Homology of Linear Groups

Progress in Mathematics Volume 193

Series Editors H. Bass J. Oesterle A. Weinstein

Kevin P.P. Knudson Knudson Kevin

Ho mo log y Homology of Linear Lin ea r Groups Groups of

Birkhiiuser Verlag Birkhiiuser Verlag Basel - Boston - Berlin Basel . Boston . Berlin

Author: Kevin P. Knudson Department of Mathematics Wayne State University Detroit, Detroit. MI 48202 USA e-mail: [email protected] 2000 Mathematics Subject Classification 200 10

Knudson, Kevin P. (patrick), 19691%9Homology oflinear groups / Kevin P. Knudson. p. em. -- (Progress in mathematics; v. 193) Includes bibliographical references and index. ISBN 3764364157 (alk. paper) -- ISBN 0-8176-6415-7 (alk. (a1k. paper) 1. Linear algebraic groups. 2. Homology theory. I. Title. Title, II. II, Progress in mathematics (Boston, Mass.) ; vol. 193. QA179 .K59 2000 512'.55--dc21

00-057147

Deutsche Bibliothek Cataloging-in-Publication Cataloging-in-Pubtication Data Knudson, Kevin P.: Homology of linear groups / Kevin P. Knudson. - Basel; Boston; Berlin: Birkhliuser, 2001 (Progress in mathematics; Vol. 193) ISBN 3-7643-6415-7

ISBN 3-7643-6415-7 Birkhliuser Verlag, Verlag. Basel - Boston - Berlin

AlJ rights are reserved, whether the whole or part of This work is subject to copyright All the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained. © 2001 Birkhliuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Member of the BertelsmannSpringer Publishing Group Printed on acid-free paper produced of chlorine-free pulp. TCF ao 00 Printed in Germany ISBN 3-7643-6415-7

To Ellen, for inspiration; and to Gus, for distraction

Contents Prefa.ce Pre:fa.ce ................................................................ .................•••................•...........................

ix IX

Chapter 1. Topological Methods...................................... 1 Methods ........•...••••.....•...•............ 1.1. Finite Fields .................................................. 1 1.2. Quillen's Conjecture .......................................... 12 1.3. Etale homotopy theory........................................ theory ........................................ 14 1.4. Analytical Methods ........................................... 19 1.5. Unstable Calculations......................................... Calculations................. ..... ................ ... 21 1.6. Congruence Subgroups ........................................ ....................................... . 23 Exercises ........................................................... 29 Chapter 2. Stability. . . . .• . . . .• .• . . . . . . . . . . . . . . . . . . . . . . . . . . . .• .• .• . . . . . . . . . . 2.1. van der Kallen's Theorem..................................... Theorem ..................................... 2.2. Stability for rings with many units ............................ 2.3. Local rings and Milnor K-theory .............................. 2.4. Auxiliary stability results ..................................... 2.5. Stability via Homotopy........................................ Homotopy ......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. The Rank Conjecture......................................... Conjecture ......................................... Exercises ...........................................................

33 34 38 46 56 59 61 63

Chapter 3. Low-dimensional R.esults .................................. .......••.•.•....••••••••.....•.•• 3.1. Scissors Congruence ........................................... 3.2. The Bloch Group ............................................. 3.3. Extensions and Generalizations ................................ 3.4. Invariants of hyperbolic manifolds ............................. Exercises ...........................................................

65 65 70 82 87 90

Chapter 4. R.ank One Groups ........................................ Groups........................................ 4.1. SL 2 {Z[1/p]). (Z[1/pl). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. The Bruhat-Tits Tree ......................................... 4.3. SL 2 {k[t]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . (kl:tl)...................................................... 1 4.4. SL2{k[t, ]) ••.•••••••••••••••••••••••••..••••••••••••••••••• SL 2 (k[t, tell) ................................................. 4.5. Curves of Higher Genus ....................................... 4.6. Groups of Higher Rank ........................................ Exercises ...........................................................

91 91 95 96 97 99 105 115

viii

Contents

Chapter 5. The Friedlander-Milnor Conjecture ••••••••••••••••.•••..• ••••••••••••••••.•••... 5.1. Lie Groups .................................................... 5.2. Groups over Algebraically Closed Fields ....................... 5.3. Rigidity ................................. ',' . . . . . . . . . . . . . . . . . . .. 5.4. Stable R.esults Results ................................................. 5.5. HI, H 2 , and H3 ............................................... Exercises ...........................................................

117 117 121 132 139 144 146

Appendix A. Homology of Discrete Groups .•...••..••..•...•••...•.. A.1. A.l. Basic Concepts ............................................... A.2. Spectral Sequences ...........................................

149 149 156

Appendix B. Classifying Spaces and K-theory .•••.•••••••.••••.•••..• ........................ B.1. B.l. Classifying Spaces ............................................ B.2. K-theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. K-theory ................................................... "

165 165 168

Appendix C. Etale Cohomology ...................................... C.1. C.l. Etale Morphisms MorphismB and Henselian Rings ........................ C.2. Etale Cohomology ............................................ C.3. Simplicial Schemes ...........................................

173 173 177 181

Bibliography ........................................................... ...........•.........•.•...••..•....•.•......•..•.•.••....• 183 Index .................................................................. .................................................................................... 191

Preface With his definition of the higher algebraic K-groups of a ring, Daniel Quillen launched a new branch of mathematics. These groups are defined as the liomotopy groups of a rather complicated space BGL(R)+. This space is It modification of the classifying space of the infinite general linear group UL(R) = Un>l GLn(R); its homology groups coincide with those of the discrete group GL(R). Quillen'S calculation of the (co)homology of GLn(lFq) and tile resulting computation of the K-groups of IF'q were the first results in this JlOW field. In the intervening 25 years, some of the world's best mathematicians have d(~voted their energies to the study of Quillen'S K-groups. From the very be.,;inning, the importance of the unstable homology groups H.(GLn(R)) was Itpparent. Explicit computations have been hard to come by, though. For example, a group as simple as 8L4(7L.) has resisted the complete calculation of it.s homology. Not surprisingly, then, the K-groups of 1£ are still mysterious (although a great deal is known; for example, the 2-torsion in K.(7L.) has been dotermined completely). This monograph presents the current state of affairs in the study of the Ilomology of linear groups. I have tried to trace the development of the theory chronologically, beginning with Quillen's results and proceeding to the present. This linearity is interrupted occasionally, however. Chapter 1 is an overview of the early results of the subject. Quillen's (~lLlculation of H· (G Ln (IFq)) is presented along with certain conjectures· about t.he structure of H·(GLn(A), IFp)(due to Quillen and Lichtenbaum), where A iH a 7L.[l/p]-algebra. These conjectures have spurred the development of a great (btl of interesting mathematics, such as the etale K-theory of W. Dwyer and K Friedlander. Chapter 1 also includes a discussion of A. Borel's calculation of the stable cohomology of arithmetic groups (e.g., 8L(I£)) and the resulting ('(}Jlsequences for the K-theory of rings of integers in number fields. There is also a brief discussion of congruence subgroups. A thorough discussion of 1IIIIch of the material in Chapter 1 would require an entire book; for this 1'1l11.'iOn, I have chosen simply to outline or omit proofs of several results. The Illajor exception to this is the calculation of H· (G Ln ('!f q)). When proofs are t.I'Jlllcated, a reference is always provided. Chapter 2 presents the known stability results for H.(GL n ). The basic idea IH t.o study the extent to which the groups Hi (GLrJ stabilize IUj n increa.seH. We

x

Preface

first discuss a theorem of W. van der Kallen, a very general result in this area, and we prove a special case, due to Y. Nesterenko and A. Suslin, for so-called rings with many units. This stability result is then used to glean information about the K-groups of local rings. Chapter 2 also contains some results about the rank conjecture for infinite fields. Chapter 3 is concerned with low-dimensional homology groups, especially the group H 3 (CL 2 ). This group has a surprising connection to the study of scissors congruence classes of polytopes in hyperbolic space, an idea studied extensively by J. Dupont and C.-H. Sah. The relation between H3 and the so-called Bloch group (due to Suslin) is also discussed, along with generalizations and extensions due to the author, S. Yagunov, and P. Elbaz-Vincent. A connection between the Bloch group and hyperbolic 3-manifolds, discovered by W. Neumann and J. Yang, is also presented. Rank one groups are the focus of Chapter 4. It turns out that the homology of S L2 and PC L2 over certain rings is computable via actions of these groups on trees. The homology of the following groups is computed completely: SL 2(Z[1/p]) (Adem-Naffah), SL2(k[tl), SL2(k[t, ell) (Knudson), PCL 2 (k[C]) where C is a smooth affine curve of the form X - {p}, where X is projective over an infinite field k (Suslin, Knudson). The generalization of these results to groups of higher rank (due to Henn, SouIe, and the author) is also discussed, along with various applications. The final chapter (Chapter 5) provides a comprehensive account of the Friedlander-Milnor conjecture concerning the homology of algebraic groups made discrete. All known cases-solvable Lie groups (Milnor); solvable algebraic groups, tori, normalizers of maximal tori (Jardine); the stable groups CL (Suslin, Jardine); arbitrary groups over iFp (Friedlander-Mislin)-are discussed. The low-dimensional cases HI, H2 (Sah, Milnor) and H3 (C L2) (Sah, Knudson) are also proved. There are three appendices. The first provides a brief overview of the homology of discrete groups. The second recalls the basic notions of classifying spaces and the definition of K-theory (topological and algebraic). The third covers the fundamentals of etale cohomology. Also, I have included exercises at the end of each chapter. These vary from the routine to the very challenging. I hope they will prove to be useful to graduate students. This book should be accessible to graduate students who have a good working knowledge of algebraic topology and the fundamentals of group cohomology. Indeed, many of the calculations presented here are interesting applications of the spectral sequence techniques introduced in Chapter VII of K. Brown's excellent book [21]. An acquaintance with the basics of algebraic groups is helpful, but if the reader prefers, the following translation may be

Preface

xi

used: reductive group: GL m semi-simple group: SL n , Borel subgroup: upper t.riangular matrices, torus: diagonal subgroup, unipotent group: upper triangular matrices with 1's on the diagonal, Weyl group: the symmetric group on n letters. Spectral sequences are used extensively; the construction of the spectral sequence associated to a filtered complex is reviewed in Appendix A. No extensive knowledge of algebraic geometry or etale cohomology is required; however, we do assume that the reader understands the basics of scheme theory.

I would like to thank all those with whom I have had the opportunity to discuss the material in this book. These include Dick Hain, Andrei Suslin, Eric Friedlander, Rick Jardine, Howard Garland, John Harer, Jun Yang, Marc Levine, Chuck Weibel, Mark Walker, Serge Yagunov, Philippe Elbaz-Vincent, Burt Totaro, Rob de Jeu, and Hans-Werner Henn. lowe a special debt to Professor Suslin for freely sharing his ideas and for being an inspirational influence overall. I am also grateful to a pair of anonymous referees whose comments significantly enhanced the quality of this monograph. The first draft of this book was written while I was supported by aNat.ional Science Foundation Postdoctoral Fellowship at Northwestern University. I thank the NSF and Northwestern for their generous support. Detroit, :vIichigan April 2000

KK

Chapter 1

Topological Methods In this chapter, we discuss the topological methods available for attacking linear group homology. Most of these techniques have their foundations in the work of D. Quillen and A. Borel. One of Quillen's major contributions to mathematics was his definition of higher algebraic K -theory and subsequent calculation of the K -theory of finite fields [99J. This computation was a consequence of a complete description of He(GLn(k), lFl) for k finite and I =1= char k, and a partial description of He(GLn{k),lFp) for p = char k. These results provided a complete description of the integral cohomology of the stable group G L( k) which in turn yielded a homotopy equivalence BGL(k)+ ~ F'I!q, where F'I!q is a certain space with easily computed homotopy groups. The method of proof was purely topological (with some representation theory thrown in). Borel took a more analytical approach [14, 15J. He studied complexes of differential forms on symmetric spaces to compute the (stable) cohomology of arithmetic subgroups of GLn(C) (such as SLn{'Z)). This led to Borel's famous calculation of the ranks of the K -groups of rings of integers in number fields, a theorem which should be viewed as a higher dimensional analogue of the Dirichlet unit theorem. These influential papers spawned a great deal of interesting mathematics including the introduction of etale K-theory by Dwyer and Friedlander and the study of the relationship between cuspidal cohomology and the cohomology of arithmetic groups. We discuss this and other related topics below. Most of the proofs in this chapter, especially those in Section 1.4, are deliberately sketchy. This chapter is included mostly for completeness.

1.1. Finite Fields In this section we survey Quillen's remarkable paper [99J. Let k be a finite field with q = pd elements and let I be a prime distinct from p. We shall describe !.lIe structure of He(GLn(k),lFl). Denote by BU the infinite complex Grassmannian (see Appendix B). This Hpace represents complex topological K-theoryj i.e., for a compact space X

kO(X)

= [X, BUJ

1. Topological Methods

2

where kO(X) is the kernel of the dimension map KO(X) --+ KO(pt) = Z and [A, B] denotes the set of homotopy classes of maps from A to B. For a positive integer q there is an endomorphism 1jJq of KO(X) called the qth Adams operation. It is characterized by the properties that 1jJq is a ring homomorphism and if x E ~ (X) is the class of a line bundle, then 1jJq (x) = x q. The map 1jJq is induced by a map -q,q : BU --+ BU. This essentially follows from Yoneda's Lemma: the functor {(O is represented by [X, BUJ and hence any natural transformation on kO is represented by a map BU --+ BU. Of course, this argument does not quite work since BU is not compact. However, BU ean be realized as an increasing union of finite CW-complexes Xm with cells only in even dimensions. In such a C8.'le, one can show that kO(BU) = lli!!kO(Xm ), so that 1jJq is representable by wq. Denote by Fw q the fiber of the map 1 - w q : BU ----+ BU

which represents id - 1jJq. LEMMA

1.1.1. The homotopy groups of Fw q are

7f2i(Fwq) = 1I'2i-l(Fw q) PROOF.

0

Zj(qi -1).

Consider the long exact homotopy sequence

'"

---+

1I'k(BU) l~q 1I'k(BU)

----+

1I'k-l(Fw q) ----+

By Bott periodicity, 1I'2i-l(BU) = 0 and 1I'2i(BU) 1 - -q,q acts as multiplication by 1 - qi on 1I'2i.

= Z.

••••

One checks easily that 0

Quillen then computes the mod 1 cohomology of Fw q by describing a set of classes which form a basis. Fix 1 =1= p and let r ~ 1 be the least integer with qr == 1 mod l. THEOREM

1.1.2. The monomials

with 0 ::; OJ and 0 ::; {3j ::; 1 form a basis for H· (Fw q , IF I). Here deg( Cjr) = 2jr and deg(ejr) = 2jr - 1. If 1 =1= 2, or if 1 = 2 and q == 1 mod 4 ( the typical case), then eJr = 0 so that

H·(Fw q, IFl) ~ PIer, C2r, ... J ® /,.Je r , e2r, ... J where P denotes a polynomial algebra and Il?IP'T'

]F,.

A denotes

an exterior algebra

1.1. Finite Fields

3

PROOF. The idea of the proof is as follows. The space Fw q is obtained via a cartesian square

BU~qPUXBU

where BUI is the path space of BU and A sends a path to its endpoints (this is the general construction of the space of homotopy fixed points of a map ¢ : X ~ X). Associated to this square is an Ellenberg-Moore spectral sequence (see e.g., [111]) E~,t

= Tor~:(BUXBU)(He(BU),He(BUI)) ===> He(Fwq).

Let Ci E H2i(BU) denote the ith universal Chern class mod 1. Then we have

= P[Cl' C2, ... J = P[c~,~, ... ,c~,~, ... J = pri(Ci) and ~' = pr2(Ci). Set A = He(BU x BU). Then as Aone has He(Bu I ) = AIM and He(BU) = AIJ, where M (resp. J) He(BU) He(BU x BU)

where ~ modules, is the ideal with generators ~ - d! (resp. qi~ - d/). Denote by A j , 1 S j S 4 the subrings of A with the following sets of generators

(1) ~ - d! (2) ~ - d! (3) qi~ - d! (4) ~

fori for i for i for i

==0 ¢. 0 ¢. 0 == 0

modr modr modr modr.

By the Kiinneth formula

E2 = TorA(AIJ,AIM) ~ TorA1 (lFl' IF,) ®IF, ®IF, ®~. ,rhus we have an isomorphism

E2 ~ P[Cr, ~r, ... J ® /\[er, e2r, ... J where Cjr E Eg,2 j r and ejr E E;l,2 j r-l. Since E2 is generated by Eg,e and 1~21,e which are killed by the differentials (this is a second quadrant spectral H(lquence), it follows from the multiplicative structure of the spectral sequence I,hat

E2 = Eoo. This provides an isomorphism

grHe(Fw q) ~ P[Cr,C2n··.J ® /\[e r ,e2r, ... J ii,r Home filtration of

He(Fw q ).

4

1. Topological Methods

This gives the additive structure of He (Fw q, lFl ). The multiplicative structure is also given by this decomposition except that instead of having eJr = 0, we only know that eJr is in the subring generated by Cr, C2r, .... We now compute the mUltiplicative structure of He(Fwq,lFz) while simultaneously computing He(BGLn(k),lFz). To do this, we first need a map BGLn(k) ~ Fwqj equivalently, we must construct an element of the reduced K-theory of BGLn(k). Given a representation a : G ~ GLN(C) of a finite group G, there is an associated vector bundle on BG given as the fiber product EG x G eN, where G acts on eN via a and EG is the universal cover of BG. Thus, we need a virtual representation of GLn(k); i.e., a formal difference of representations of GLn{k) in the complex representation ring R(GLn(k)). This is achieved via Brauer lifting. Let k be an algebraic closure of k and choose an embedding -x

p: k

---+

ex.

Let G be a finite group and let a : G ~ GL(E) be a finite dimensional representation of Gover k. Let {Ai(g)} be the eigenvalues (counted with multiplicity) of a(g) E GL(E). Define the Brauer character of E to be the complex-valued function XE(g) = LP(Ai(9)). i

THEOREM 1.1.3 ([48], Theorem 1). The function XE is the character of a unique virtual complex representation pE E R( G). 0 Now suppose that E is a representation of Gover k and set E = E ®k k. One can define Adams operations on the representation ring R(G) (sec, for example, [8], Section 5.9). Since the set of eigenvalues {Ai{g)} is stable under the Frobenius map x 1--+ x q and since ('I/IqX)(g) = X(gq), we see that 'I/Iq(pE) = pE. Thus Brauer lift gives rise to a map Rk(G) ---+ R(G)t/J q sending E to pE. Now, the composition R(G) ~ KQ(BG) ~ f(Q(BG)

=

[BG,BU]

commutes with the operation 'I/Iq and hence we have a well-defined map R(G)t/J q ---+ [BG, BU]t/Jq.

By the Atiyah Completion Theorem (see, e.g., [9], p. 48), we have [BG, U] = 0, and hence, by Lemma 1 of [99], [BG, Fw q] ~ [BG, BU]t/J q.

1.1. Finite Fields

5

Thus, given a representation E of Gover k, we have associated a homotopy class of maps

E#: BG - - Fw q • Define mod 1 classes cjr(E),ejr(E) E He(G,lFl) by cjr(E) = (E#)*(cjr),

ejr(E) = (E#)*(ejr)

where Cjnejr E He(Fwq,IF,) are the generators. Now let k n denote the standard representation of GLn(k) on kn. Recall that qr == 1 mod l. THEOREM

1.1.4. The classes cjr(k n ), ejr(k n ) satisfy the following proper-

ties. 1. cjr(k n ) = ejr(k n ) = 0 for jr > n. 2.

The monomials er(kn)oq ... Cmr(kn)c>mer(kn),Bl '" emr(kn)C>m with 0 ~ Cij and 0 ~ {3j :::; 1, m = [n/r], form a basis for the ring He (GLn(k), 1F1).

In the typical case, then, we have an algebra isomorphism He (GLn(k), IFl) 9:! P[er (k n ), ... , cmr(k n )] 0 A[er(kn ), ... ,emr(kn )]. PROOF. If C denotes a cyclic group of order qr - 1, then C has an irreducible representation of dimension r over IF q via the isomorphism e ~ lF~r. Write n = mr+e where e < r. We can then embed C m into GLn(k) as r copies of e along the diagonal together with an e-dimensional trivial representation. We claim that the restriction map

He(GLn(k),lF l )

__

He(cm,lFt}

is injective. Assume first that 1 is odd. If G is a group, recall that the wreath product Esl G consists of elements (O";gl,'" ,gs) with multiplication defined by

(a'; g~, . .. ,g~)(a; gl, ... , gs) = (0"' 0"; g~(I)gl' ... , g~(s)gs). Now factor the embedding of C m in GLn(k) as em

~

Em ZGLr(lFq) ~ GLn(k).

