer-Verlag, New York (1977). M. Levine: Modules of finite length and K-groups of surface singularities, Compos. Math. 59 (1986) 21-40. M. Levine: Localization on Singular Varieties, Invent. Math. 91 (1988) 423-464. J.-L. Loday: K-th6orie alg6brique et repr4sentations de groupes, Ann. Sci. E.N.S. 9 (1976) 309--377.
340
Algebraic K-Theory
J.-L. Loday: Cyclic Homology, Grundlehren Math. 301, Springer-Verlag, New York (1992). S. Mac Lane: Categories for the Working Mathematician, Grad. Texts in Math. 5, Springer-Verlag, New Yrok (1972). J. P. May: Simplicial Objects in Algebraic Topology, Midway Reprints, Univ. Chicago Press (1982). J. P. May (with contributions by F. Quinn, N. Ray and J. Tornehave): Eoo ring spaces and E~o ring spectra, Lect. Notes in Math. 533, Springer-Verlag, New York (1976). A. S. Merkurjev, A. A. Suslin- K~-cohomology for Severi-Brauer varieties and the norm residue homomorphism, Math. USSR lzv. 21 (1983) 307-340 (English translation). A. S. Merkurjev: K2 of fields and the Brauer group, in Applications of Algebraic K-Theory to Algebraic Geometry and Number Theory, Part II, Contemp. Math. 55, A.M.S. (1986). J. S. Milne" Etale Cohomology, Princeton Math. Ser. 33, Princeton (1980). J. Milnor: The geometric realization of a semi-simplicial complex, Ann. Math. 65 (1957) 357-362. J. Milnor: On spaces having the homotopy type of a CW complex, Trans. A.M.S. 90 (1959) 272-280. J. Milnor: Introduction to Algebraic If-Theory, Ann. Math. Studies 72, Princeton University Press, Princeton (1971). J. Milnor, J. (3. Moore: On the structure of Hopf algebras, Ann. Math. 81 (1965) 211-264. P. Olum: Obstructions to extensions and homotopies, Ann. Math. 52 (2950) 1-50. A. I. Panin: On Algebraic K-theory of Generalized Flag Fibre Bundles and their Twisted Forms, in Algebraic K-Theory, ed. A. A. Suslin, Advances in Soviet Math. Vol. 4, A.M.S. (1991) 21-46. nmer. Math. Soc. (1991). S. Priddy: On ~~176176176 and the infinite symmetric group, Proc. Syrup. Pure Math. 22, A.M.S. (1971) 217-220. D. Quillen: On the cohomology and K-theory for the general linear group over a finite field, Ann. Math. 96 (1972) 552-586. D. Quillen: Higher Algebraic K-Theory I, Lect. Notes in Math. 341, SpringerVerlag, New York (1973). A. A. Rojtman: The torsion of the group of 0-cycles modulo rational equivalence, Ann. Math. 111 (1980) 553-569. J. Rosenberg: Algebraic K-Theory and Its Applications, Grad. Texts in Math. 147, Springer-Verlag, New York (1994). G. Segal: Classifying Spaces and Spectral Sequences, Publ. Math. I.H.E.S. 34 (1968) 105-112. J.-P. Serre: Local Fields, Grad. Texts. in Math. 67, Springer-Verlag, New York (1979). C. Sould: K2 et le groupe de Brauer (d'apr~s A. S. Merkurjev et A. A. Suslin), Sdm. Bourbaki 601, Nov. 1982, Astdrique 105-106, Soc. Math. France (1983).
Bibliography
341
V. Srinivas: Zero Cycles on a Singular Surface I, II, J. Reine A ng. Math. 359 (1985) 90-105 and 362 (1985) 4-27. N. Steenrod: A convenient categroy of topological spaces, Mich. Math. J. 14 (1967) 133-152. M. Stein: Surjective stability in dimension 0 for K2 and related functors, 7kans. A.M.S. 178 (1973) 165-191. A. A. Suslin: Algebraic K-theory and the norm residue homomorphism, J. Soviet Math. 30 (I 985) 2556-261 I. A. A. Suslin: Torsion in K2 of fields, K-Theory I (1987) 5-29. R. Swan: K-theory of quadric hypersurfaces, Ann. Math. 122 (1985) I13-153. J. Tate: Relations between K2 and Galois cohomology, Invent. Math. 36 (1976) 257-274. R. W. Thomason: Beware the phony multiplication on Quillen's A - ~ A , Proc. Amer. Math. Soc. 80 (1980) 569-573. R. W. Thomason, T. Trobaugh: Higher Algebraic K-Theory of Schemes and of Derived Categories, in The Grothendieck Festschrift, Volume III, Prog. in Math. 88, Birkh~iuser, Boston (1990). W. van der Kallen: T h e / ( 2 of rings with many units, Ann. Sci. E.N.S 10 (1977) 473-515. Sh. Wang: On the commutator subgroup of a simple algebra, Amer. J. Math. 72 (1950) 323-334. A. Weil: Basic Number Theory, Grundlehren Math. 144, Springer-Verlag, New York (1974). G. W. Whitehead: Elements of Homotopy Theory, Grad. Texts in Math. 61, Springer-Verlag, New York (1978).