Since 1 is odd, the index of the wreath product in GLn(k) is prime to 1 and hence the restriction map in mod I cohomology is injective. By a theorem of Nakaoka [90], the cohomology of a wreath product Em ZG is detected on Gm and Em X G. Also, the Sylow l-subgroups of Em are iterated wreath products of cyclic groups of order l. It follows that the cohomology of GLn(k) is detected on I-subgroups of exponent dividing qr -1. Since these are conjugate to subgroups of em, the re/mit followH.

1. Topological Methods

6

Now if 1 = 2, then r = 1 and m = n (recall that r is the multiplicative order of q modulo l). The index of the above wreath product is not odd, so we must find a different subgroup. In this case, consider the subgroup E[n/2]lGL2(IFq) x (IF; )n-2[n/2]. This subgroup has odd index. To complete the proof, we need only check that the mod 2 cohomology of GL2(IFq) is detected on abelian 2subgroups of exponent dividing q - 1. For this, one need only consider the Hochschild-Serre spectral sequence of the extension

1 - - t SL 2(k)

k X --t 1 and note that since the Sylow 2-subgroup of SL 2 (k) is quatemion, the cohomology of SL 2 (k) is periodic of order 4. This gives an upper bound for the cohomology of GL 2 (k) which in turn gives the result. ~ow,

--+

GL2(k)

--+

consider the maps

H·(Fwq,IFI)

--+

H·(GLn(k),IF I) - - t H·(Cm,JF I ).

The normalizer of C in GLr(lFq) contains a cyclic group of order r acting on C by raising elements to the qth power. Thus, the image of the second map above lies in the invariants of Em l7l/r. Thus we have

H·(Fwq,JFt)

--t

H·(GLn(k),IFl)

--t

((H·(C,lFq)Z/r)®m)~",.

The second map is injective, so we need only show that the composition is surjective to complete the proof of Theorems 1.1.2 and 1.1.4. The characteristic classes Cr and er of the r-dimensional representation of Cover IF q are nonzero; this follows by considering the ring structure of H·(C,JFl). Using the formula for computing the characteristic classes of a direct sum of representations [99], one sees that the classes Cr, ... , Cmr and er , ... , e mr of the n-dimensional representation of cm generate the invariants of the Em 17l/r action on H·(Cm , lFl)' It follows that the composite map above is surjective and that the second map is an isomorphism. We also see that we may evaluate the squares e;r by

re~~~~~. If 1 is odd, then H·(C,lFz) is generated by an element in degree 2 and an element in degree 1 which squares to O. If 1 = 2, then C has order q - 1. If q == 1 mod 4, then C has order divisible by 4 and the cohomology of C has the same structure as in the case of 1 odd. If q == 3 mod 4, then C is the product of a cyclic group of order 2 and a cyclic group of odd order so that H·(C,lFz) is a polynomial ring on a single generator of degree 1 [8], p. 67. Thus, in the typical case, we have eJr = 0 and in the exceptional case, we have eJr = Cb C2jr-l-b· This completes the proof of Theorem 1.1.4. 0

Et:o

This also completes the proof of Theorem 1.1.2. By passing to the limit, we obtain a map (unique up to homotopy)

BGL(k)

--t

Fw q,

0

I·. 1

1.1. Finite Fields

7

THEOREM 1.1.5. The above map induces an isomorphism

H-(Fw q, IFl) ~ H-(BGL(k), IFl).

o The mod p cohomology of GLn(k) is much more difficult to compute. Quillen did prove the following vanishing results. Recall that k has q = pd dements. THEOREM 1.1.6. Hi(GLn(k),JFp) = 0 for 0 < i < d(p - 1) and all n. PROOF. Let U be the subgroup of GLn(k) consisting of upper triangular matrices with ones on the diagonal and denote by T the diagonal subgroup of OLn(k). The group U is a Sylow p-subgroup of GLn(k), hence the restriction lIlap Hi(GLn(k)) - t Hi(U) is injective. Moreover, U is normalized by T so t.hat the image of the restriction map is contained in the subgroup Hi(U)T of invariants. Thus, it suffices to show that Hi(U)T = 0 for 0 < i < d(p - 1). Let k be an algebraic closure of k. The Grothendieck group of representat.ions of T over k may be identified with the integral group ring of Hom(T, kX). I}cnote by cl(V) the class of a representation V and write cl(V) = 2: nu u, where nu is the mUltiplicity of the character u in V. Define the Poincare series ()r H- (U) as a representation of T by

PS(He(u)) =

L cl(Hi(U) ®lFp k)zi i~O

where z is an indeterminate. This series has the form 2: niuuzi with i 2:: 0 and 'It E Hom(T, kX). We must show that nig = 0 for 0 < i < d(p - 1), where £ is I.he trivial character. -x Let a : T ---t k be a character and denote by ka the abelian group k with T-action t(x) = a(t)x. As an abelian group ka is an elementary abelian I'-group; its cohomology is given by

He(k ) a

~ {!\(k~[l]) ®lFp S(k~[2]) S(k~[l])

P -:f 2 p

=

2

where k~[il denotes the subspace of Hi(ka) isomorphic to

k:

= HomlFp(ka,JFp)

!\ and S denote the exterior and symmetric algebra functors over IFp' We It"V(~ a ring isomorphism k ®lFp k ~ (k)d with components x ® y ~ x Pb y where II .:: b < d. So as a representation of T over k, we have /l.lld

kn

®1F'1'

k = EB L(a Pb ) iJ

1. Topological Methods

8

where L( u) denotes the one-dimensional representation of T over k with charb acter u. Hence, k: 0lFp k ~ EBbL(a-P ) and therefore,

where on the right hand side, all ®, /\ and S are over k. Since the Poincare series of a tensor product is the product of the Poincare series of its factors, we see that for p f= 2 d-l

PS(He(ka)) =

II (1 +

a-pb

z)/(l -

a-pb

z2)

b=O

while for p

= 2 we have PS(He(ka))

=

II 1/(1 II(1 +

a- 2b z)

b

=

a- 2b z)/(l

-

a- 2 b+l Z2)

b

II(1 +

a- 2b

z)/(l- a- 2b z2)

b

since a2d = a. We now use the fact that the group U is an iterated extension of the onedimensional ka to obtain an upper bound on PS(He(u)). Let A+ be the set of positive roots. These are the homomorphisms t I-t ti/tj for 1 S j < i S n where ti is the ith entry of the diagonal matrix t. Order A + by (i', j') S (i, j) if either i' < i or if i' = i and j' S j. For a E A +, let Ua be the subgroup of U generated by the one-parameter subgroups corresponding to the roots greater than a. We have an extension of groups

1 - ka -U/Ua -U/Ual - 1 where a' denotes the element of A + immediately preceding a. This is a central extension and hence the associated spectral sequence has the form

This shows that

where « means that each coefficient on the left side is less than or equal to the corresponding coefficient on the right. Thus if we let a vary over A + and

1.1. Finite Fields

Os b < d,

9

then a

II(1 + a-pb a,b L(II

= =

z)

La- jpb z2j j~O

a- M " (J))zD(I)

1

a

where I runs over families (mab' nab) of integers with 0 S mab S 1, 0 S nab and

Ma(I)

L(mab + nab)pb

=

b

L(mab + 2nab). a,b

D(I)

Now, to show that Hi(U) does not contain the trivial character of T for 1) it suffices to show that for any family I satisfying

o < i < d(p -

II

a- M " (I)

=£

a

I.hat either D(I) = 0 or D(I) roots. Then we may write

~

d(p - 1). Let ai(t) = tHt/ti be the simple

i=l

where Cai = 1 for h S i < j and 0 otherwise (a(t) pmduct becomes

== tjlth). Then the above

n-l

II a

-e'•

i

= £,

ei

= L Cai(mab + nab)pb. a,b

i=l

The map T ~ (kX )n-l with components ai is surjective. Since k X is cyclic of order pd - 1, we have, for each 1 SiS n - 1,

LCai(mab +nab)pb == 0 mod (pd -1). a,b We now need the following number-theoretic fact. Let jb, 0 S b < d be lIollllcgative integers such that d-l

Ljbpb h:;:o

== 0 mod

(pd -1).

10

1. Topological Methods

If not all the jb are zero, then 'Ebjb ~ d(p - 1). Given this, suppose that the family I = {mab, nab} satisfies D(I) > O. Then for some a, b we have rnab + nab> 0 so that for some b and i we have 'Ea cai(rnab + nab) > O. Thus by the statement above we have

L

D(I) ~ ~)mab + nab) ~ Cai(mab + nab) ~ d(p - 1). a,b ab This completes the proof of Theorem 1.1.6. REMARK

o

1.1.7. An alternate proof of Theorem 1.1.6 is outlined in the

exercises. COROLLARY

1.1.8. If k' is an infinite algebraic extension ofJFp , then

Hi(GLn(k'),JFp) = 0 for all i

> O.

PROOF.

We have

He(GLn(k'), JFp ) =

ll!!! kCk,He(GLn(k), IFp) o

where k runs over all finite subfields of k'. COROLLARY

1.1.9. Hi(GL(k),JFp ) = 0 for all i > O.

PROOF. Recall that the mod 1 cohomology of GLn(k) was computed via "characteristic classes" for the standard representation. Let i > 0 and suppose x E Hi (G L (k), IFp). Interpret x as a characteristic class for representations over k as follows. A representation E of G with dim E = n determines a map G ~ GLn(k). Hence, associated to any E we get a homomorphism E# : G ~ GL(k) (unique up to conjugation) and we set x(E) = (E#)*(x) E Hi(G, IFp). Since Hi(GL(k),JFp) is the inverse limit of the Hi(GLn(k),IFp), it suffices to show that x(E) = 0 for all E to show that x = O. We proceed by induction on i, the case i = 1 being obvious since Hl(GL(k),JFp) ~ Hom(kX, JFp) = O. Suppose that Hj(GL(k), JFp) = 0 for 0 < j < i. Then x is a primitive class in He(GL(k),JFp). If E and E' are two representations of G, then we have

x(E EB E') = x(E)

+ x(E').

Choose an extension k' of k of degree d' with (d',p) = 1 and dd'(p - 1) > i. Them by Theorem 1.1.6, Hi(GLn(k'),JFp) = 0 for all n, hence there are no nOll-trivial i-dimensional characteristic classes for representations over k'. In particular, if a representation E' over k' is regarded as being over k, then :r:(E') = O. Now if E is any representation over k, we have () = x(E ®k k') = x(EG)tl') ShW(1 d' is prime to p, :r:(E) = O.

= d'x(E).

o

1.1. Finite Fields

This corollary allows call that there is a map

US

11

to compute the homotopy type of BGL(k)+. Re-

cp : BGL(k)

---t

F'Il q.

Since 7rl(F'Ilq) is abelian, this map can be extended to a map cp+ : BGL(k)+ ---t F'Il q. THEOREM 1.1.10. The map cp+ is a homotopy equivalence. PROOF. By the Whitehead theorem, it suffices to show that cp+ induces an isomorphism on integral (co) homology. In turn, to show this it suffices to show that cp+ induces a (co)homology isomorphism with '01, IFp , and IFl coefficients (1 i= p). Since H.(BGL(k),Z) ~ H.(BG~(k)+,Z), it suffices to show this for cp. By Theorem 1.1.5, the result holds for IFI-coefficients. By Theorem 1.1.6, the lFp-(co)homology of BGL(k) vanishes. The rational (co)hcimology of BGL(k) is also trivial since GL(k) is a union of finite groups. On the other hand, since F'Il q is a simple space with finite homotopy groups of order prime to p, its rational and mod p (co)homology are also trivial via the p-Iocal version of the Hurewicz theorem (see [85], Chapter 10). 0 COROLLARY 1.1.11. If k is a finite field with q elements, then for all i

K2i(k) K 2i - 1 (k)

=

~

1,

0

= Zj(qi -

1).

Moreover, if k ----+ k' is a homomorphism of finite fields, then the induced map K.(k) ----+ K.(k') is injective. PROOF. The calculation of the K-groups is clear. For the second statement, we may assume that k' is a subfield of k and that the map k ----+ k' is the standard inclusion. Let d be the degree of this field extension; then k' has q' = qd elements. Since I

[F'Il q, F'Il q 1 = [F'Il q, BUjW q

I

we see that there is at most one map F'Il q ----+ F'Ilql lying over a given endomorphism of BU. Since the Brauer lift of a representation E depends only on E ® k, the map BGLn(k) ----+ BGLn(k') lies over the identity of BU. Since

[BGL(k)+,BUl

= [BGL(k),nUj = \!!!!n[BGLn(k),BU],

the map BGL(k)+ ----+ BGL(k')+ corresponds to the unique map t : F'Il q F'Ilql lying over the identity of BU. Consider the identity d-l

1- 'ljJql = L'ljJqb(l - 'ljJq). h=O

----+

1. Topological Methods

12

This implies that we have a map of fibrations l-\II q

Fwq-BU-BU

I!

~ FW q' -

,!Et:~

\IIqb

l-\II q

BU -

BU.

Then the long exact homotopy sequences l_ q i

0-Z -

1T2i-l(Fw q) -

Z-

!t.

I) . )Ebqbi l_ q dt

0-Z -

0

,

Z -1T2i-l(Fwq) - - 0

o

show that t. is injective. COROLLARY

1.1.12. Let F be an algebraic closure ofIFp. Then for i

K2i(F) K 2i- 1 (F)

~

1,

0 =

EBQz/ZI. l#p

PROOF. Note that K.(F) is the direct limit of K.(k), where k runs over the finite subfields of F. Thus, K2i(F) = 0 is clear. From the previous corollary, the I-primary component of K2i-l(F) (1 =f. p) is a union of cyclic groups of order In with n - 00, hence it is isomorphic to Ql/Zl. 0

1.2. Quillen's Conjecture The particularly simple description of the mod 1 cohomology of GLn(IFq) is nppealing, but it is unrealistic to expect the structure of H·(GLn(R)) for other rings R to be as nice. Quillen's study of equivariant cohomology [98]100 him to make a conjecture which spurred the development of a great deal of mathematics. We discuss this in this section. In [98], Quillen proves that under certain hypotheses there is a one-toone correspondence between the minimal prime ideals of the cohomology ring H· (G, IFp) and conjugacy classes of maximal elementary abelian p-subgroups of G. Here, G is a compact group or a group with finite IFp cohomological dimeIlsion. Typical examples are finite groups and S-arithmetic subgroups of reductive groups over Ilumber fields. This result has certain consequences which led to QUillCIl'H conjoctural deHcription of H·(r, 1F1,) for r = GLn(A), where A iH t.he ring of 8-int.egors iII tl. numher field.

1.2. Quillen's Conjecture

13

Now, the cohomology ring H·(GLn{lF1d),IFp ) is the tensor product of a polynomial ring and an exterior algebra. Suppose that H· (r, IFp) has this form. Then this ring has only one minimal prime ideal and hence the group r has only one conjugacy class of maximal elementary abelian p-subgroups. However, in many cases, this does not happen (e.g., A = Z). Often there are maximal elementary abelian p-subgroups of different ranks and hence H· (r, IFp) cannot, in general, be a finitely generated free module over a polynomial subring. However, if p is a nnit in A, then things improve. For simplicity, assnme that A is the ring of S-integers in a number field containing a primitive pth root of unity, (p (e.g., A = Z[l/p, (pl). If A is an elementary abelian p-subgroup of r, decompose An into eigenspaces An = EBEi

corresponding to the characters of A. Since A is a Dedekind domain and each Ei is a projective A-module, we see that Ei is a sum of invertible modules. If A is maximal, it follows that Ei is invertible and that A ~ J.t;, where J.tP is the group of pth roots of unity. Moreover, the ith factor of A acts by multiplication on Ei and trivially on the other summands. Thus, all maximal A have rank n. We can now state Quillen's conjecture. Let Ci E H2i(r, IFp) be the image of the ith universal Chern class in H2i(BGLn (C)to P ,lFp ) under the map induced by the embedding A - C. (This image is obtained via the map induced by the composition GLn(A) - GLn(C) ~ GLn(C)to p , where £ is the continuous map A ~ A from the discrete group GLn(C) to the Lie group GLn(C)to P • We recall that H·(BGLn(C)to p , IFp) ~ H·(BU(n), IFp) ~ H·(G(n), IFp) ~ IFp[cl, ... , en], where G( n) is the Grassmannian of n-planes in Coo, see e.g., [49].) The elements CI,"" Cn generate a polynomial subring of H·(r, IFp) over which H·(r, IF p ) is a finitely generated module. CONJECTURE

1.2.1 ([98], p. 591). H·(r, IFp) is a free module over the sub-

ring IFp[ct, ... ,en].

This implies the following conjecture in many cases. If G is a group with subgroup K, we say that the cohomology of G is detected on K if the induced map H·(G, IFp) - H·(K, IFp) is injective. (,CONJECTURE 1.2.2 ([57], p. 51). The modp cohomology ofGLn(A) is detected on the subgroup Dn (A) of diagonal matrices.

This conjecture has been proved in a few cases and disproved in infinitely many others, but remarkably little else is known. The case A = Z[1/2] was proved for n = 2 by Mitchell [84] and for n = 3 by Henn [56] via explicit computations of the cohomology of GLn(A). Voevodsky recently announced a proof of the mod 2 LichtenbaumQuillen conjecture which implieH Conjecture 1..2.2 for A = Z[1/2] in the Htable mnge.

14

1. Topological Methods

A clever disproof of Conjecture 1.2.2 for GLn{A), A = Z[1j2J, n = 32, was given by Dwyer [32J. We sketch the line of proof. Let P and G be groups. Two homomorphisms a, (3 : P ~ G are conjugate if there is an element g E G with gag- 1 = (3. Let Pnl : GLn{A) ~ GLn(lR) and PF3 : GLn{A) - GL n (lF 3 ) be the obvious maps. Two maps a, (3 : P ~ GLn{A) are said to become conjugate over lR (resp. over !F3) if plR.a and Pnl(3 (resp. PF 3a and PF3(3) are conjugate. Dwyer proves the following result. THEOREM 1.2.3. Suppose the mod 2 cohomology ofGLn{A) is detected on diagonal matrices. Let P be a finite 2-group with homomorphisms a, (3 : P G Ln (A). Then a is conjugate to (3 if and only if a becomes conjugate to (3 over Rand !F3.

This leads to a disproof of Conjecture 1.2.2 for n = 32 as follows. Let J..ln denote the group of 2n th roots of unity. The smallest n such that the ideal class group of A{J..ln) (the integral closure of A in Q{J..ln)) is nontrivial is 6, and the rank of A(J..l6) as a A-module is 4>(26 ) = 25 = 32. Let P = J..l6 C A(J..l6) x and let I be a nonprincipal ideal in A{J..l6)' One uses the action of P on I and on A(J..l6) to construct two nonconjugate homomorphisms P ~ GL 32{A) which become conjugate over IR and over!F3 (this follows since I ® k ~ A(J..l6) ® k for k = JR, !F3). REMARK 1.2.4. Recently, Henn and Lannes have improved Dwyer's computation to n = 14. Moreover, Henn [55J proved that if Conjecture 1.2.2 is false for n = no, then it is false for all n ~ no. Also, using methods similar to Dwyer's, Anton [3J has disproved Conjecture 1.2.2 for A = Z[1/3, (3J and n ~ 27. He also provided a proof of 1.2.2 for n = 2.

The proof of Theorem 1.2.3 is homotopy theoretic in nature and relies on the construction of a space Xn and a map Xn : BGLn{A) ~ Xn satisfying (~crtaill properties. The space Xn is constructed using etale homotopy theory

[40j.

• 1.3. Etale homotopy theory A complete discussion of etale homotopy theory would lead us too far afield, bllt we can say a few things. This theory provides a covariant mechanism for assigning a (pro-)space X et to a (simplicial) scheme X. We provide some examples of etale homotopy types. EXAMPLE 1.3.1. Let k be a field and denote by 7r the Galois group over k of the separable algebraic closure of k. Then Spec{k)6t is a pro-space of type K(7r, 1). If S is a. complete loeal ring with rCHiciue field k, then the map Spcc(k)(\t. ~ Spec(S)"t iH Hon nquivl-l.lellcc. ThuH SpOC(C)"I. iH cont.ra.ct.ible, as

1.3. Etale homotopy theory

15

is Spec(oo. Both Spec(lFp)et and Spec(Zp)et are equivalent to the profinite completion of 8 1 • EXAMPLE 1.3.2. The etale homotopy type of a number ring is rather complicated in general. We give a partial description for the ring A = Z[lj2] which we shall use in the proof of Theorem 1.2.3. Choose an embedding lFa ~ C. Then we have a commutative diagram of pro-spaces spec(C)et - - spec(IR)et

1

spec(lFa) -

1

spec(A).

Since spec(C)et is contractible, we have an induced map spec(lFa)et V spec(IR)et --+ spec(A)ct. Choose an equivalence BZj2 ~ spec(IR)et and a map 8 1 ~ spec(lFa)et which sends the generator of 7[1(81) to the Frobenius automorphism of Fa over lFa. Then we have a map 8 1 V BZj2 --+ spec(A)et which by [34] induces an isomorphism in mod 2 cohomology. EXAMPLE 1.3.3. Let k be an algebraically closed field of characteristic zero. Then (BGLn,k)et is equivalent to the profinite completion of BGLn(CtoP ). EXAMPLE 1.3.4. If k is a field of characteristic zero with algebraic closure

k, then the sequence (BGLn,lC)et

--+

(BGLn,k)et

--+

Spec(k)et

is a fibration of pro-spaces. This yields an etale approximation to BGLn as follows. Let R = Z[ljl] and let A be an R-algebra. According to [33], 4.2, the space BGLn(A) can be identified with the basepoint component of the space of maps spec(A) ~ BGLn,R over spec(R). Denote by BGL~t(A) the basepoint component of the space of maps from spec(A)et to the fibrewise I-completion of (BGLn,R)et over spec(R)et. Func'toriality yields a natural map ¢n,A : BGLn(A) --+ BGL~t(A).

[n fact, one can perform the same construction over any affine group scheme G. PROOF OF THEOREM 1.2.3. Define the space Xn to be the space

BGL;;I(A)

ancllct

Xn: BGL.,.(A)

--t

X"

1. Topological Methods

16

be the natural map. Denote by G n the group GLn(A). Let BD., BG. and X. denote the spaces I1 BDn , I1 BG n and I1 X n , respectively. There are maps BD. ~ BG. ---S X •.

Under matrix block sum, these spaces are homotopy associative H-spaces; the maps above respect the multiplications. Note that BDl ~ B(Z x Z/2) ~ S1 x BZ/2.

Let e be the generator of Hl(S1) and (3k the generator of H k (BZ/2) (all homology in this proof is with Z/2 coefficients). Denote the classes e ® (3k-1 and 1 ® (3k in Hk(BD 1) by ak and bk respectively. Set

ar

= i*(ak),

br

= i*(bk), a~ = (xi)*(ak), b~

= (Xi)*(bk).

Since H.(BD.) is the free Z/2-algebra on the elements ak, k ~ 1 and bk, k ~ 0, and bf, and that the the image of i* is the subalgebra generated by the image of (Xi)* is generated by the classes a~ and b~. By the work of Mitchell [84], we have the following result.

af

1.3.5. The algebra H.(X.) is the free commutative Z/2algebra on the classes a~ and b~ subject to the following relations: 1. (a~)2 = 0 for k odd, and PROPOSITION

2. a:b~

+ a~_l bf + ... + afb~_l =

0 for k even.

0

ar br

Mitchell's results also imply that the classes and in H.(BG.) satisfy the analogous relations. From this we conclude the following. 1.3.6. If the mod 2 cohomology of G n is detected on D n } then : H· (Xn) --4 H· (BG n ) is an isomorphism. LEMMA

X;.

The map (Xnin)* is surjective and its kernel is equal to the kernel of (in)... Dualizing, we see that the cohomology map (Xnin)* is injective with image equal to the image of i~. Thus i~ is injective if and only if X~ is an isomorphism. 0 PROOF.

Now let P be a finite 2-group. If f is a discrete group, denote by {P, f} the set of conjugacy classes of homomorphisms P --4 f. There is an injective lIlap [32], 3.6, [BP, Xn] -

[BP, BGL n (F 3 )] x [BP, BGL~OP(~)]

where' denotes 2-completion. Moreover, a theorem of Carlsson [24] implies that the natural map {p,r} ~ [BP,Brl- [BP,Br] iK a hijection for

r

of virtually finite cohoIllologicnl ciinlOnKioll.

1.3. Etale homotopy theory LEMMA

17

1.3.7. If X~ is an isomorphism, then for any finite 2-group P, the

map Xn' (-): [BP,BGnl-- [BP,Xnl is a bijection. PROOF. The space Xn is 2-complete, so if X~ is an isomorphism, then Xn

is equivalent to BGn • Since Gn has virtual finite cohomological dimension, the above result finishes the proof. 0 LEMMA 1.3.8. Let P be a finite 2-group with homomorphisms a, (J : P ~ G n . Then Xn . (Ba) is homotopic to Xn' (Bf3) if and only if a becomes conjugate to f3 over IR and IF3'

PROOF. Consider the commutative diagram B(Ja

BPFs

[BP, BGLn(lR)l- [BP, BGnl- [BP, BGLn(IFg)l

!

!

!

[BP,BGL~t(lR)l- [BP,Xnl- [BP,BGL~t(IF3)1.

The right and left vertical arrows are bijections and the maps in the bottom row are injective. The result follows. 0 0

Lemmas 1.3.6, 1.3.7 and 1.3.8 together imply Theorem 1.2.3.

DEFINITION 1.3.9. Let A be a finitely generated R-algebra of finite mod I etale cohomological dimension. The etale K-theory space BGLet(A) is defined as BGLet(A) = colimn{BGL~t(A)} and the ith mod III etale K-group of A is defined to be Ktt(A, '£,/lll) = 1i"i(BGLet (A), '£,/111), i > 1.

Taking the limit of the maps ¢n,A defined above produces a map

¢ : BGL(A) __ BGLe\A) which can be lifted to a map

¢ : BGL(A)+ _

BGLet(A).

Then we have induced maps on homotopy:

¢r :Ki(A, '£,/ll/) -

Ktt(A, '£,/lll).

¢r

1.3.10 ([33]). The maps satisfy the following. 1. If A is a finite field of order q where 1 does not divide q, then isomorphism for all II > O. THEOREM

¢r is an

18

1. Topological Methods

2. If A is the ring of S -integers in a global field (and contains a primitive 4th root of unity if 1 = 2), then for i > 0 and v > 0 the map f is surjective. 3. If A is a local field or the coordinate ring of a smooth affine curve over 0 a finite field, then f is surjective for i > 0 and v > O. In a certain sense, then, the etale K-theory space BGLet(A) provides a plausible analogue of PiJ!q for the ring A. Conjecture 1.2.1 is equivalent to the conjecture that the maps f in 2 above are isomorphisms. Dwyer and Friedlander [35] used the spaces BGL~t(A) to obtain a lower bound for the cohomology of BGLn(A) for the ring A = Z[ljl, (d. They prove the following result. PROPOSITION 1.3.11. Let A be a noetherian Z[ljl]-algebra such that the Picard group Pic (A) has no infinitely I-divisible elements. Denote by Tn the diagonal subgroup of GLn. Then the natural map

induces an isomorphism

Note that the cohomology algebra H·(BTn(A), Zjl) is easily computable via the Kiinneth formula and the identification of Tl(A) with the group A x. This result is then used to prove the following. PROPOSITION 1.3.12. Let 1 be an odd regular primej that is, 1 does not divide the order of the class group of Z[(d. Let A denote the ring Z[ljl,(d. Then the natural map BGLn(A) ~ BGL~t(A) induces a monomorphism

The proof of this relies on the existence of a "good mod 1 model" of Ai details may be found in [35]. Dwyer and Friedlander also compute the graded ring associated to a filtration of H·(BGL~t(A),Zjl). It is isomorphic to the (1 + 1)j2-fold tensor power Os ®s ... ®s Os

where Os is the deRham complex of S = lFllcI, ... , cn ], dcg(cd = 2i. The above injectivity holds also for the stable group BGL(Z[ljl]).

1.4. Analytical Methods

19

1.4. Analytical Methods At around the same time as Quillen's calculation of K.(lFq), A. Borel, building upon previous work of Garland [44, 46J, Matsushima [78J, and Raghunathan [101J, was studying the cohomology of arithmetic subgroups of semisimple groups defined over Q. Let r be such a subgroup of the semisimple group G and denote by X the symmetric space GjK, where K is a maximal compact subgroup of the group G(JR) ofreal points of G. Then there is an isomorphism H·(O~,JR) ~ H·(r,JR),

where O~ is the complex of f -invariant smooth differential forms on X. This follows from the deRham theorem, since n~ computes the real cohomology of f\X, and since X is contractible the cohomology of r coincides with that of f\X up to torsion. Now let Ie be the complex of forms on X which are invariant under the identity component G(R)O of G(JR). Denote by If; the sub complex consisting of f-invariant forms. Since Ie consists of closed (in fact harmonic) forms, we obtain a homomorphism

r: If; ~ H·(f,JR).

If G(R)jf is compact, then standard results of Hodge theory imply that j. is injective in all dimensions. If G(JR)jf is not compact, it is still possible to show that j. is injective in a certain range explicitly computable from the algebraic structure of G. Results of Garland and Matsushima show that j. is surjective in a suitable range so that we have an isomorphism

r : If; -+ H·(r,JR) for • sufficiently small. THEOREM

the map

1.4.1 ([14]). There exist constants c(G) and m(G(JR» such that

i : (I~)i ~ Hi(f, JR)

is injective for i ~ c(G) and surjective for i ~ min{c(G), m(G(JR»)}.

0

The constant c( G) is defined in terms of the weights of G and is easily computed from this data. The constant m(G(JR» is defined using the Cartan decomposition of the Lie algebra g: 9

= ! El1 p,

where ! is the Lie algebra of K, along with the Killing form and a certain quadratic form on 1\ 2 p defined using the curvature tensor. Borel showed that ('(G),m(G(JR» ~ (rankQ(G»j4-1. This theorem allowed Borel to compute the real cohomology of the stable linear groups SL, 0, Sp, (ltc. over rings of integers in number fielcl.H. Utilizing

1. Topological Methods

20

Cartan's calculation of the homotopy of symmetric spaces he obtained the following result. Let k be a number field with ring of integers Ok. Let S be the set of archimedean places of k and let d = ri + 2r2 be the degree of k over Q. For v E S, denote by kv the completion of k at v; we have kv = JR or C depending on whether v is real or complex. Let G n be a semisimple group over Q with arithmetic subgroup fn- Assume G n '---t Gn+l and rankQGn - t 00 as n ----> 00. Denote by G~, 1 the corresponding group over k. Denote by G the direct limit li!n G n and by f the limit li!n f n·

f:

THEOREM 1.4.2. There is an isomorphism

He(f', JR) ~

0

Iv

vES

for G' and f' as in the following table.

G'

f'

SL(k) Sp(k) Here,

!\

Iv

v real !\(Xi), deg(xi) = 4i + 1 P(Xi), deg(xi) = 4i - 2

Iv v complex !\(Xi), deg(x;) = 2i + 1 !\(Xi), deg(xi) = 4i - 1.

denotes an exterior algebra and P denotes a polynomial algebra.

EXAMPLE 1.4.3. Suppose k = Q, Ok = IZ. Then we sec that

He(SL(IZ),JR) ~ !\(xi),deg(xi)

= 4i

+ 1.

In [15]' Borel considered the computation of He(f, E), where E is a nontrivial (complex) representation of G. The case HI had been considered previously by Raghunathan [101] who showed that HI (f, E) = 0 provided rankQ(G) > 1. Let E be a finite dimensional representation of G. The constants c(G) and m(G(JR)) can be modified to obtain constants c(G,E) and m(G(JR), E). Borel prOVC/l the following. THEOREM 1.4.4. Suppose that E contains no nontrivial subspace on which G(JR)O acts trivially. Then Hi(f,E) = 0 for i ~ min{c(G, E), m(G(JR), E)}.

EXAyrPLE 1.4.5 ..For the group r = SLn(IZ), the constants c and mare bounded by [(n - 1)/4]. Thus, H2(SLn(IZ), E) = 0 for n ;::: 9. Application: The real K-theory ofnurnber rings. Quillen defined the K-groups of a ring R as the homotopy groups of the H-space BGL(R)+: Ki(R) = 7r.i(BGL(R)+). Note that the universal cover of the space BGL(R)+ is (usually) BSL(R)+. Thus, Ki(Jl) e! 7ri(BS'L(R)+) for i :;::. 2,

1.5. Unstable Calculations

21

By the Milnor-Moore theorem[82], there is an injection 7ri(BGL(R)+) ® lR

------t

Hi(BGL(R)+, JR)

with image equal to the space of indecomposable elements. Thus, the calculation above gives the following. THEOREM 1.4.6. Let k be a number field with ring of integers Ok. Then fori;::: 2, dim(Ki(Ok)®JR) is periodic of period 4 and is equal to 0, rl +r2, 0, r2 depending on whether i == 0,1,2,3 mod 4. 0

A complete description of the torsion subgroups of K-groups of number rings remains elusive. Recent calculations by Rognes-Weibel [104] give the 2-torsion.

1.5. Unstable Calculations Twenty years after publishing [14], Borel, in collaboration with J. Yang, successfully computed the real cohomology algebras of SLn(k) for all number fields k. This calculation is carried out in [17] and relies on a hard analytical theorem of Blasius-Franke-Grunewald [10]. Before describing the result, we need to establish some notation. Let k be a number field, Voo (resp. VI) the set of archimedean (resp. non-archimedean) places of k and V = Voo U VI' If S c V, define Sf = S n Vf. The completion of k at v is denoted by kv. Let G be an affine algebraic group over k. If S is a finite subset of V, define r( 5, G) to be the sum of the kv ranks of G for v E 5. Set

Goo = G(k ®Q JR) =

II G(kv), vEVoo

which is viewed as a real Lie group. If M is a real Lie group and E = JR or C, then H~t(M, E) denotes the ith continuous cohomology group of G with coefficients in E; i.e., cohomology based on continuous (or differentiable) cochains (see e.g., IX in [16]). Let M8 be M viewed as a discrete group and let

f : H:t (M, E)

------t

He (M 6 , E)

be the natural map. The main result of [17] is the following. THEOREM 1.5.1. Let G be a connected, simply connected, almost absolutely simple k-group of strictly positive k-rank rk (G). Then the natural map

J-t:

H:t(Goo,JR)

---t

He(G(k),JR)

which is a composition of f with the restriction map He(G~, JR)

'/8 an isomorphism.

---t

He(G(k), JR)

22

1. Topological Methods

PROOF. Let S be a finite subset of V containing V00 and at least one finite place (Sf =I- 0). Let rs be a congruence S-arithmetic subgroup of G(k); e.g., if G <.....+ GL N , we may take

rs = G(Os) := G(k) n GLN(Os), where Os is the ring of S-integers of k. By Theorem 1 of [10],

He(rs, 1R)

~ H~t(Goo, 1R) ffi H~usp(rs, 1R).

The second summand is cuspidal cohomology. The relevant property of H~usp(rs,IR) is that it vanishes in degrees less than r(S" G). As S runs through a strictly increasing sequence of subsets of V, r(Sf, G) --+ 00. Thus if i E N, we have

F(rs,IR) ~ mt(Goo,IR) if S is sufficiently large. Since k = limscvOs, we have G(k) = limscvG(Os). Thus, for a fixed i, duality implies that

Hi(G(OS),IR)

~

Hi(G(k),IR)

for S sufficiently large (depending on i). Dualizing again, we see that

Hi(G(k),JR) ~ Hi(G(OS),JR) for large S and hence

F(G(k),IR) ~ mt(Goo,IR). One checks easily that this isomorphism is the one claimed in the statement of the theorem. 0 Theorem 1.5.1 can be used to compute the algebra He(SLn(k),IR) as follows. Let Gn = SLn be viewed as an algebraic group over k. Then

H:t(Gn,oo,JR) ~

® H~t(SLn(kv),JR). vEVoo

It is known, via [16], that

He (SL (k ) JR) = {He(SUnl SOn, JR) ct n v, He (SUn' JR)

v real v complex.

The values of the right hand side were computed by Borel in [13J. They are nodd neven

1.6. Congruence Subgroups

23

where the second sUbscript of Xn,j denotes the degree and deg(e n ) if m > n, then Xm,i f-+ Xn,i under the induced map

F(SLm(k v ), 1R)

----+

By Theorem 1.5.1, H-(SLn(k),IR)

= n.

Also,

Hi(SLn(k v ), 1R).

~ H~t(SLn,oo,IR).

H-(SL2(Q(i)), 1R) H- (SL 3 (Q( \I'2)),]a;)

=

For example,

A. (e2) A. (X2,3) A. (X3,5).

1.6. Congruence Subgroups The methods used in Section 1.4 can be applied equally well to subgroups of finite index in SLn('Z). In particular, we can consider the principal congruence subgroups f(n, k) defined by the short exact sequence 1 ----+ r(n, k)

----+

SLnCZ) m~k SLn(Zjk)

----+

1.

With rational coefficients, the cohomology of the groups r(n, k) and SLn(Z) are related by the simple formula

Hi(SLn(Z), Q)

~

Hi(f(n, k), Q)SLn(Zjk)

(this follows from the Hochschild-Serre spectral sequence). This is one motivation for studying the cohomology of these groups. A thorough study of congruence subgroups could fill an entire book. There are deep connections with the theory of modular forms, for example, which would take us too far afield. Instead, we present two results which are in keeping with the general theme of this monograph. 1.6.1. Principal congruence subgroups and K 3 (Z). Let p be a prime and consider the principal congruence subgroup r(n,p) C SLn(Z). Denote by.s£n(lFp) t,he Lie algebra of trace-zero matrices over IFp. THEOREM 1.6.1 ([75]). Let n

~

3. There is a homomorphism

cp : r(n,p)

----+

.srn(lFp)

whose kernel is the commutator subgroup. Thus, H 1 (r(n,p),Z)

~

.s£n(lFp).

PROOF. If X E r(n,p), we may write X = 1+ pA, where A is an n x n integral matrix and I is the identity. Define cp(X) = A mod p. This is a Ilomomorphism:

cp((I + pA)(I + pB))

=

= =

cp(J + p(A + B) + p2 AB) A+B modp cp(I + pA) + cp(I + 1)8).

1. Topological Methods

24

Moreover, if X = 1+ pA, then A has trace zero. Indeed, we have 1 = detX

=

det(I + pA)

==

1 + trace(A)

mod p2,

from which it follows that trace(A) == 0 mod p. The surjectivity of'P is shown as follows. Let eij be the matrix with 1 in the i, j position and zeroes elsewhere. Then s[n(lFp) has basis {eijhh U {di}z:ll where di = eii - eHl,Hl' Then

'P( I

+ peij) =

eij

and

'P(I + p(ei,Hl - eHl,i + di )) = ei,Hl - eHl,i + di· Thus, 'P hits all the basis elements. It follows that we have an exact sequence

1 - + r(n,p2)

-+

r(n,p) ~ S(n(IFp)

-+

O.

Let EnCl.) c SLn(Z) be the subgroup generated by all 1+ teij, i =I- j, t E Z. Let E( n, p) be the normal subgroup of r( n, p) generated by all 1+ teij, i =I- j, t == 0 mod p. By the work of Bass-Milnor-Serre [4], we have

r(n,p2) = E(n,p2). Moreover, we clearly have

E(n,p2) ~ [E(n,p),E(n,p)],

n ~ 3.

Thus,

r(n,p2) = E(n,p2) ~ [E(n,p),E(n,p)]; that is, r(n,p2) = [r(n,p),r(n,p)].

o

Lee and Szczarba also obtain an explicit description of the SL3(1F 3)-module H2(r(3,3)). These results arc applied in [76] to show that the group K3(Z) is cyclic of order 48. This stood as the largest n for which Kn(Z) had been computed completely until 1998 when Rognes [103] showed that K4(Z) = O. 1.6.2. Congruence subgroups over polynomial rings. Let R be a ring such that

SLn(R) is generated by elementary matrices (e.g. R Euclidean or local) and consider the exact sequence 1

-+

K(R)

-+

SLn(R[t]) ~ SLn(R)

-+

1.

The group K (R) consists of those matrices which are congruent to the identity modulo t. Define a sequence of groups

K(R) = Kl(R)

:J

K2(R)

:J ...

by Ki(R)

= {A

E K(R) : A

== I

mod ti}.

25

1.6. Congruence Subgroups

One checks easily that [Ki, Ki] ~ Ki+i and the graded quotients satisfy

Ki(R)1 Ki+l (R) ~ .5Cn(R). This is proved exactly as in the case above. Thus, we have an exact sequence

K2 (R) -----+ K (R) -----+ .5 Cn (R) -----+ 0, and hence, K2(R) ~ r2 K(R), where r e denotes the lower central series. Suppose R = F is a field. We studied the group Hl(K(R)) in [68]. The group K(F) acts on the Bruhat-Tits building associated to SLn(F[t]) (see 1 -----+

Section 4.6) and a fundamental domain can be computed. Via this approach we proved the following. THEOREM 1.6.2. Let ~ s[3(F).

F =

lF2

or

lF 3 .

H1 (K(F),Z)

If n = 3, we have an isomorphism 0

The technique of proof is much too complicated to work for larger finite fields. Indeed, the case n = 3, F = lF3 requires the computation of the rank of a 42 x 25 matrix. However, we expect that the following holds. CONJECTURE 1.6.3. If F is a finite field, then for all n isomorphism HI (K(F), Z) ~ .5[n(F).

~

3, there is an

This is false in general. For most fields F, there is a surjective map K 2 / r2 -----+

01,

01

where is the module of differentials; this is nonzero in general. Thus, the group HI (K(F)) is strictly larger than .5Cn(F) since we have an exact sequence 0-----+

K 2/r 2 -----+ HI (K(F))

-----+

sCn(F)

-----+ O.

01

Since = 0 for finite fields, this obstruction is not an issue. Also, one can compute Hl(K(F)) when n = 2 via the free product decomposition [66J K =

llsESL2(F)/B(F)SCS- 1

where C c K is the upper triangular subgroup and B(F) is the upper triangular subgroup of SL2(F). In particular, Hl(K(F)) is an infinite-dimensional P-vector space. 1.6.3. Malcev completions. Let r be a group. The Malcev completion of r is a prounipotent group M defined over Q together with a homomorphism r --+ M satisfying the obvious universal mapping property: If r --+ U is another map into a prounipotent group, then there is a unique map M --+ U making the cliagram

r ------ M

~lu

26

1. Topological Methods

commute. If G is nilpotent, the Malcev completion G can be identified with the set of "group like" elements of the completed group algebra QG (completed with respect to the augmentation ideal). Quillen [96] characterized G by the following three properties: 1. 2. 3.

G is nilpotent and uniquely divisible; The kernel of j : G -+ G is the torsion subgroup of G; If x E G, then xn E im(j) for some n =1= O.

If G is an arbitrary group, denote by Gr the nilpotent group GjrrG. Define the Malcev completion of G to be

This group is easily seen to satisfy the universal mapping property. A quick check of the definitions shows that if H1(r,Q) = 0, then theMalcev completion of r is trivial. Deligne suggested the following notion of relative completion to cover this situation. Suppose S is a semisimple group over Q and that p : r -+ S is a representation with Zariski dense image. The completion of r with respect to p is a proalgebraic group g over Q which is an extension of S by a prounipotent group P, and a lifting p : r -+ g of p with Zariski dense image. The group g is required to satisfy the obvious universal mapping property. If S is the trivial group, then this reduces to the usual Malcev completion. The basic properties of relative completion were worked out by Hain [52]. Ho used this idea in [53] to study the relative completions of the mapping class groups r g,r of orient able surfaces. With this approach, he was able to provide a lower bound on the second cohomology of the Torelli group Tg,r, which is the subgroup of r g,r consisting of elements that act trivially on the first homology of the surface. While he was able to compute the completion gg,r of r g,r with respect to the map r g,r -+ SP2g(7l.), the homomorphism r g,r -+ gg,r is a bit mysterious. In [72], we computed the completions of the groups SLn(7l.[t]) and SLn(7l.[t, rl]) with respect to the homomorphisms

and

The basic idea is to compute the ordinary Malccv completion of the kernel K(R) (R = 7l.[tJ, Z[t, t-I]); denote this complet.ion hy M(R). 11' we demot.e by

1.6. Congruence Subgroups

27

Q(R) the relative completion of SLn(R), then we have a commutative diagram

1 - - K(R) - - SLn(R) -

SLnCZ) -

1

111

1-

P(R) --~ Q(R) --~ SLn(Q) - - 1.

The universal mapping property of M(R) implies that we have a unique map M(R) - t P(R) through which K(R) - t P(R) factors. Note that there is an obvious candidate for Q(R); namely, the proalgebraic group SLn(Q[[T]]). We have a commutative diagram

where SLn(R) - t SLn(Q[[TJ]) is induced by t f-+ T for R = Z[t] and by t f-+ 1 + T for R = Z[t, rl]. The group U is the prounipotent subgroup of matrices congruent to I modulo T. Using the basic properties of relative completion in [52] and some results of Bousfield [19], we obtain the following results. PROPOSITION

1.6.4. If n

~

4, the map K(R)

-t

U is the Malcev comple-

tion. PROOF.

The first step is to show that for each r, the map K(R)jKT(R) ~ UjUT

(where U· is the T-adic filtration in U) is the Malcev completion of the nilpotent group K(R)jKT(R). This is straightforward from Quillen's criteria. Now, the kernel of the map Kjr TK - t UjUT is the abelian group KT jrT K. It remains to show that this group is torsion (it certainly contains the torsion subgroup of Kjr TK). For n ~ 4, we have an isomorphism (see e.g., [122]) KTjEn(R,mT) ~Kl(R,mT),

where En(R, mT) is the normal subgroup of SLn(R) generated by the matrices T + eij(x), i =I- j, x E mT (m = (t), resp. (t - 1)). We have En(R, mT) ~ P'K ~ K T, By the work of Roberts [102], the group Kl(R,mT) is torsion. This completes the proof. 0 1.6.5. This result is false for n = 2. The proof breaks down in this case because the Lie algebra S(2(Z) is not perfect. It is probably true for 'II. = 3, but we do not have a proof. REMARK

1. Topological Methods

28

COROLLARY 1.6.6. lfn ~ 4, then H 1 (K(R),Q)

= 's[n(Q).

PROOF. We have an exact sequence

o

The kernel is torsion. THEOREM 1.6.7. If n ~ 4, then SLn(R) with respect to the map SLn(R) --7 SLn(Q).

--7

SLn(Q[[T]]) is the completion

PROOF. Let P denote the prounipotent radical of the group Q(R). The map U --7 P is surjective [52] and its kernel is central in U. Since U has trivial center, the map is an isomorphism. Since Q(R) is the semidirect product of SLn(Q) and P, we must have Q(R) ~ SLn(Q[[T]]). 0 These results can be used to obtain information about the groups Q). One expects that these groups vanish. Indeed, if n ~ 5, then using van der Kallen's stability theorem 2.1.3 and the fundamental theorem of algebraic K-theory, we see that H2(SLn(R), Q) = o. The cases n = 3,4 were treated in [72] (the case n = 3 requires the conjectural result that U is the Malcev completion of K). Consider the Hochschild-Serre spectral sequence associated to the extension 1 ~ K(R) ~ SLn(R) ~ SLn('2..) ~ 1.

H 2 (SL n (R),

To show that H 2 (SL n (R),Q)

1. H2 (SLn ('2..) , Q) = OJ 2. Hl(SLn('2..), Hl(K(R), 3. HO(SLn('2..), H2(K(R),

= 0, it suffices to show

Q» Q»

= 0; =

o.

The first statement is clear since H2 (S Ln ('2..), '2..) is torsion for all n. The second follows from [101] since Hl(K(R), Q) is the adjoint representation S(n(Q). To obtain partial information about the third statement, we use continuous cohomology [51]. Let 1f be a group and define

H:tB (1f, Q) = lliQ H· (11" /rr1l", Q). The group H~tB (11", Q) coincides with Hk (11", Q) for k map

= 0, 1 and the natural

H~tB(1I",Q) ~ H2(1I",Q)

is injective. A group 11" is called pseudonilpotent if its continuous and ordinary cohomology coincide. ExampleM of pseudonilpotent groups include free groups and the pure braid groups.

Exercises

29

If P is the Malcev completion of 7r and P is the Lie algebra of P, then if HI (7r, Q) is finite dimensional, the natural map

H:ts (7r,Q).---+ H:ts(p,Q) is an isomorphism. Note that this is the case for the group K(R). THEOREM 1.6.8. lIn ~ 3, then HO(SLn(Z),H;ts(K(R),Q))

= O.

PROOF. Let u be the Lie algebra of U. The algebra u is the inverse limit of the nilpotent Lie algebras ujul . To prove the theorem, it suffices to show that for each I,

HO(SLn(Z), H2(K(R)j KI(R), Q)) = O. To do this, one uses Lie algebra cohomology and induction on I; the case I = 2 is obvious since

H2(K(R)jK2(R),Q) = H 2(.5Cn(Z),Q) =

A.

2

.5Cn(Q),

and direct computati9n shows that this group has no invariant sub module. The general case follows by considering the Hochschild-8erre spectral sequence for Lie algebra cohomology associated to the central extension

o .---+ ul ju!+!

----t

uju!+!

----t

uju!

----t

O.

o Exercises 1. Show that 1 - wq acts as multiplication by 1 - qi on 7r2i(BU). 2. Let G be a finite group and let E be a representation over k = IFq' Let E = E Q$)k k. Show that the eigenvalues {Ai(g)} are stable under the Frobenius map x f-+ x q • 3. Show that if I is odd, then the index of r;m l GLr(lF q ) in GLn{lF q ) is prime to I. 4. Verify the inequality for PS(H-(U)) given in the proof of Theorem 1.1.6. 5. Let jb, 0::; b < d be nonnegative integers such that d-l

Ljbpb

== 0 mod (pd -1).

b=O

If not all the jb are zero, show that L,bjb ~ d(p - 1). (Hint: choose a solution {jb} with L,jb minimal and positive. Argue that each jb < p and use the uniqueness of the p-adic expansion to show that jb = P - 1 for each b.)

1. Topological Methods

30

6. (Quillen, [39]) This exercise provides an alternate proof of Theorem 1.1.6 which also works for the groups 80 2m , 8P2m and U2m. First prove the following. Let V be an lFq-vector space and let A in IF; act by multiplication on V, where Al/b generates IF; and b divides p - 1. Then the induced map A* : Hn(V,lFp) -----+ Hn(V,lFp) has no invariants for 0 < n < d(p - l)/b (here, q = pd). (Hint: The structure of H*(V, IFp) is given in the proof of Theorem 1.1.6. If A,. has a fixed point, then the induced map A* on Hn(V0Fp lFp,lFp) has 1 as an eigenvalue. Examine the action of A* on l\i(V 0Fp lFp) ® 8 i (V ®Fp lFp) (i + 2j = n) to obtain a relation like the one in the previous exercise. The previous exercise finishes the proof.) Now, to prove Theorem 1.1.6, it suffices to prove the vanishing of the IFp-cohomology of Bn (IF q), the subgroup of upper triangular ma- trices. Proceed by induction on n, the case n = 1 being trivial since Bl(lFq) = GLl(lFq) = IF; is a group of order prime to p. For the general case, consider the short exact sequence

1 -----+ Rn (IF q)

-----+

Bn (IF q)

---+

B n- 1 (IF q)

-----+

1

where Rn(lFq) is the upper row subgroup of Bn(IFq). One has

Hi(Rn(lF q), lFp) = HO(lF; , Hi (V, lFp )) since Rn (IF q) is the semidirect product of IF; and V ~ lF~-l . The action of IF; is by multiplication, so the first part of the exercise finishes the proof. Now, to prove vanishing for the other groups, one needs to construct similar exact sequences of maximal solvable subgroups. These take the form 1 -----+ R(802m (lFq)) -----+ ~(802m(lFq))

1 -----+ R(8P2m(lFq)) 1

-----+

R(U2m (IFq))

---+

Bm(IFq)

-----+

1

-----+ ~(8P2m(lFq)) -----+

-----+ ~(U2m(lFq)) -----+

Bm(lFq) -----+ 1 Bm(IFq2) -----+ 1

where ~(G) denotes the maximal solvable subgroup of G. This group is defined by taking those matrices which preserve the standard maximal isotropic flag of lF~m. The action of A is by multiplication by A2. This gives vanishing of the IFp cohomology of these groups up to dimension d(p-l)/2 for 802m (p odd), and for 8P2m for allpj and up to dimension d(p-l) for U2m,' Verify nIl these assertions. (For Chm, identify this group

Exercises

31

with the subgroup of GL2m (lF q2) consisting of matrices M such that

M

7.

8.

9. 10.

(0

1m

1m) (M(q»)T = (0 1m) 0 1m 0

where M(q) is obtained from M by raising each entry to the qth power.) Show that for the filtration Ke(R) of the group K(R) of matrices congruent to the identity modulo t, the graded quotients Ki(R)jKi+1(R) are all isomorphic to .s[n(R). Let G be a group, let Gr be the nilpoterit group G jrrG and let Gr be the Malcev completion of Gr. Prove that M = ll!!! Gr is the Malcev completion of G. Show that if H 1 (G,Q) = 0, then the Malcev completion of G is trivial. Show that for each r, the map K(R)jKr(R) ~ UjU r

is the Malcev completion (this is true for n 11. Show that free groups are pseudonilpotent.

~

2).

Chapter 2

Stability Suppose we have a sequence G 1 C G2 C ... C Gn C . .. of groups. A natural question to ask is the following: for a fixed k, is there an integer n(k) such that the map Hk(G i ) ~ Hk(Gi+l) is an isomorphism for i ~ n(k)? The answer is certainly no if for example Gn is the free abelian group of rank n, but there are many examples for which stability does happen. For example, there are stability results for the sequence {En}n~l of symmetric groups [89] and also for the mapping class groups {rg,r }g,r~O of orient able surfaces [54] (9 = genus, r = number of boundary components); Hk stabilizes at n = 2k in the first case and at 9 = 3k in the second. In this chapter we study the stability question for the sequence

GL1(R) C GL 2 (R) C ... C GLn(R) C ... where GLn(R) is included in GLn+l (R) via

A~ (~ ~). This question has obvious implications for K-theory. If one can show that Hk stabilizes at some n(k):

Hk(GLn(k)(R),Z)

~

Hk(GL(R),Z),

then one need only concentrate on computing Hk(GLn(k)(R)). Granted, this may still be a very difficult problem, but in Chapter 4 we shall see that useful techniques are available for studying H.(GLn(R)) for finite n. Let us outline the general strategy for proving stability, since all such arguments are essentially the same. Suppose we have a sequence Gl C G2 C ... C G n C ...

of groups. The basic idea is to construct a simplicial complex Tn on which G n acts such that (1) Tn is highly connected, (2) G n acts transitively on simplices of fixed low dimension, and (3) for some k < n, Gk is the stabilizer in Gn of some ()..simplex.. One then uses the spectral sequence associated to this set-up to relate the homology of Gn to the homology of the stabilizers. Often the

2. Stability

34

sequence gives an isomorphism Hi(Gk) ~ Hi(G n ) for i small (depending on n). It is useful to keep the following example in mind. Consider the fibration SOn(lR.) ---t SOn+l(lR.) ---t sn and the resulting long exact homotopy sequence

... -- 7l'Hl(sn) -- 7l'i(SOn(lR.»

-+

7l'i(SOn+l(lR.» -- 7l'i(sn) -- ....

Since sn is (n - 1)-connected, we have an isomorphism

for i

~

n - 2 and a surjection 7l'n-l(SOn(l~»

->

7l'n-l(SOn+l(lR.».

To compute homology, we have to replace the long exact sequence by some sort of spectral sequence, but the philosophy is the same. The most general result proved with this method is due to van der Kallen [63]. We shall not give the complete proof (which occupies more than 30 journal pages), but we shall provide a proof in the case of rings with many units due to Nesterenko-Suslin [92]. This special case actually yields a better range of stability than van der Kallen's general result. We then discuss applications of the latter result to the Milnor K -theory of local rings. We also provide some auxiliary stability theorems. Another approach to proving homology stability is to consider the analogallS question for homotopy groups. The homology groups then may be studied via the Hurewicz map. Suslin [123] obtained such a result which yields a better stll.bility range than van der Kallen's theorem. We discuss this at the end of the chapter.

2.1. van der Kallen's Theorem The starting point for van der Kallen's theorem is some unpublished work of Quillen, where a certain simplicial complex is introduced. Quillen showed that for local rings the complex is highly connected and he conjectured that the same is true for finite dimensional noetherian rings. The main result of [63] proves this conjecture. In this section we describe Quillen's complex and sketch the proof of van der Kallen's result. Let R be an associative ring with unit. A vector (all"" an)T in K" is unimodular if the ideal generated by the ai is R. DEFINITION 2.1.1. The stable mnk of R, denoted sr(R), is the least natural number n such that for every unimodulll.r column (ao, ... , anf there are b1 , .•. , bn E R such t.hll.t (iLt + aob 1 , ••• , an + aobn)T is also unimodular.

35

2.1. van der Kallen's Theorem

For example, any local ring has stable rank 1. Let Roo denote the free right R-module with basis e], e2, ... and let R!" be the submodule generated by el," . , en. A collection of vectors Vb • •. , Vrn in Roo is called jointly unimodular if VI, •.• , Vrn form a basis of a free direct summand. DEFINITION 2.1.2. Let V be a set. Define O(V) to be the partially ordered set of ordered sequences of distinct elements of V with partial ordering defined in the following manner: (Vl"'" Vrn) ~ (WI, ... , w n ) if there is a strictly increasing function ¢: {1, ... ,m} - {1, ... ,n} with Vi = W¢(i)' A subset F of O(V) satisfies the chain condition if for each (WI,"') W n ) E F, we have (VI,""V rn ) E F whenever (VI, ... ,Vrn ) ~ (WI, ... ,Wn ). If F is such a subposet, F. denotes the simplicial set whose d-simplices are the (vo, ... , Vd) in

F. Let U be the subposet of O(ROO) consisting of jointly unimodular sequences. Such an element (VI,'" , Vk) is called a k-frame. THEOREM 2.1.3. Let R be a ring and set r = sr(R). Let fJ be 0 or 1. Then O(R!" + en+lfJ) nU is (n - r - I)-connected (that is, the geometric realization of the associated simplicial set is (n - r - I)-connected). PROOF. Proving the acyclicity ofthis set (i.e., the vanishing ofthe reduced homology groups) is not difficult if R has "many units", but the theorem also asserts that the geometric realization is simply connected. While acyclicity suffices for the purpose of computing homology, we sketch the proof of the stronger statement. The proof follows from some technical lemmas. We first introduce some terminology. If F is a poset and S a subset of F, then for x E F we write S+(x) for {y E S: y 2': x} and S-(x) for {y E S: y ~ x}. The link of x in S, denoted Links (x) is Links(x) = Links (x) U Link! (x), where Links(x) = {y E S : y < x} and Linkt(x) geometric realization of the link is the join: ILinks(x)1 = ILinks(x)1 We also have, for x tj. S, that IS over ILinks(x) I.

u {x}1

= {y

E S : y

> x}. The

* ILinkt(x)l·

is obtained from 181 by adding a cone

LEMMA 2.1.4. Let F be a poset, and let S be a subset such that for each x E F, the poset S-(x) has a supremum in itself. Then S i.~ a deformation retract of F.

36

2. Stability

Define r : F ~ S by r(x) = sup(S-(x». Then r(x) = x for ~ F is the inclusion, then i(r(x» ::; x for all x E F so that we have a natural transformation of functors ir ~ idF. This implies that on the level of geometric realizations, we have a homotopy equivalence i*r* ~ id, where 0* denotes the induced map on realization. 0 PROOF.

XES. If i : S

Recall that U is the set of frames in Jl:"O. Two frames (VI"," vn ) and (WI,"" Wm) are transversal if (VI,,'" V n , WI, ... , Wm) is also a frame. If F ~ U and (Vb ... ,Vn ) is a frame, we write FCvl, ... ,Vn) for the set of frames (Wb"" wm) with (WI, ... Wm, VI, ... , V n ) E F. 2.1.5. Let F ~ U satisfy the chain condition. Let d be an integer and (VI,''''V m ) E F be such that for all (Wl, ... ,wn ) E F+((vl, ... ,vm » the poset FCW1, ... ,w n ) is (d - n)-acyclic. Then the poset LinkF((VI,'''' vm » is (d - 1) -acyclic. LEMMA

PROOF. By hypothesis, for 0 ::; k ::; m - 2 there are natural one-to-one correspondences between k-simplices of Linkp ((Vb ... , V m », ~ubsets of size k + 1 of {I, ... , m} and k-dimensional faces of the standard (m - 1)simplex. It follows that ILinkF (( VI, ... , V m » I is the barycentric subdivision of the boundary of the standard (m - 1)-simplex and is hence an (m - 2)-sphere. Thus we have

ILink:F((Vl,"" vm»1 ~m-lILinkt((vl"'"

* ILinkt((Vl,"" vm»1 vm»I,

where E denotes the suspension functor. To complete the proof, it remains to lihow that Linkt (( VI, ... , V m » is (d - m)- acyclic. This is proved by induction 011 m. Denote by L the poset Linkt((Vl,'" ,vm ». Let Po = {(WI,'''' W n ) E L : Wn = vm } and let PI = {(WI,"" W r ) E F: for some n with 1 ::; n ::; r, (WI,"" W n ) E Po}. Evidently, Po is a deformation retract of Pt. If m = 1, then Po is isomorphic to F(Vl)' which is (d-l)-acyclic by hypothesis. If m > 1, then Po is isomorphic to Linkt(vm) ((VI, ... ,vm ». By the induction hypothesis, this is (( d-l) - (m-l) )-acyclic. Thus for m ~ 1, the poset PI is (d-m)-acyclic. The complement of PI in L consists of the elements (VI, ... , Vm , ZI, ... , Zq) in F with q ~ 1. Let Qr = PI U {(VI," .,Vm,ZI,'" ,Zq) E F: 1::; q::; r}. Then L is the union of the Qr and it suffices to show that each Qr is (d - m )-acyclic (the case r = 0 is already known). Observe that Qr+I - Qr is discrete; hence one passes from IQrl to IQr+II by adding a cone over the link in Qr of each (VI, ... , Vm , ZI, ... Zr+l) in F. By the argument above, this link is isomorphic to the r-fold suspension of Linkt ((Vi, ... , Vm , Zl, ... Zr+1»' Call this latter poset L'. When m = 1, L' h88 ali deformation retract

37

2.1. van der Kallen's Theorem

This poset is isomorphic to F(Vl,%1, ... ,%r+l),and hence is (d - r - 2)-acyclic (by hypothesis). Thus L' is (d - r - 2)-acyclic. Similarly, one can show that if m > 1, then L' is (d - m - I)-acyclic. Thus, one passes from IQrl to IQr+l1 by adding cones over (d - m - 1)-acyclic links. Since Qo is (d - m )-acyclic a 0 Mayer-Vietoris argument shows that Qr is (d - m)-acyclic. LEMMA 2.1.6. Let F ~ U satisfy the chain condition and let X ~ Roo. Assume that O(X) n F is d-connected and that for all frames (VI! ... , Vk) E F - O(X), the poset O(X) n F(Vl, ... ,V/,) is (d - k)-connected. Then F is dconnected. PROOF. Let Po = {(VI,"" Vk) E F : at least one Vi E X}. Then by Lemma 2.1.4, O(X)nF is a deformation retract of Po; hence Po is d-connected. Now let Pq = Po U {(WI! ... wr ) E F : r:::; q}. Consider LinkPq((wI! ... ,Wq+l)) for some (WI"'" Wq+1) E Pq+l - Pq. Let

Q = O(X U {(WI, ... , Wq+l)}) n F. Then Link pq ((WI, ... , wq+l)) has LinkQ ((WI, ... , Wq+l)) as a deformation retract and this retract is (d - I)-acyclic by Lemma 2.1.5. Thus, one passes from IPql to IPq+l1 by adding cones over (d - 1)- acyclic links. It follows easily that Pq is d-acyclic for all q ~ 0, and when d ~ 1 the van Kampen theorem shows that Pq is simply connected (hence d-connected). 0 Now note that F = Uq~O Pq • We now complete the proof of Theorem 2.1.3. Let F = O(Rn +en+Io) nu and set d = n-r-l (r = 8r(R)). We must show that F is d-connected. When n = 0, this is clear. Ifn > 0, choose X = (~-l +en+lo) U(Jr- 1+en +en+lo). By the induction hypothesis O(X) n F is d-connected since X is the same as (~-l + enol U (~-I + eno + en+l) up to a change of coordinates. One checks easily that all the conditions of Lemma 2.1.6 are satisfied and hence F is d-connected. 0 Van der Kallen then uses this result to study the stabilization of the h
=

max(I,8r(R) - 1). Then

and

Hm(En(R),Z) -- Hm(En+I(R),Z) are surjective for n

~

2m + c - 1 and injective for n

~

2m + e,.

o

38

2. Stability

We shall not prove this here. The proof involves a careful analysis of the spectral sequence obtained from the action of GLn{R) and En(R) on the geometric realization of the poset O{~) n U. In the next section, we provide a proof for rings with many units which gives a better range of stability than the above result. The argument used to prove Theorem 2.1.7 is essentially the Hame as the proof given below except that it works in greater generality. Van der Kallen also studied homology with twisted coefficients. For example, one has the standard representation ~ of GLn{R) or the adjoint representation grn(R). If Pn is a representation of GLn(R), then van der Kallen proves that Hm(GLn(R), Pn) stabilizes at approximately n = 2m + r - 1 + k, where k is the "degree" of the coefficient system. For example Pn = (~)®k has degree k. Details may be found in Section 5 of [63].

2.2. Stability for rings with many units DEFINITION 2.2.1. A ring A is an 8(n)-ring if there are al,"" an E AX such that the sum of each nonempty subfamily is a unit. If A is an 8{n)-ring for all n, then we say that A has many units.

I.;

I

Some examples of rings with many units are the following. 1. Infinite fields. Indeed, any field is clearly an 8{1 )-ring. Suppose F is an infinite field and assume inductively that F is an 8(n) ring. Let at. ... , an be a family of nonzero elements of F such that the sum of each nonempty subfamily is also nonzero. Since F is infinite, we can choose a nonzero element an+! such that -an+l is not equal to the sum of any subfamily of {all"" an}. Thus F is an 8{n + I)-ring. 2. Local rings with infinite residue fields. For each n we construct a set al, ... ,an with the requisite property by lifting such a set from the residue field. 3. Algebras over rings with many units. This is left as an exercise. In [92], Nesterenko and Sustin prove a strengthened version of Theorem 2.1.3 for rings with many units. This is based upon a computation of the homology of certain affine groups of the form

where Mnm (A) denotes the ring of n x m matrices over A. We firHt prove this result and then prove the stability theorem.

.~

2.2. Stability for rings with many units

39

2.2.1. Homology of affine groups.

THEOREM 2.2.2. Let A be a ring with many units. Then the inclusion i : GLn(A) ~ Affnm(A)

induces an isomorphism on integral homology. REMARK 2.2.3. This theorem is almost trivial if A has characteristic zero. Indeed, we have the Hochschild-8erre spectral sequence E~,q = Hp(GLn(A), Hq(Mnm(A)))

==?

Hp+q(Affnm(A))

and it suffices to show that E;,q = 0 for q > O. If char(A) = 0, then Mnm(A) is a torsion-free abelian group so that Hq(Mnm(A)) = I\q Mnm(A). The central subgroup AX C GLn(A) then contains an element a which acts as multiplication by a q on Hq(Mnm(A)) and hence in the same way on E~,q for all p ~ O. However, since A x is central, this action must be the identity and so Hp (GLn (A) , Hq(Mnm(A))) is annihilated by a q - 1. Since this space is a rational vector space (A is a Q-algebra) it must vanish for q > O. That this same argument works in positive characteristic is much more delicate. We need several lemmas. LEMMA 2.2.4. Let 'PI, ... ,'Pn : A -+ B be ring homomorphisms such that = 1 for all x E AX. Then A is not an Sen + 1)-ring.

I1~=1 'Pi(X)

PROOF. Suppose there are elements aI, ... , a n +I E A x such that for any 1 ~ ji ~ ... ~ jk ~ n + 1 the element ajl + ... + ajk E A x. Set I = {1, ... ,n + 1}. For any s = (ml,' .. ,mn ) E In, set 7I"(s) = I1:=1 'Pi(amJ and p(s) = {mt, ... ,mn} ~ I. Let J = {il < ... < jk} ~ I. Then by hypothesis n

1=

II 'Pi (ail + ... + ajk) = ~ 7I"(S). sEJn

i=l

We prove by induction on k = IJI that Lp(s)=J7I"(s) = (_1)k+ 1 . If k = 1, then p(s) = J for any S E ~ and the formula above proves the result. In the general case, the induction hypothesis gives

L

~

7I"(s)

7I"(s) -

sEJn

p(s)=J

L L

1- L(-1)IKI+I KcJ

(!) (_1)m

=

1 + };

=

(1 - 1)k - (_1)k (_l)k+1.

=

'r(s)

KCJ p(s)=K .

2. Stability

40

Applying this to I, we obtain a contradiction:

o=

~ 7r(s) = (_I)n. p(s)=I

(The first equality follows since no sequence s

= (m}, ... , mn)

E

In satisfies

pes) = I.)

0

From now on, assume that A has many units. If B is an abelian group, denote by Sn(B) the group (B®n)I:n of symmetric tensors (En denotes the symmetric group on n elements). Denote by r( B) the algebra of divided powers in B ([21], Chapter V). The ring reB) is a graded commutative algebra concentrated in even dimensions. Moreover, r 2n (B) = Sn(B) with multiplication IT(V®W) V'W=

(v E r2n(B), W E r2m(B)) and the system of divided powers is given by

=

L

IT(v ® ... ® v) uEI:n./T where T ~ Eni is the subgroup of permutations which permutes the blocks {1, ... , n}, {n + 1, ... , 2n}, ....

V(i)

LEMMA 2.2.5. Let cp : A --t B be a ring homomorphism and consider the induced AX-action on B®n: a(b 1 ® ... ® bn ) = cp(a)b1 ® ... ® cp(a)bn . Then Ho(Ax,B®n) = O.

PROOF. Denote by I the ideal of B®n generated by elements of the form cp(a)® ... ®cp(a)-l. Then Ho(Ax,B®n) = B®njI. Suppose I is a proper ideal and denote by CPi the map A --t B®n /1 given by CPi(a) = 1®·· ·®cp(a)®···® 1. Then for each a E AX, I1~=1 cpi(a) = 1, which contradicts Lemma 2.2.4. 0

Now assume that B is an algebra over Q or Fp so that the tensor and exterior powers of B over Z coincide with those over Q and F po 2.2.6. Let cp : A

LEMMA

--t

B be a ring homomorphism. Then

Ho(AX, Sn(B)) PROOF. S~l,. .. ,kr the

= O.

Let kl, ... , kr be natural numbers with E k i = n. Denote by subspace generated by tensors of the form

~ UEI:n/I:"l

x .. · xI:"r

IT(bl®·:·®bl,,® .. ·®~). kl

kr

Then S!l, ... ,kr is an AX-invariant subspace of Sn(B) and Sn(B) = E S!l, ... ,kr • Set S~i) = Er~n-i S~h ... ,k,.. Then the S~i) form an AX-invariant filtration.

41

2.2. Stability for rings with many units

One checks easily that if r B r ---* S~i) / S~i-l) given by O"E~n /~"l

= n-

i and n = kl

+ ... + kr

then the map

x··· X~kr

is multilinear and hence gives an AX-equivariant map B®r Thus we obtain an A x -equivariant surjection

EB

B®r

----->

---*

S~i) /S~i-l).

S~i) / S~i-l).

kl+ .. -+kr=n

Since Ho is a right exact functor, we have a surjection

Ho(AX

,EB B®r)

----->

Ho(A x ,S~i) / S~i-1));

Lemma 2.2.5 then shows that

Ho(A X, S~i) / S~i-l») = By induction on i, we see that Ho(AX, S~i»)

o.

= 0 for any i.

D

2.2.7. Letnlln2 ~ 0 withn = nl+n2. LetP andQ beB-modules. (p) ® Sn2(Q) has a natural Sn(B)-module structure.

LEMMA

Then

1\n

1

PROOF. Clearly, p®nl ®Q®n 2 is a B®n-module, hence an Sn(B)-module. Moreover, if U E ~nl X En2 , then u(vw) = u(v)lJ'(w) for all v E B®n, w E p®nl ®Q®n 2 • It follows that p®nl ®Sn2 (Q) is an Sn(B)-submodule of p®n 1 ® Q®n2. Since the kernel of the projection

p®nl ®Sn2(Q)

----->

I\n 1 (p)®Sn2(Q)

is generated by tensors w E p®nl ® Sn2(Q) satisfying w(ij) = w for some transposition (ij), it is naturally an Sn(B)-submodule. This proves the lemma. D LEMMA 2.2.8. Let A be a ring, M a A-module, G an abelian group and 'P : G ---* AX a homomorphism. If Ho(G, A) = 0, then Hp(G,M) = 0 for all p~ O.

PROOF. We have Ho(G, A) = A/I where I is generated by all 'P(g) - 1, 9 E G. By hypothesis, I = A. The group G acts on M via A-linear maps 1'10 that the groups Hp(G,M) have a natural A-module structure. The map (7g , Ug) : (G, M) ---* (G, M), where 7 g is conjugation by g and u g : M ---* Mis lIlultiplication by g, induces the identity on homology. On the other hand, since (j is abelian, 7 g = id so that the induced map on homology is multiplication hy 'P(g) EA. Thus, Hp(G,M) is annihilated by the generatorH of the ideal I and hence vanishes Aince I = A. D

2. Stability

42

REMARK 2.2.9. This last proof is a generalization of the center kills argument given in Appendix A.

COROLLARY

2.2.10. Under the above hypotheses,

PROOF. By Lemma 2.2.6, Ho(AX,Sn(B)) = O. Since I\n 1 P® Sn2(Q) is an Sn(B)-module by Lemma 2.2.7, the result follows. 0

Recall the structure of the homology of an abelian group N (see Appendix A). With rational coefficients we have Hp(N, Q) = I\P(N ® Q). With Fpcoefficients, the situation is more complicated, but we have an isomorphism

where pN denotes the p-torsion subgroup of N. Unfortunately, this isomorphism is not canonical since it depends on a choice of section of the map H2(N,Fp) ----> pN. Define a filtration on Hj(N,Fp) by setting

Hy)

=

EB Aj - 2\N/P) ®r

2i (pN).

i<::;r

One checks easily that this filtration is independent of the choice of section and there is a canonical isomorphism

PROPOSITION

j

2.2.11. Let k be a prime field. Then for all i

~

0 and all

> 0, we have Hi(A x, Hj(AS, k)) = 0 where A x acts on AS diagonally.

Set B = AS ® k. Then Hj(AB, k) is either I\l(B) or has an AX_ invariant filtration whose successive factors are I\j- 2i CB)®r 2i (M), where M = pAR is a B = AS /p-module. In either case, the result follows from Corollary 2.2.10. 0 PROOF.

We now prove Theorem 2.2.2. It suffices to check that i induces an isomorphism on homology with coefficients in any prime field k. For this it suffices to show that Hp(GLn(A),Hq(Mnm(A),k)) = 0 for q> 0 (see Remark 2.2.3). Identify AX with the scalar matrices in GLn(A). Then the Hochschild-Serre spectral sequence

along with PropoHition 2.2.11 finiHheH the proof.

o

2.2. Stability for rings with many units

43

2.2.2. The stability theorem. Denote by U(An) the simplicial set whose ksimplices are the unimodular frames (vo, . .. , Vk), Vi E An, with the usual face operators Oi: (vo, ... ,Vk) f-+ (vo, ... ,Vi, ... ,Vk). Let SU(An) be the simplicial subset consisting of special unimodular frames; that is, those frames (vo, . .. , Vk) which can be completed to a basis of An. The following result is easily proved by induction on k. LEMMA 2.2.12. Let (VI,"" Vk) be a unimodular frame in An with n ~ k+ sr(A). Then there is a matrix a E En(R) with a(vl, ... , Vk) = (el," ., ek). D

Lemma 2.2.12 implies that for k ::::; n-sr(A) -1, the k-simplices of U(An) coincide with those of SU(An): SU(An)k' = U(Ank LEMMA 2.2.13. The simplicial set SU(An) is (n - sr(A) - I)-acyclic. PROOF. Set r = sr(A). By Theorem 2.1.3, the sct U(An) is ('{I- - r - 1)acyclic; this immediately implies the (n - r - 2)-acyclicity of SU(An). We need only show that Hn _ r _l(SU(An)) vanishes. For this it suffices to show that if u = (vo, ... ,Vn - r ) E U(An) is a unimodular frame, then the (nr - I)-cycle du is a boundary in SU(An). Using Lemma 2.2.12 we reduce to the following special case: u = (el,"" en - r , el + .. , + en - r + v), where v E Aen - r+1 + ... + Aen is unimodular. Denote by w the frame (el, ... , en - n el + ... + en - r + V, en - r +1)' This frame is not unimodular in general, but we do have u = On-r+1(w). Moreover, for i = 0, ... , n - r, Oi(W) is clearly a special unimodular frame. Consider the formula

0= d2 (w)

= dC~\-I)iOi(W)).

From this we conclude that dtt = (_I)n+r

d(~( -1)iOi (W))

is a boundary in SU(An).

D

Denote by C.(An) the augmented complex associated to SU(An) with dimensions shifted by 1; thus, Co = Z and Ck = Z[SU(An)k_lJ. By Lemma 2.2.13, we see that Hi(C.(An)) = 0 for i ::::; n - sr(A). The group GLn(A) acts naturally on Ck(An) (on the left), turning C.(An) into a complex of GLn(A)modules. Consider the two spectral sequences associated to the action of GLn(A) on C.(An) (see Appendix A). The first sequence has

E;,q = Hp(GLn(A), Hq(C.))

=0

2. Stability

44

for q ~ n - sr(A). Thus, the common limit of the spectral sequences is 0 in dimensions less than n - sr(A). From now on, assume that A has many units and set r = sr( A). The (transposed) second spectral sequence has E~,q = Hp(GLn(A), Cq). The group GLn(A) permutes transitively the elements of C q and the stabilizer of the frame (e n- q, ... , en) is the affine group Aff (A) = (GLn_q(A) n-q,q Mq,n_q(A)

0) I

.

By Shapiro's lemma and Theorem 2.2.2, we obtain the following result. LEMMA

2.2.14. E~,q

Hp(GLn(A) , Cq) Hp(Affn_q,q(A) , Z) Hp(GLn_q(A), Z).

We now compute the differential d 1 : E~,q - t E~,q_l (recall that we have transposed the spectral sequence; i.e., in Appendix A, we have E~,q = Hq(GLn(A), Cp)). This map has the form

Hp(GLn-q(A). Z)

---->

Denote by i the embedding GLn_q(A) LEMMA

Hp(GLn_q+l (A), Z). -t

GL n- q+1(A).

2.2.15. The homomorphism d 1 is zero if q is even and equals i*

if q is odd.

The homomorphism d 1 is induced by d : C q - t C q - 1 . Since d = Ek:'~( _l)kok' it suffices to check that for any k, the map (Ok)* coincides with i •. By construction the isomorphism PROOF.

Hp(GLn(A), Cq) ----> Hp(GLn_q(A), Z) is induced by the morphism

(GLn_q(A),Z) (in~q) (GLn(A),Cq) where i n- q is the standard embedding and uq maps 1 to the frame (e n- q+1, ... , en). The composition (lG, ok)(i n- q, uq) coincides with the map (i n- q, OkUq), while the composition (in-q+l,Uq-l)(i, 1z) agrees with the map (in-q,Uq-l). We must prove that these two compositions induce the same map on homology. Observe that the frame OkUq = (en-q+l, ... , en- q+k+1,"" en) can be obtained from Uq-l by multiplication by the permutation matrix (J corresponding to the cyclic permutation (n - q + k + 1,n - q + k, ... , n - q + 1). Since (J commutes with i n - q , we have

45

2.2. Stability for rings with many units

where c( (j) is conjugation by (j. But conjugation induces the identity on homology. D COROLLARY 2.2.16.

E2 = {ker(Hp(GLn_q(A),Z) -----+ Hp(GLn- q+1(A),Z) p,q coker(Hp(GL n _ q_ 1(A),Z) ---+ Hp(GLn_q(A),Z) PROPOSITION 2.2.17. The differentials d", for r 2 - E oo E p,q - p,q'

~

q odd q even.

2, are trivial. Hence,

PROOF. Proceed by induction on n. For n = 0, 1, the sequence is concentrated in the first two columns so that the higher differentials vanish. Let n ~ 2. If G. is a complex, denote by G.[m] the shifted complex with Gk[m] = GHm' Define a homomorphism of complexes 'IjJ : G.(An-2)[-2] ---+ G.(An) by

'IjJ(V1' ... ,Vq-2)

= (VI,' .. ,Vq-2, en-I, en) - (V1, . .. ,Vq-2, en-1, en-1 + en) + (V1, . .. ,Vq-2, en, en-1 + en).

The map 'IjJ respects the embedding GLn_2(A) ---+ GLn(A) and hence induces a map of spectral sequences 'IjJ* : E(An-2)[o, -2] ---+ E(An). Denote the first sequence by E. We have E~,q = E(A n- 2)p,q_2 = Hp(GLn_q(A),Z)

for 2 ~ q ~ n and E~,q = 0 for q = 0, 1. An argument similar to the proof of Lemma 2.2.15 shows that for 2 ~ q ~ n the map -1

'IjJ* : Ep,q = Hp(GLn_q(A))

-----+

1

Ep,q = Hp(GLn_q(A))

is the identity. It follows that E;',q ~ E;',q for 2 ~ q ~ n. We now prove that di = 0 by induction on i ~ 2. We know that J,i vanishes by our induction hypothesis on n (i.e., we have assumed that di = 0 for k < n). Moreover, the differentials starting in the first two rows vanish for dimension reasons (i.e., they map into the fourth quadrant). If q ~ 2, then by the induction hypothesis on i, E;,q = E;',q = E;',q = E;,q. Consider the commutative diagram

This shows that di =

o.

[]

2. Stability

46

THEOREM 2.2.18. If A is a ring with many units, then the natural map

Hp(GLn-l(A),Z)

---?

Hp(GLn(A),Z)

is surjective for p ~ n - sr(A) and bijective for p ~ n - sr(A) - 1. PROOF. The limit of the spectral sequence is 0 in dimensions Now using 2.2.16 and 2.2.17 we get

~

n-sr(A).

coker(Hp(GLn_l(A),Z) ~ Hp(GLn(A),Z)) = E~,o = E~o = 0 for p ~ n - sr(A) and

ker(Hp(GLn_l(A),Z) ~ Hp(GLn(A),Z)) = E~,l = E~l = 0 for p + 1 ~ n - sr(A).

D

2.3. Local rings and Milnor K-theory Suppose that A is a local ring with infinite residue field k. Since sr(A) = 1, Theorem 2.2.18 implies that Hn(GL(A)) = Hn(GLn+l(A)) and that the map Hn(GLn(A)) ~ Hn(GLn+1(A)) is surjective. In fact, the latter map is injective and the quotient Hn (G Ln (A)) j H n (G L n- l (A)) has a particularly nice description. Denote by T.(A X) the tensor algebra of the abelian group A x. Note that To (A X) = Z and Tl (A X) = A x. Let I be the homogeneous ideal generated by all elements of the form a® (1- a) for a, 1- a E A x. The algebra T.(A X)j I is denoted by K~ (A) and is called the Milnor ring of A. We have Kr (A) = Z and Ktt (A) = A x. If we denote the image of al ® ... ® an in K!, (A) by {al' ... , an}, then multiplication is given by

{all ... , an}' {bl , ... , bm} = {al, ... , an, bl"" bm}. Moreover, K!, (A) can be presented as the abelian group with generators {a1, ... , an} and relations 1. {al,"" aia~, . .. ,an} 2. {al, ... , an} = 0 if ai

= {al,' .. ,ai, . .. ,an} + {al,' .. ,a~, ... ,an}

+ ai+l

= 1 for some i.

For any n, m we have natural homomorphisms

Hn(GLn(A)) ® Hm(GLm(A))

---? ----1

Hn+m(GLn(A) x GLm(A)) Hn+m(GLnTm(A))

induced by the composition of the homology product with the map induced by the inclusion GLn,(A) x GLm(A) ~ GLn+m(A). In particular, we have a map

e: AX ® ... ~ AX = H\ (GLI(A)) ® ... ~ H\(GL\(A)) -- Hn(GLn(A)).

'

47

2.3. Local rings and Milnor K-theory

Composing with the quotient homomorphism, we obtain a map

8: A x ® ... ® A x

---+

Moreover, if ai + aj = 1 for some i induces a map, still denoted bye, K~ (A)

Hn(GLn(A))/ Hn(GLn-I(A)).

i=

j, then 8(al ® ... ® an)

= O. Thus, 8

Hn(GLn(A))/Hn(GLn-l (A)).

---+

THEOREM 2.3.1. Let A be a local ring with infinite residue field. Then 1. The map Hn(GLn(A), Z) 2. The map 8: K~(A) phism.

---+

---+

Hn(GLn+l (A), Z) is an isomorphism;

Hn(GLn(A))/Hn(GLn-l(A)) is an isomor-

PROOF. Recall the simplicial sets SU(An) ~ U(An). Since A is local, every projective A-module is free, and hence every unimodular frame in An is special; that is, SU(An) = U(An). Let V, W be finitely generated free A-modules. If V has rank n, then we say that vectors VI, ... , Vs E V are in general position if any min(n, s) of them are jointly unimodular. Define simplicial sets U(V, W) c GP(V, W) as follows. The p-simplices of U(V, W) (resp. GP(V, W)) are the frames Vi E

V,Wi E W

such that the vectors vo, ... , vp are jointly unimodular (resp. are in general position). Denote by C.(V, W) (resp. C.(V, W)) the associated augmented chain complexes with dimensions shifted by 1. If V s=' An, then Ci (V, W) = Ci(V, W) for i ~ nand Ci(V, W) = 0 for i > n. LEMMA 2.3.2. For any V, W, the complex

C. (V, W)

is acyclic.

PROOF. Let

be a p-cycle. Since the residue field k is infinite, we can find V E V such that (denoting by x E Vim s=' k n the image of x E V modulo the maximal ideal m) for any j, (v, vi, ... , v~) and hence (v, vi, ... ,V~) are in general position. Now pick any W E Wand set

One chcckH eaHily thnt rlz

= y.

D

2. Stability

48 COROLLARY

2.3.3. If V

~

An, then C.(V, W) is acyclic in dimensions

D

~n-1.

REMARK 2.3.4. Note that this is a much simpler proof a weak version of Theorem 2.1.3 in this special case (Theorem 2.1.3 asserts the simple connectivity of the associated space). This approach does not work for general rings, however.

Now, if W = 0, denote by C.(An) the complex C.(An, 0) and by Bn(A) the module Ho(GLn(A) , Hn(A)), where Hn(A) is the nth homology group of C.(An). Note that Hn(A) is the only nonzero homology group of C.(An). Evidently, we have Ho(A) = Z. Thus, Bo(A) = Z. Moreover, the group Hl(A) is the augmentation ideal I (A) of the group ring Z [A x ]. It follows that Bo (A) = Ho(A x, I(A)) = A x. Now, for each n 2 2, there is an exact sequence

Since the coinvariants functor is right exact, we obtain an exact sequence

The group GLn(A) acts freely on the sets of (n+ I)-frames and (n+2)-frames in general position. Each orbit can be represented by an element of the form· (el' ... , en, v) and (el' ... , en, v, w), respectively. The vectors in each frame are in general position if and only if v = L: aiei, where ai E A x and w = L: >'iaiei, where the>. E A x satisfy ~i =f:. ~j for i =f:. j (here, ~ denotes the image of >. in tho residue field). Denote the orbit of the frame (el,' .. , en, L: ai ei) by p( a) and the orbit of (el"'" en, L: aiei, L: >'iaiei) by pea, >.). Then

a

and

Ho(GLn(A), Cn+ 2 (A n )) =

ED z· pea, >.). a,A

The induced map in the above exact sequence is then

(2.1) n

+ L:) _1)i-l p «>'l - >'i)al, ... , (>'~i)ai"'"

(>.,. - >'i)an, >'i)'

2.3. Local rings and Milnor K-theory

49

Hence, the group Sn(A) is generated by elements d(p(a)) = [al,"" an], ai E A x, subject to the relations [Alal, ... ,Ananl - [al' ... ,anl

=

n

:~:)-l)i+n[(Al - Ai)al, ... , (Ai~ai"'" (An - Ai)an, Ail i=l where Ai E AX and 'Xi =f:. 'Xj for i =f:. j. Since the generators of K~(A) also satisfy these relations, we obtain a homomorphism ipn : Sn(A) ~ K~ (A) sending [all"" an] to {al, ... , an}. Observe that we can multiply elements

as follows. Let V be a finitely generated free A-module. Say that x = (Vl' ... , v n ) E Cn(V) and y = (Wl, •.. ,wm ) E Cm(V) are in general position if the vectors Vl, ... , V n , Wl, .. • , Wm are. In this case, set xy = (Vl"'" Vn , WI, •.. , W m ) E Cn+m(V). This extends to a linear pairing C.(V) ® C.(V) ---+ C.(V).

Now, if V is decomposed into a direct sum V = V l E9 V2, then there are embeddings of complexes C.(VI, V2) ~ C.(V) and C.(V2) ~ C.(V). If x E Cn(V1 , V2) and y E C m (V2), then x and y are in general position and xy E Cn +m (V). This gives a pairing

C.(Vll V2) ® C.(V2)

---+

C.(V)

and an induced pairing

All these pairings respect the GLn(A)-action and hence we have an induced pairing This pairing is associative, and we therefore obtain a graded ring

S.(A)

= $Sn(A). n~O

PROPOSITION 2.3.5. The map

ip. : S.(A) is a homomorphi,9m of graded rings.

---+ K~ (A)

2. Stability

50 PROOF.

We first describe how to represent an arbitrary element of Sn (A)

as a sum of generators. Let

x

= L: ni(vL .. ·, v~) i

be an arbitrary cycle. Find a vector x

E Hn(A)

vin general position with each v;. Then

= (-l)nd(xv) = (_l)n L: n i d(v1, ... , v;).

Each frame (vL ... ,v~) defines a matrix a. E GLn(A). Write a,lv = (aL ... ,a~)T. Then the frame (vL ... ,v~,v) is GLn(A)equivalent to the frame (el,"" en, E j a~ej). Thus x == (_l)n

L ni[aL .. · ,a~]

mod GLn(A).

Now, let x=

L: ni

8, E

Hn(A),

where the 8. are unimodular n-frames in An and the Tj are unimodular mframes in Am. Find vectors v E An and w E Am in general position with all the 8. and Tj respectively. Each frame 8. (resp. Tj) determines an invertible matrix in GLn(A) (resp. GLm(A», which we denote by the same symbol. Set ' ... ,ain )T and T -1 w= (1.1 L1 )T • Thevecor t (11)" 8,-1 v= (al, CTl""'V"m w Ismgener al j position with the frames (8i' rj) and (8i, Tj)-l (~) = (aL .. . , a~, ai, ... ,blm)T. Thus we have x

L: n.[aL ... , a~l mod GLn(A), (_l)m L: mj[a{, ... ,him] mod GLm(A),

== (_l)n

y -

j

xy

_

(_l)n+mL:nimj[aL ... ,a~,bi, ... ,him]

mod GLn+m(A),

i,j

nncl hence, i,j

= This completes the proof.

'Pn(x)'Pm(Y)· D

We now construct a homomorphism 8. :K~(A)~S.(A). Since Kf1(A) = AX = SI(A), the identity map Kf/{A) -+ SI(A) will extend to the desired map of graded rings, provided we show that the defining relations of K~ (A) hold in S.(A). For thiH, it Huffir.eH to show the following

51

2.3. Local rings and Milnor K-theory

= 0 in S2(A).

LEMMA

2.3.6. If a E A x with a =I- 1, then [a][1 - a]

PROOF.

First consider the following fact: if aI, ... , an

E

A x , then in Sn (A)

we have

(-I)k[al' ... , 1, ... , 1, ... ,an],

[al]'" [an] =

where the 1's are located in the il,"" i k positions. To see this, note that [a] = ((a) - (1)) mod GLI(A) so that the left hand side coincides with the image in Sn (A) of the cycle

((aler) - (er))((a2e2) - (e2))'" ((ane n) - (en))

=

L

(_I)n-k(el,"" ail eil"'" aikeik"'" en).

l~iI< .. -
To represent this as a sum of generators in Sn (A), take the vector v to be L7=1 aiei' This is in general position with all the vectors in the sum. The procedure above yields the result. Now, to prove that [aHl - a] = 0 in S2(A), we express it as a sum

[a][l- a] = [a,1 - a]- [a, 1]- [1,1 - a]

+ [1,1].

We now manipulate the defining relation of S2(A):

[Alal,A2a2]- [a1,a2] = [(Al- A2)al,A2]- [(A2 - A1)a2,Al]' Via obvious choices for ai and Ai, we obtain the following relations:

= [(1 -

A)a, A]- [A - 1,1],

:.\ =I- 1

[(1 - A)a, A]- [a, A] = [-Aa, I][A2, 1 - AJ,

:.\ =I- 1.

[a, A]- [a, 1] and

Substituting the second into the first, we have

+ [-Aa, 1]- [A2, 1- A]- [A - 1,1]; [-Aa, 1] + [a, 1] = [A2, 1- A] + [A -1,1]. Thus, the element [a, 1] + [b, t] [a, A]- [a, 1] = [a, A]

that is, with a =I- -1) depends only on ba- 1 . Similarly, we have

[1- A, A] = [(A - 1) a, 1] + [a, 1]- [Aa, 1.]. Since A-I =I- -1, we have

[(A - l)a, 1]

+ [a, 1] =

[Aa, IJ

+ [A(A -

I)-la, 1].

It follows that [1- A, A] = [A(A -1 )-la, IJ. Since a is arbitrary in this formula, we see that [1- A,A] = [a, 1] = [1,1]. Using the defining relation again, we find

[1, aA] - [1, a] = [1 - A, A] - [(A - l)a, 1].

52

2. Stability

This, along with the previous formula then shows that [1, a>..] - [1, a] = O. It follows that [1, a] = [1,1] for any a. This proves the lemma. D COROLLARY

2.3.7. The composition

K~ (A) ~ 8.(A) ~ K~ (A) 'i8 the identity. PROOF. The existence of e. follows from Lemma 2.3.6. Moreover, it is clear that en ({ aI, ... , an}) = [ad ... [an]. By the proof of Lemma 2.3.6, we have 'Pn([al]'" [anD = {al,"" an}. D

We identify K~ (A) with the subring of 8. (A) generated by 8 1 (A) = A x, Denote by Sn(A) the subgroup of decomposable elements in Sr,(A). By definition, this is the image of the map

8 1 (A) 129 8 n - l (A) EB ... EB 8 n - l (A) 129 8 1 (A) Clearly, for any nontrivial decomposition An

H(V, W) 129 H(W)

---->

=V

Hn(A)

---->

8 n (A).

EB W, the image of the map

---->

8 n (A)

lies in Sn(A). We now show the following. LEMMA

2.3.8. Ifn 2:: 2, then for all aj,>.. E AX and all 1

[al,"" >"ai,"" an] -lal, ... , an]

E

<::::

i <:::: n,

Sn(A).

PROOF. This is clear for the subgroup of Sn(A) consisting of elements in K ~ (A). That it is true in general is more delicate. Suppose 1 < i < n. Introduce the notation (s) for the class of a cycle 8 E Hn(A) modulo the nction of GLn(A). Let a = I: aiei' Then

[nJ,' .. ,>"ai,' .. ,an] - [al,' .. , an]

= d((el""

,en,alel + ... + >"aiei + ... + anen) - (el,'" ,en, a)) = d( (el' ... , >.. -1 ei, ... , en, a) - (e 1, ... , en, a)) = d( ((el,"" ei-I)((>" -lei) - (ei))(ei+l, ... , en, a))) + (d( el, ... ,Ci-I)( (>.. -lei) - (ei))( ei+l, ... ,en, a) -

+ (-I)i(el"'" =

ei_I)((>..-lei) - (ed)d(ei+l,.'" en, a))

(z).

Now, let u = aIel

+ ... + aiei.

Then we have

(ei+l,"" en, a) = d(u, ei+l, .. ·, en, a) and

+ (u)d(ei+l"'"

en, a),

2.3. Local rings and Milnor K-theory

53

Substituting these formulre into the above equation, we write

Z = Zl - Z2

+ Z3.

We now show that each Zi is decomposable. Denote by Vl and V2 the submodules of An generated respectively by el, ... , ei and ei+l, ... , en. Then the cycles

and

d(el, ... , ei_I)((A-Iei) - (ei))(u)

+ (U)((A-lei) -

(ei))

are in H(VI) and d(ei+l, ... , en, a) is in H(V2' VI)' Thus, Z2 and Z3 are in the image of H(V2' VI) Q9 H(VI) --+ Hn(A). This shows that (Z2 - Z3) lies in Sn(A). Also, if WI is the submodule of An generated by u, ei+l, ... ,en and if W 2 is any direct complement of WI, then

d(u, ei+l, ... ,en, a) E H(Wt} and

d(el, ... , ei_t)((A-led - (ei)) E H(W2' Wt}. Thus, (Zl) lies in Sn(A). The cases i = 1, n are similar and are left to the reader.

o

As a consequence we have the following result. COROLLARY 2.3.9. lfn ~ 3, then Sn(A) = Sn(A). PROOF. First observe that Lemma 2.3.8 implies that [al, ... ,an ]- [1, ... ,1] is decomposable for any n ~ 2. Thus to prove the corollary, it suffices to show that if n ~ 3, then the cycle [1, ... ,1] is decomposable. Consider the defining relation of Sn(A). If n is odd, the left hand side ofthe relation is decomposable and the right is congruent to [1, ... ,1] modulo the decomposables. If n is even, consider the cycle

This cycle has the form E niTi where the Ti are unimodular n-frames and 1. Expressing each Ti as a sum of generators, we obtain

E ni =

[1, ... ,1][1,1] =

L nda1, ... , a~].

But each term in the sum is congruent to [1, ... ,1] modulo the dccomposables and since E ni = 1, the result follows. 0

2. Stability

54

This corollary implies that the ring S.(A) is generated. over K~(A) by [1,1]. It is easy to see that [1,1] is central in S.(A). It follows that we have a direct sum decomposition

Sn(A) = K::! (A) EB [1, IjSn-2(A).

(2.2)

Now denote by C.(G) the bar resolution of a group G (see Appendix A). Since resolutions are unique up to homotopy, we obtain a unique homomorphism of complexes ... -+

Cn(GLn(A»

----> ••• -+

Co (GLn (A»

----> ••• -+

Cl(An)

! 0-+

-+

Z

-+

Z

0

-+

O.

II

!

Hn(A)

-+

Factoring out the GLn(A)-action and passing to homology, we obtain a map en: Hn(GLn(A),Z)

~

Ho(GLn(A), Hn(A» = Sn(A).

Note that the map Cl(An) -+ Co(An) = Z has a GLn_l(A)-equivariant - . section given by m 1-+ m(en ). Thus, the restriction of en to Hn(GLn-l(A» is trivial. Thus we get a map Moreover, the following diagram commutes

Hn(GLn(A» ® Hl (GLl (A» e n ® e1

l

~

Hn+1(GLn+1(A» e n +1

Sn(A) ® Sl(A)

)I

l

(2.3)

Sn+1(A).

Since el : A x -+ A x is the identity, we see that the image of en contains K,";f(A). To finish the proof of Theorem 2.3.1, consider the action of GLn(A) on C.(An) and the associated. spectral sequences. The first sequence collapses to !lhow that = 0 for m < nand

E:::

= Hp (GLn (A) , Hn(A». Note that E:;:> = Ho (GLn(A) , Hn(A» = Sn(A) and there is an increasing E~p

filtration such that

oo .. = E2n-tttS n (A)(i)jSn (A)(i-l) = E n-t,t .. Now, by Corollary 2.2.16,

Sn(A)(O)

= Hn(GLn(A»jHn(GLn-l(A»,

(the isomorphism being obviously induced by en) and

Sn(A)(l)jSn(A)(O) = E.~I,I

=

ker(Hn ._l (GL n._.l (A»

-+

Hn-1(GLn(A»).

2.3. Local rings and Milnor K-theory

55

We now proceed as in the proof of Proposition 2.2.17. Define a map '0 : C.(An-2)[-2] ~ C.(An) by multiplying by the cycle (Cn-l, en )- (en-I, Cn-l + en) +(e n , en-l +en ). This map commutes with the GL n- 2 (A) action and hence induces a map of spectral sequences. As before, .1. • '1/*'

E- 2

pq

"'"E2

--7

pq

for q ;::: 2. This implies that gXJ = Boo EEl (Eoo)(1). Now, B:: the map '0* : Sn-2(A) = - - 7 E': = Sn(A) is obviously multiplication by [1, 1]. Thus

= Sn-2(A) and

B::

Sn(A)

=

[1, 1]Sn-2(A) EB Sn(A)(l).

I3ut, we also have and K~(A)

Thus, K~ (A)

c Sn(A)(O).

= Sn(A)(O) = Sn(A)(l). This shows that K~(A)

= Hn(GLn(A))/Hn(GLn-I(A))

and

o = E~-l,l = ker{Hn-1(GLn-I(A))

--7

Hn-I(GLn(A))}.

o

This completes the proof of Theorem 2.3.1. REMARK 2.3.10. The statement of Theorem 2.3.1 asserts that the map

e : K:~ (A)

--7

Hn(GLn(A), Z)/Hn(GLn-1(A), Z)

is an isomorphism. The proof actually asserts that

En : Hn(GLn(A), Z)/Hn(GLn-dA), Z)

--7

K~ (A)

is an isomorphism. These two maps are evidently inverses. Theorem 2.3.1 has the following applications. For any ring A, the natural map A x ~ Kl(A) induces a homomorphism K;\1 (A) ~ K.(A). If A is local with infinite residue field, then we define a map cp : Kn(A) ---+ K;~ (A) to be the composition

Kn(A) = 7r n (BGL(A)+) Hu~icz Hn(BGL(A)+) = Hn(GL(A)) - - 7 Hn(GL(A))/Hn(GLn_I(A)) = K~(A). THEOREM 2.3.11 ([92], Theorem 4.1). The composition

K:;t (A) ~ K",(A) ~ K/,! (A) coincides with multiplication by (-1)" - I (n - I)!.

2. Stability

56

Consider the particular case n = 2. LEMMA 2.3.12. Let A be a commutative ring with SKI (A) local ring). Then the natural map

=

°

(e.g. any

K2(A) - H 2(GL(A))/H2(GL 1 (A))

is an isomorphism. PROOF. By hypothesis, SL(A) = E(A) so that K 2(A) = H2(SL(A)). The result now follows by considering the Hochschild-Serre spectral sequence associated to the (split!) extension 1

----7

SL(A)

----7

GL(A)

----7

Ax

----7

1

and noting that the AX-action on H.(SL(A)) is trivial.

D

COROLLARY 2.3.13. If A is a local ring with infinite residue field, then K2(A) = Kr (A). PROOF. This follows since H 2(GL(A))/H2(GL 1 (A))

= Kr(A).

D

REMARK 2.3.14. Corollary 2.3.13 was proved originally by van der Kallen

[64].

2.4. Auxiliary stability results 2.4.1. Orthogonal groups. In [131]' K. Vogtmann considered the question of stability for the orthogonal groups On,n(k), char k = 0, where On,n(k) denotes tho orthogonal group of the quadratic form

(~ ~)

on k2n (I = n x n

id(mtity matrix). THEOREM 2.4.1. If k is a field of characteristic zero, then the natural map i* : Hm(On,n(k),Z)

is surjective for n

~

----7

Hm(On+l,n+l(k),Z)

3m + 1 and bijective for n

~

3m + 3.

o

The proof of Theorem 2.4.1 is virtually identical to the proof of Theorem 2.2.18. The idea is to construct a complex X on which Gn = On,n(k) acts and utilize the resulting spectral sequence. The stabilizers of the various simplices are analogous to the affine groups of Section 2.2. Since char k = 0, the usual center kills argument shows that homologically these groups are the same as GLp(k) x G n - p' This allows one to use an induction procedure to prove the theorem. Moreover, one expects that by using a suitable modification of the arguments in Section 2.2, the theorem holds in positive characteristic as well. The simplicial complex X is defined as followH. If k is any field, let V be a 2n-dirnensional vector Hpace over k with qlll1dmtic form ( ). Choose a

2.4. Auxiliary stability results

57

polar basis {el' ... , en, !1, ... , f n} of V; that is, in this basis the matrix of the form is

Let. denote the associated inner product. DEFINITION 2.4.2. A subspace W C V is totally isotropic if v • w all vectors v, wE W.

= 0 for

The set of all nontrivial totally isotropic subspaces of V is partially ordered by inclusion; denote by X the geometric realization of this poset. A p-simplex of X is a chain of (p + 1) totally isotropic subspaces of V:

Wo

C

W 1 c···

C

Wp.

Since every maximal totally isotropic subspace of V has dimension n, the complex X has dimension (n - 1). PROPOSITION 2.4.3.

The complex X is homotopy equivalent to a wedge of

(n - 1) -spheres. PROOF. Let E C V be spanned by ell ... , en. For 0 :::; k :::; n, define a Hubcomplex X k by

Xk

=

union of all maximal simplices such that dim (An n E)

=

U

~

Al C ...

cAn

n- k

st(An)

dim(A .. nE)~n-k

where st(An) denotes the closure of the star of An. Thus, Xo = st(E) and Xn = X. Via a proof similar to that of Theorem 2.1.3, one shows that X k - 1 is a deformation retract of X k for 1 :::; k n - 1. Since Xo is contractible, X n - 1 iH contractible. Note that X n - 1 is X minus the stars of maximal isotropic subspaces An such that An n E = O. For such an An, the link, Link(An) is eontained in X n - 1 . Thus, by contracting X n - 1 to a point we obtain

s:

X ~

V

~(Link(An)).

A .. nE=O

But Link(An) is the Tits building for An since every subspace of a totally isotropic space is totally isotropic. By the Solomon-Tits theorem [45], Link(An) is homotopy equivalent to a wedge of (n - 2)-spheres. Thus,

IIJol

claimed.

D

58

2. Stability

2.4.2. Dedekind domains. A Dedekind domain is a noetherian normal domain of dimension 1. Typical examples include the ring of integers in a number field and the coordinate ring of a nonsingular irreducible affine curve over a field. Theorem 2.1.3 applies to such rings, but R. Charney [27] proved a stability theorem for GL n of such rings by contructing a different complex on which GL n acts. The crucial step in the proofs of Theorems 2.2.18 and 2.4.1 is showing that the map i. : H.(GLn(A), Z)

---+

H.(Affnm(A), Z)

is an isomorphism (i.e., we can "kill" the unipotent piece of Affnm(A)). If A is a Dedekind domain with many units, then there is no problem; the results of Section 2.2 apply. But if A is the ring of integers in a number field, then we are out of luck. Charney circumvented this difficulty by inventing split buildings. Let A be a Dedekind domain and let W be a rank n projective A-module. Define a partially ordered set S(W) by S(W) = {(P, Q) : P ffi Q =

w, P -I- 0, Q -I- O}

with ordering (P,Q) :::; (P',Q') if P ~ P' and Q ;;2 Q'. Denote by [W] the geometric realization of S(W). THEOREM 2.4.4. If W is a rank n projective A-module, then [WI has the 0 homotopy type of a wedge of (n - 2) -spheres. The proof of this is similar to Vogtmann's proof that the space built from total isotropic subspaces of k2n is a wedge of (n - 1)-spheres. The usefulness of this construction is that the stabilizers of simplices in ' [WI have the form

( GLp

o

0 ) GL n - p

so that we no longer need to get rid of a unipotent piece. The usual stability argument then goes through to yield the following result. THEOREM 2.4.5. If A is a principal ideal domain, then 1. Hk(GL n + 1 (A), GLn(A); Z) = 0 for n ;:::: 3k;

2. Hk(GL n+1 (A), GLn(A); Z[1/2]) = 0 for n;:::: 2k. The first statement is true also for SL n .

o

This range of stability is not quite as good as Theorem 2.1.3. However, if we ignore 2-torsion, then it is hasically the sallIe as van der Kallen's result.

2.5. Stability via Homotopy

59

2.5. Stability via Homotopy The Hurewicz homomorphism hi : 7Ti(X) -----; Hi(X, Z)

is an important tool in the study of both homotopy and homology. Consider the particular case X = BGLn(R)+. Denote the ith homotopy group of X by K~n(R) ("Q" for Quillen). Then the Hurewicz map is a homomorphism

K~n(R) -----; Hi(GLn(R),Z). Thus, if one can prove stability for K~n (R), then the above map will yield '. stability for Hi (G Ln (R), Z). Unfortunately, the groups K~n (R) are difficult to work with. Moreover, for i = 1,2, they need not agree "v'ith the "classical" definitions: K1,n(R)

=

GLn(R)j En(R)

K 2 ,n(R)

=

ker{Stn(R) -----; En(R)}

(here, St n (R) is the nth Steinberg group of R; see [80]). To circumvent this, one can work with the unstable Volodin groups Ki,n

= 7Ti-l(Vn(R)),

where Vn(R) is a certain simplicial complex defined below. Suslin [123] proved the following stability result for the groups Ki,n(R). THEOREM 2.5.1. Let R be a ring and set r = sr(R). Then the canonical map Ki,n (R) -+ Ki,n+! (R) is surjective for n ;::: r + i - I and bijective for n;::: r + i. Suslin then constructed maps Ki,n(R) -----; K~n(R)

for n ;::: 2i + 1 and showed that they are surjective for n ;::: max(2i + 1, r + i-I) and bijective for n ;::: max(2i + 1, r + i). Thus, we have the following result. COROLLARY 2.5.2. The canonical map

K~n(R) -----; K~n+l (R) /8 surjective for n ;::: max(2i+ 1, r+i -1) and bijective for n ;::: max(2i+ 1, r+i).

o COROLLARY 2.5.3. The canonical map H i ( GLn(R), Z) -----; Hi(GLn+dR), Z) i.8 surjective for n ~ max(2i+1,r+i-l) and bijectiveforn ~ lllflX(2'i+l,r+i).

60

2. Stability PROOF.

Consider the commutative diagram

1T'i+l(BGL~+1,BGL-:;) - - - ) K~n(R) ---~) K~n+l(R)

1

1

1

Hi+1(BGL~+1,BGL-:;) - - Hi(GLn(R)) - - H i (GLn+1(R))

The result follows from the fact that if the relative homotopy groups vanish, then the relative homology groups Hj(X, Y) vanish as well.

1T'j (X,

Y)

0

The proof of Theorem 2.5.1 rests on the construction of certain spaces. If G is a group and {GihEI is a family of subgroups, define a simplicial complex V(G) as follows. The vertices of V(G) are the elements of G and a p-simplex of V (G) is a collection 90,' .. ,gp of distinct elements of G such that all the gjgk 1 lie in some G i . This construction is functorial with respect to maps (G,{GihEl)

----->

(H,{Hj}jE.J).

Define a second simplicial complex W (G) as follows. The p-simplices are collections (gO, .. . , 9p) of (not necessarily distinct) elements of G such that the 9j9k 1 all lie in some G i . Denote by S.(V(G)) the simplicial set of singular simplices of V(G); the geometric realization of S.(V(G)) is homotopy equivalent to V(G). Define a map W(G) ---+ S.(V(G)) by

(gO, ... ,gp)

f---+

{Vi

f---+

gil,

where Vi is the ith vertex of the standard p-simplex. This map is a homotopy equivalence. Now, if R is a ring and (J is a linear ordering of {1, 2, ... , n}, define T:;(R) t.o 1>0 tho ~mbgroup of GLn(R) consisting of matrices (akl) with aii = 1 and (f

0 for i f. j. Such subgroups are called triangular. Denote by Vn (R) the MPltc(~ V (G Ln (R), {T~ (R)}) and define

(tj.J ""

Ki,n(R)

= 1T'i-1(Vn(R)).

The connected component of the identity matrix in Vn(R) is the space V(En(R)); its universal cover is V(Stn(R), {i'~(R)}) (i';[(R) is the lift of T;[(R) to the Steinberg group). Thus we have Ki,n(R)

= 1T'i-dV(Stn(R))) = 1T'i-1(W(Stn (R))),

i;:::

3.

Theorem 2.5.1 follows by proving homology stability for the groups Hi(W(Stn(R))): for then the relative homology groups vanish and the rel-

ative Hurewicz theorem shows that the relative homotopy groups vanish as well. The proof of stability for Hi(W(Stn(R))) is tiimilar to the proofs above and we omit it. The interested reader is urged to consult Suslin's paper [123] for full details.

2.6. The Rank Conjecture

61

2.6. The Rank Conjecture Let A be a commutative ring and denote by Kn(A)Q the vector space Kn(A)®z Q. By the Milnor-Moore theorem, we have an injection

Kn(A)Q

~

Hn(GL(A),Q)

with image equal to the space of primitive elements (H. (GL(A) , Q) is a Hopf algebra). Define the rank filtration of Kn(A)Q by

TmKn(A)Q = {Im(Hn(GLm(A), Q)

~

Hn(GL(A), Q))} n Kn(A)Q.

This is an increasing filtration. By Theorem 2.2.18, if A = F is an infinite field, then and

TnKn(F)Q/Tn-lKn(F)Q

= K:! (F).

There is another filtration, the ,),-filtration [115]' defined as follows. We first define A-operations on the K-groups of A. Consider the Grothendieck group Ko(Z[GLn]). A representation of GL n gives rise to a map GL n -+ GL, and using the H-space structure on BGL(A)+, we get a map

Ko(Z[GLn]) ~ [BGLn(A),BGL(A)+]

= [BGLn(A)+,BGL(A)+].

The latter object is an abelian group, and the map above is a homomorphism of abelian groups. It is also compatible with the inclusion GL n -+ GL n+1, so we get a homomorphism

\!!!!nKo(Z[GLnD ~ \!!!!n[BGLn(A)+, BGL(A)+]. Note that \!!!![BGLn(A)+, BGL(A)+] = fug[BGL(A)+, BGL(A)+]. Denote by idn the standard n-dimensional representation of GL n , and by n the trivial n-dimensional representation. The collection {idn - n}n is an element of lli!!Ko(Z[GLnJ). Define A-operations on Ko(Z[GLnD by Am(v) = Amv if V is a representation, and by demanding that the map At : Ko(Z[GLnD -+ Ko(Z[GLnD defined by

At(x) = 1 + Al(x) + A2(x) + ... be a homomorphism of abelian groups. These operations are compatible with the maps Ko(Z[GLn+lD -+ Ko(Z[GLn]). Then we can consider the maps Am as elements in \!!!![BGLn(A), BGL(A)+] by taking {Am(idn - n)}n in this group. We then get induced operations, still denoted by Am on the homology groups H.(GL(A), Z) and on the homotopy groups K.(A). We now use the A-operations to define Adams operations. We have an involution on Ko(Z[GLnD defined by sending a representation to its dual;

2. Stability

62

denote this map by x I-t X. Then if k is a nonzero integer, we define the kth Adams operation by demanding

1ji(x)

= 'IjJ-k(x),

k~1

and Now define

K!P(A) = ker('ljJk - ki'IjJl : Kn(A)Q

--+

Kn(A)Q),

the jth eigenspace of 'ljJk. This gives us a decreasing filtration on Kn(A)Q, n m = ffi KU) (A) F'Y,n Q7 n ,

j=m

called the 'Y-filtration. CONJECTURE 2.6.1 (Rank Conjecture). Let F be an infinite field. Then for all n 2: 1 and all m, we have a direct sum decomposition

rmKn(F)Q EB F:;:;tl(F) = Kn(F)Q. A recent result of R. de Jeu [28] shows that for any commutative ring with identity, there is an inclusion m

j=l

and hence the rank conjecture is equivalent to the assertion that m

j=l

The rank conjecture for number fields. Let k be a number field. The rank conjecture for k is implied by two assertions about the homomorphism

ji,m,n : Hi(GLm(k),lR) ------t Hi(GLn(k),lR) (m :5 n) induced by the standard inclusion; namely, (1) im(hm-l,m,n)::J PH2m- 1 (GL n (k), lR), (2) im(hm-l,m-l,n) nPH2m- 1 (GL n (k),lR) = {a}, for n » m 2: 2, where PHi is the space of primitive elements. In other words, we need to prove that

(1)' rmK 2m - 1 (k)Q = K 2m - 1 (k)Q, and (2)' r m- 1 K 2m -l(k)Q = a. But there is an isomorphism (n

» m ~ 2)

I

Exercises

63

so that it suffices to prove (1) and (2) with GL n replaced by SL n . This is easily obtained via the following result. THEOREM 2.6.2. For any field E of chamcteristic zero, 1. j:",n : PHi(SLn(k), E)

PHi(SLm(k), E) and 2. ji,m,n : PHi(SLm(k), E) -----+ PHi (SLn(k), E) are isomorphisms for i odd, i

~

-----+

2m - 1.

3. j!~1,~(PH2m-l(SLn(k), E))

= {o}.

PROOF. This is a consequence of the following formulre:

j:",n(Xn,i) = Xm,i, j:",n(Xn,i) = 0,

i ~ 2m - 1 i > 2m - 1

j~,n(en) = 0,

n even.

The first two equations follow from the fact that if m > n, then Xm,i maps to Xn,i under the induced map

H:t(SLn(kv),JR.) (see Section 1.5). That j~,n(en) dimensional cohomology class.

-----+

H:t(SLm(kv),JR.)

= 0 follows from

the fact that en is an even0

The rank conjecture now may be deduced as follows. Assertion (1) follows from part (1) of the theorem. By duality, the second assertion is equivalent to

j!7n- 1 (PH 2m - 1 (SLn(k), E))

=0

which follows from (3).

Exercises 1. Prove that a local ring has stable rank 1. 2. Describe the structure of the partially ordered set O( {I, 2, ... ,n}) (see Definition 2.1.2). 3. Check that the conditions of Lemma 2.1.6 are satisfied by the poset

F = O(Rn + en+18) nU.

4. Modify the proof of Theorem 2.2.18 to obtain a proof of Theorem 2.1.7. 5. Prove that an algebra over an S(n)-ring is also an S(n)-ring. 6. Show that the map Br --... S~i) / S~i-l) given by

is multilinear.

2.

Stability

7. Prove that the filtration of Hj(N,IFp), where N is an abelian group, defined by

is independent of the choice of section of the map

H2(N,IFp) -+ pN. 8. Prove Lemma 2.2.12. 9. Show that the map 'l/!. defined in the proof of Proposition 2.2.17 is the identity. 10. Complete the proof of Lemma 2.3.2 by checking that dz = y. 11. Check the formula in equation (2.1). 12. Show that the generators of K:! (A) satisfy the relation (2.1). (Hint: proceed by induction on n, the case n = 1 being an obvious consequence of the definition of K:! (A). For the general case, make use of the facts that in K!;! (A): and

13. 14. 15. 16.

{aa(l)"'" aa(n)} = sgn(cr){al"'" an}. To prove these last two facts it suffices to show that {a, -a} = 0 in Kr(A) and {al,a2} = -{a2,al}; this can be done by direct computation.) Finish the proof of Lemma 2.3.8. .j Prove that if z E Sn(A), then [a]z = (-l)n z[a]. ',I Prove that the diagram (2.3) commutes. Prove that the maps

e: K~(A)

-+

Hn(GLn(A),Z)/Hn(GLn-l(A),Z)

and

17. 18. 19. 20. 21.

are inverses. Show that every maximal totally isotropic subspace of a 2n-dimensional vector space V has dimension n. Show that every subspace of a totally isotropic space is totally isotropic. Show that K1,n(R) = GLn(R)/En(R). Show that the map W(G) -+ S.(V(G» is a homotopy equivalence. Show that the univel'Hal cover of V(E7I(R» is V(Stn(R».

Chapter 3

Low-dimensional Results The main focus of this chapter is the computation of low-dimensional homology groups of GLn- The results of the previous chapter show, for example, that H 2(GL 2(A),Z) ~ H 2(GL(A),Z) for A local with infinite residue field. Thus, one need only consider the former group. In this case, Suslin completely described the structure of H 2 (GL 2 (A),Z)-it surjects onto the second Milnor K-group Kt£(A) and the kernel of this map is the image of H 2(GL 1 (A),Z) in H 2 (GL 2 (A),Z). That begs the question: What about H3(GL2(A),Z)? There is a natural map H3(GL 2(A» ---t H3(GL3(A)) = H3(GL(A), but what more can be said? More generally, what about Hn+1(GLn(A))? There are no easy answers to these questions, but we can present a few results.

3.1. Scissors Congruence We begin with a problem which is seemingly unrelated to the study of the homology of linear groups, namely the scissors congruence question. This is an extended version of Hilbert's third problem and has been studied primarily by Dupont, Sah, Wagoner and Cathelineau [105, 106, 107, 31, 30, 25, 26]. We present it in its most general terms. Let X be a set with a specified family of distinguished subsets called cells (or n-cells if n is the dimension). Two cells A and B are interior disjoint if 1. A n B contains no nonempty cells; 2. If C is a cell contained in Au B, then C <:;:; A if and only if en B contains no nonempty cells. A finite union of pairwise interior disjoint cells is called a polytope (the empty set is allowed). If P = UPi where the Pi are cells (resp. polytopes), then the finite collection of Pi'S is called a cell (resp. polytope) decomposition of P. If A = B U C (A, B, C cells), then we say that Band C form a simple pasting of A. If P = U Pi is a cell decomposition, then a simple subdivision (resp. pasting) of P is understood to be one involving one (resp. two) of the finite number of cells Pi. Any finite iteration of simple subdivisions (resp. pastings) beginning with P = U Pi is callcel a Hllbelivision (resp. pasting) of P. A finite sequence of HlibelivisionH anel pa.'1t.ings will he called n cut. anel

3. Low-dimensional Results

66

paste. This evidently defines an equivalence relation among cell decompositions P = II Pi of a given polytope P. We need to assure ourselves an ample supply of small cells. Thus we impose the following axiom: Let P = II Pi and Q = II Qj be cell decompositions. Then there exist cut and paste processes leading to cell decompositions P = IIu Ru and Q = IIv Sv such that each Ru (resp. Sv) is either interior disjoint from Q (resp. P) or coincides with one of the Sv (resp. Ru). EXAMPLE 3.1.1. Consider the classical Euclidean, spherical, or hyperbolic spaces. In these cases we can take cells to be convex closures of finite numbers of points. Then there is a well-defined notion of dimension. We now define the notion of geometric scissors congruence. Fix a group G of bijective maps (called motions) of X. We assume further that the set of cells of X is closed under the action of G.

'j

DEFINITION 3.1.2. A G-scissor between polytopes P and Q, denoted P fVG -~ Q or P '" Q mod G, consists of cell decompositions P = IIi Pi, Q = IIi Qi, "j 1 :;; i :;; t such that ~ and Qi are G-congruent for each i. The G-scissors congruence problem is the following: Determine necessary and sufficient conditions for polytopes P and Q to be G-scissors congruent. DEFINITION 3.1.3. An equivalence relation == on the set of polytopes of X is subtractive if whenever P = pi II P" and Q = Q' II Q" are polytope , uecompositions, then P == Q and pi == Q' together imply pI! == Q". The geometric scissors congruence problem has an algebraic formulation.

Define a group 'P(X, G) as follows. First define a group P(X, {I}) as the abelian group with generators the cells in X subject to the relations

A - B - C = 0 when A = B II C is a simple subdivision. The dru;s of A will be denoted by [Aj. Now define P(X, G) as

P(X,{l})/([aAj- [Aj: a

E

G,A is a cell inX).

The group P(X, G) is called the group of G-scissors congruence classes of polytopes on X. P

THEOREM 3.1.4 ([105]). Assume that the relation "'G is subimctive. Then Q if and only if [P] = [Q] in P(X, G). 0

"'G

Thus, the study of G-scissors congruence classes is equivalent to the study of the group P(X, G). Let ?in denote n-dimensional hyperbolic space and uenote by 'Fi:' the extended hyperbolic space obtained by adding the ideal points on 8rt n , We write 'P(rt n ), 'P('Fi:') for the scissors congruence groups where G is the group of 1\11 isonwtricH of rt n , 'Fl.".

3.1. Scissors Congruence

67

THEOREM 3.1.5 ([31], Theorem 2.1). The natural inclusion 1-I,n ...... ~ in-

duces an isomorphism

PROOF. According to Dupont [29], there is an alternate description of

p(1-I,n) and p(1-I,n). Let T(1-I,n) (resp. T(1-I,n)) be the Tits complex of flags of proper geodesic subspaces of 1-I,n (resp. ~) (this is the simplicial complex whose k-simplices are flags Vo C VI C ... C Vk , where each Vi is a proper geodesic subspace). Let G( n) be the group of isometries of 1-I,n. The complexes T(1-I,n) and T(~) have homology concentrated in dimension n - 1 (see Appendix A). Put and

St(1t)

= Hn- I (T(~), Z).

Then there are natural isomorphisms

and P(~) ~ Ho(G(n), St(1t)t),

where the upper index t indicates that the action is twisted by the determinant (= ±1). For p E 8'H n , let T(1-I,n,p) be the complex of flags of proper subspaces of 1-I,n passing through p. One checks that T(1-I,n,p) is isomorphic to the affine Tits complex AT(lRn-l) of flags of proper affine subspaces of lRn - 1 (note: 81-1,n ~ lRn - 1 U {oo}). The homology of this space is concentrated in dimension n - 2. Set

St(1-I,n,p) ASt(lRn - l )

Hn_ 2 (T(1-I,n ,p), Z), n ~ 2 Hn_2 (AT(lRn- I ), Z), n ~ 2.

The group St(1-I,n,p) is a module over the isotropy group G(n)p and the group ASt(lRn - 1 ) is a module for the group of affine transformations of lRn - l . In our model, G(n)oo acts on lRn - 1 C 81-1,n as the group Sim(n - 1) of similarities of lRn - l . One now checks that there is an exact sequence of G(n)-modules

0--+ st(1-I,n)

--+

St(1-I,n)

--+

E9 pEO'H."

as follows.

St(1-I,n,p)

--+

0

3. Low-dimensional Results

68

Consider the long exact homology sequence for the pair (TCF{'\ T(1-I,n)). Clearly there is an isomorphism of complexes

C.(T('it\ T(1-I,n)) ~

EB

C.- 1 (T(1-I,n),p).

pE8'H n

Each factor of the summand has homology only in dimension n - 2, and each space in the pair (T(1i"') , T(1-I,n)) has homology concentrated in degree n - 1. Thus the only relevant portion of the long exact sequence is

0 - Hn_l(T(1-I,n))

---+

EB

Hn_l(T(1-I,n)) -

Hn_2 (T(1-I,n),p) -

O.

pE8'H n

The group Sim(n) is a semidirect product

= T(n) . Simo(n),

Sim(n)

where T(n) ~ JRn is the subgroup of translations and Simo(n) is the group of linear similarities. It is easy to see (via a center kills argument) that

H.(Simo(n),Zt) ~ H.(Sim(n),Zt). The crucial step in the proof is showing the following: LEMMA

3.1.6. For all n

> 0,

H. (Sim(n), ASt(JRn)t) = O. PROOF. The proof is by induction on n. For n Mequence of Sim(I)-modules

0 - ASt(JRl) -

EB Z(P) ~ Z

= 1, we have the exact ---+

0

pERl

where e is the augmentation. Shapiro's lemma shows that H.(Sim(I),

{EB Z(p)}t) ~ H.(Simo(I), zt), pERl

the isomorphism being induced by e and the inclusion Simo(l) --... Sim(I). Since the right hand side of the above equation is isomorphic to H.(Sim(I), zt), the long exact coefficient sequence associated to the above sequence of modules shows that H.(Sim(I), ASt(JR 1 )t) = O. Now assume that n 0 - ASt(JR n ) -

> 1. There is an exact sequence of Sim(n)-modules

EB ASt(V V"

I

n - 1) - ' "

-

EBASt(VO) ~ z VII

0

'3.1. Scissors Congruence

69

where the sum over Vi runs over all j-dimensional affine subspaces of lRn. The stabilizer of lRi C JRn is a product Sim(nhtj ~ Sim(j)

X

O(n - j,lR)

where O(n - j,lR) is the orthogonal group of lRn - i . By Shapiro's lemma, we have

H.(Sim(n),

{EB ASt(Vi)}t) ~ H.(Sim(j) x O(n - j,lR), ASt(lRi)t ® zt). Vj

Using the induction hypothesis and the Kiinneth formula, we see that the right hand side of this equation vanishes for 0 < j < n. Similarly,

H.(Sim(n), {EBASt(VO)}t) ~ H.(Simo(n), zt). va Now split the above exact sequence of modules into short exact sequences

o

----t

Zo

----t

EB ASt(VO)

----t

Z

----t

0,

va

o

----t

Zj

----t

EB ASt(Vj)

----t

Zi-l

----t

0,

Vi

vn-l

Then we see

Hk(Sim(n), ASt(lRn)t)

~

Hk+1(Sim(n), Z~_2)

~

... ~ Hk+n-l(Sim(n), Z~)

o for all k

~

0

O.

Given this, the proof of Theorem 3.1.5 is finished via Shapiro's lemma and the long exact coefficient sequence:

H.(G(n),

EB

St(1in,p)t)

pE8'H. n

=

o.

o Dupont and Sah [31] then prove that the groups 'P(81i 3 ) , 'P(1i 3 ) , and 'P(1i 3 ) are divisible. This was later improved by Sah [107] who showed that 'P(1i 3 ) is uniquely divisible. The method of proof is the following. Let F be

3. Low-dimensional Results

70

a field and define an abelian group p(F) as the group with generators [x], x E F X - {I}, and relations

[x] - [y]

+ [y/x] + [(1 -

x)/(I- y)] - [(1 - x-I)/(I - y-l)]

for x =I- y. Using explicit technical calculations in p(F), Dupont-Sah prove the following result. THEOREM

ible.

3.1.7. IfF is algebraically closed, then the group p(F) is divis[]

From this we deduce the divisibility of P(1t 3 ) and P(1t\ According to Sah [107], Theorem 4.16, there is an isomorphism

P(1t 3 ) ~ p(q-, where p(C)- is the negative eigenspace under the action of complex conjugation. The divisibility of P(1t 3 ) (and by Theorem 3.1.5 that of P(1t 3 )) followsat once. The study of p(F) is intimately related to the computation of the lowdimensional homology of SL2(F). The following theorem is an unpublished result of Bloch and Wigner. THEOREM 3.1.8 ([31]). Let F be an algebraically closed field of characteristic zero. Then there is an exact sequence

0-+ J.l.F(2)

-+

H 3(SL 2(F))

-+

p(F)

~ -+

A2

I \z(F X )

sym

-+

K 2(F)

-+

O.

Here, J.l.F(2) is the group J.l.F of roots of unity in F with Aut(F) acting via the quadratic character. For z E F X - {I}, A([Z]) = Z /\ (1- z) and for u, v E F X , 1f1lm(u,'u) = {u,v}, the Steinberg symbol in K2(F). [] The map H 3 (SL 2 (F)) -+ p(F) is induced by sending the homogeneous 3-chain (go, gb g2, g3), gi E SL 2(F) to [z], where Z is the cross-ratio of the points go (00 ), g1 (00 ), g2 (00 ), g3 (00) E ]pI (F) (recall that the cross-ratio of the complex numbers zo, Zl, Z2, Z3 is (zo - Zl)(Z3 - Z2)/(ZO - Z2)(Z3 - zt}). This map is independent of the choice of 00 as a base point. We omit the proof of this theorem. Instead we shall summarize Suslin's study of a certain subgroup B(F) c p(F) in the next section.

3.2. The Bloch Group 3.2.1. Relation with H 3 (GL 2 (F)). Let F be any field and denote by V(F) the free abelian group with basis [z], where z E F X - {I}. Define a homomorphism t.p :

V(F)

--+

F X IX> F X

3.2. The Bloch Group

71

by rp([z]) = z ® (1 - z). An easy calculation shows that (3.1)

x®

(~ =:) + (~ =:) ® x.

Let a be the involution of F X ®Fx defined by a(x®y) = -(y®x) and denote by (F X ® FX)O' the quotient of F X ® F X by the action of a. Then (3.1) shows that rp induces a homomorphism ,\ : p(F)

---+

(F X ® FX)O'.

Note that ,\ is the homomorphism ,\ in the exact sequence of Theorem 3.1.8. DEFINITION 3.2.1. The Bloch group of F, denoted by B(F), is the kernel of the homomorphism ,\. Thus, we have an exact sequence

0---+ B(F)

---+

p(F) ~ (F X ® FX)(1

---+

K 2(F)

---+

O.

In the case where F is an algebraically closed field of characteristic zero, Theorem 3.1.8 asserts that there is an exact sequence

o ---+ I1F(2) ---+ H3(SL 2 (F), Z) ---+ B(F) ---+ O. Suslin [128] produced an exact sequence involving the groups B(F) and H3(GL2(F), Z) for any infinite field F. We describe this result now. Denote by Tn (resp. Bn) the subgroup of GLn(F) consisting of diagonal (resp. upper triangular) matrices and let GMn(F) be the subgroup of monomial matrices. We have an exact sequence 1

where

~n

---+

Tn

---+

GMn{F)

---+~n ---+ 1, "

denotes the symmetric group on n letters.

THEOREM 3.2.2 ([128], Theorem 2.1). Let F be an infinite field. Then there is an exact sequence

PROOF. Let C. be the complex with Cp the free abelian group with basis (xo, ... , xp), where the Xi are distinct points of pl(F) (this is the same as t.aking projective equivalence classes of unimodular vectors in F2). By Lemma 2.3.2, the augmented complex C. _ Z - 0 is acyclic. The group GL 2 (F) acts 3-transitivcIy on points in pI (F); that is, it acts transitively on Co, C1, C2 • Every orbit of the action on C p for 7J ~ 3 hH~

3. Low-dimensional Results

72

a unique representative of the form (0,00, 1, Xl, ... , X p -2), where the px - {I} are distinct. Denote this orbit by [Xl, ... ,Xp -2]. Then

Xi

E

x;f-XjEPX -{l}

and the differential d : (Cp )CL 2 (P)

(Cp - I )C L 2(P) is

---->

[ 1 - Xl , ... , 1 - Xl ] 1- X2

+

1- Xp-2

[X2, ... , XP -2]

Xl

Xl

p-2

+ 2:)-I)i[XI, ... ,Xi"",Xp -2]. i=l

(3.2)

Consider the spectral sequence a..<;sociated to the action of GL2(F) on C•. This spectral sequence converges to H.(GL 2(P)) and has El-term E~,q = Hq(GL2(F), C p ).

In particular, the row q = 0 is the complex (C.)CL 2 (P)' It has the form

Z

t-

Z

t-

Z

t-

EB

Z[x]

xEPx -{I}

ThUH wo have E2 _ p,D -

t-

EB

Z[XI, X2]

t- ....

Xl#X2

{z 0

p=o p= 1,2;

and

E~,D

ffi W Z[xl/ ([I-Xl] -_- x . 1 X2

[1-X-~lI ] + [X2] - - [X2] + [Xl] ) 1 - X2

Xl

p(F). Now, the stabilizer of 0 is the subgroup B 2 , the stabilizer of (0,00) is T2 = FX EEl FX, and the stabilizer of (0,00,1) is the subgroup px of scalar matrices. Thus,

E6,q = Hq(B 2), Ei,q = Hq(T2), E~,q = Hq(P X ). We know that the inclusion T2 ----> B2 induces an isomorphism in homology; that is,

3.2. The Bloch Group

73

Let (J' be the automorphism of H.(F X EBFX) induced by (a, b) an easy calculation shows that

dL :Hq(T2) is the map then

(J' -

id. Also, if

~

: FX

~

~

(b,a). Then

Hq(B2)

FX EB FX is the diagonal embedding,

is the map ~*' Since ~ has a left inverse, ~* is injective so that E~,. = O. Also, H1(FX EB FX) = F X EB FX and (J' transposes terms. Furthermore,

H 2(F X EB FX) = H 2(F X) EB H 2(F X) EB (FX ® F X), (J'

y

interchanges the copies of H2(FX) and (J' coincides with the involution x ® -(y ® x) on FX ® FX. Thus the E2-term of the spectral sequence is

f---)

H 3 (T2 )C1 H2(FX) EB (FX ® F X)C1

o

FX

z

o

p(F).

A long computation shows that d 3 : Elo = p(F) ~ H2(FX) EB (FX ® F X)C1 = Eg,2

is defined as d3 ([x]) = x 1\ (1 - x) - x ® (1 - x). Thus,

.

Eao = ker(d 3 ) = B(F) . Now, this spectral sequence determines a filtration of H 3 (GL 2 (F)) with H3(GL 2(F))(O) = im(H3(T2))' There is a canonical surjection

r /~*(H2(FX)) =

H 2(T2)C1 _ H 2(T2

(FX ® FX)C1 _ H 3(GL 2(F))(1)/(O).

Moreover, we have an isomorphism

H 3(GL 2(F))/H3(GL 2(F))(2) ~ B(F). To finish the proof of the theorem, we must show that H 3 (GL 2 (F))(2) = im(H3(GM2(F))). Let C. (G) be the standard resolution of the group G and let pg be the homotopy denoted by h. : C.(G,M) ~ C.(G,M) in Proposition A.1.9. LEMMA 3.2.3. Let u E H 2 (T2 )C1, and let h be a representing cycle for Denote by T the automorphism of H 2 (T2 )" which tran8poses terms and write T( h) - h = db for some 3-chain b of T 2 . Then the image of u in H:l( GL 2 (F))/ H 3 (T2 ) i.~ the clas .., of the 3-cycle b - PII(h), whc1Y! .'I =

'/1,.

(? (\) .

74

3. Low-dimensional Results

PROOF. Our spectral sequence is the spectral sequence of the bicomplex C.(GL 2 (F)) ®GL 2(F) C•. The augmentation £ : C. ~ Z induces a quasiisomorphism

C.(GL2(F)) ®GL 2(F) C. ~ C.(GL2(F))GL2(F)' Consider the diagram b - P8(h) ~ b® (0) - Ps(h) ® (00) d GL2 (F)

!

h ® ( ( 00) - (0))

de. ~E- -

h ® (0,00).

Note that d(b® (0) - Ps(h) ® (00))

= =

db® (0) - dps(h) ® (00) r(h) ® (0) - h ® (0) + h ® (00) - Sh8- 1 ® 8(00) r(h)®(O)-h®(O) + h ® (00) - r(h) ® (0) h ® ((00) - (0)).

Thus, h ~ b - Ps(h) E H 3 (GL 2(F))/H3(T2).

0 "

"

Now consider the Hochschild-Serre spectral sequence associated to the .;', ()xtemlion 1 -----t T2 -----t G M2 (F) -----t E2 -----t 1. This yields a filtration of H3(GM2(F)) satisfying 1. Ha(GM2(F))(O) = im(H3(T2));

H 1 (E 2,H2(T2)) = H 2(T2)CT /(1 + U)H2(T2) - H3 (GM2 (F))(l)/(O); 3. H3(GM2(F))(2)/(1) = 0 (since H.(E 2, T2) = 0); 4. H3(GM2(F)) = H 3 (GM2(F))(2) EB H3(E2); (the direct sum decomposition in 4 follows since the extension is split). The preceding remarks show that the homomorphism

2. H2(T2)CT

-

H 3(GM2(F))

-----t

H3(GL2(F))

maps H 3(GM2(F))(O) onto H 3(GL 2(F))(O). Lemma 3.2.3 tells us the image of E H 2(T2)CT in H 3(GM2(F))/H3(T2). Thus, H 3(GM2(F))(l) is mapped onto H 3 (GL 2 (F))(l) and we have

1.t

im(H:J(GM2(F)))

= im(Ha(E2)) E9 H:1(GL 2 (F))<2)

3.2. The Bloch Group

75

(since the first and second filtration levels coincide). Now, since the matrix s is similar to (6 ~l) E B 2 , we have im(H3(E2)) C im(H3(B2)) = im(H3(T2))

= H 3(GL 2(F))(0).

Therefore, im(H3(GM2(F))) = H 3(GL 2(F))(2), and the theorem is proved.

o

REMARK 3.2.4. The proof of Theorem 3.2.2 follows the argument used by Dupont-Sah [31] to prove Theorem 3.1.8. In this form, it works for any infinite field. The proof in [31], while giving a slightly stronger statement, relies heavily on the fact that F is algebraically closed.

3.2.2. Relation with K-theory. Since H3('GL2(F)) influences K3(F), we might expect some relationship between K3(F) and B(F). It was S. Bloch [11] who first noticed this connection (hence the name "Bloch group" ). Suslin made this relationship precise and we briefly sketch his results. The first step is to extend the homomorphism H3(GL2(F)) -> B(F) to a map H3(GL3(F)) -> B(F). This is accomplished by considering the complex where is the free abelian group with basis (xo, ... , xp), where the Xi E P2(F) are in general position. Define two differentials d, d' : G; -> G;_l by

G;,

G;

p

d(xo, ... , x p) =

2) -l)i(xo, ... , Xi,""

x p)

i=O

and

p-l

d'(xo, ... , x p) =

2:) -l)i(xo, ... , Xi,···, x p). i=l

Then an easy calculation shows that the complexes (G., d) ~ Z (G., d') ~ Z -> 0 are acyclic. Define a map A : -> by

G; G;

p

A(Xo, ... , x p) =

2) -l)Pk(xk, Xk+l,···, Xk+p). k=O

One checks easily that d' A = Ad. Consider the map of complexes

Gr Al C?

d

~

d'

~

G22

+---

Al c'J22

d'

d

G32

d ~

Al G2

~:l

ti'

<----

...

->

0 and

3. Low-dimensional Results

76

Let D. be the cone; that is, Do by the matrix

= Cl, Dp = C; EEl C;+l

with differential given

(-~ ~,). Define the augmentation Do

~

Z as the composition

2d 2E'7I D 0= Cl~CO~~'

°

Then it is easy to see that D. ~ Z ~ is acyclic. The group GL3(F) acts on D. and determines a map H3(GL3(F)) ---- H 3((D.)CL 3 (F»). Direct computation shows that

H3((D.)CL3(F») = coker ( EeZ [~] EEl EeZ

( -=-d Q) [~tJ ~ Z EEl EeZ [~])

where a =F x, b =F y E FX - {I} and ay - bx =F 0. By setting p([~]) = [a] E p(F), we get a well-defined homomorphism H3((D.)CL 3(F») ~ p(F). Hence by composing with the canonical map H3(GL3(F)) ~ H3((D.)CL 3(F»), we obtain a map H3(GL3(F)) ~ p(F). The restriction of this map to H 3 (GL 2 (F)) coincides with the map H 3 (GL 2 (F)) ~ p(F) defined above. This is proved as follows (see Lemma 3.4 of [128]). Denote by C~ the subgroup of C;+1 generated by elements (xo, ... ,Xp+1), where Xp+l = (O,O,l)T. Since d' does not affect the last coordinate, C~ is a GL 2 (F)-subcomplex of D •. Thus, the composition

H3(GL 2 (F))

~

H3(GL3(F))

~

H3((D.)CL 3(F»)

can also be written as the composition H 3(GL2(F)) ~ H3((C~)CL2(F») ~ H3((D.)CL3(F»)'

Iuductl f\. homomorphism of GL 2 (F)-complexes C~ ~ C. by projecting JPl2(F) ____ JPll(F) from (0,0, l)T and discarding the last coordinate. The com-

position

H3(GL2(F)) ~ H3((C~)CL2(F») ~ H3((C.) CL3(F»)

= p(F)

is evidently the homomorphism {) defined above. The proof is finished by noting that the following diagram

(3.3)

commutes.

77

3.2. The Bloch Group PROPOSITION

3.2.5. We have an isomorphism

+ H3(T3»

H3(GL3(F»/(H3(GM2(F»

Cr

The group H 3 (T3 ) lies in the kernel since the augmentation C:D. Z has a T3 -equivariant section:

PROOF. -->

~ B(F).

n~n·O

n

:

By [124], we know that H3(GL3(F» is generated by H 3(GL 2(F» and H3(T3)' Thus, the image of H3(GL3(F» --> p(F) coincides with that of H3(GL 2(F»; that is, the image is B(F). Since we have an isomorphism

H3(GL2(F»/ H3(GM2(F»

~

B(F), D

the result follows. Denote the map H3(GL3(F»

-->

B(F) by

H3(GL3(F»

~

a. Since

H3(GL(F»

(Theorem 2.3.1), we can view a as a map H3(GL(F» Hg(GMn(F» the kernel of the split surjection

H3(GMn(F» LEMMA

-----t

-->

B(F). Denote by

H3(En).

3.2.6. There is an exact sequence

Hf(GM(F»

-----t

H3(GL(F»

B(F)

-----t

-----t

O.

PROOF. The first step is to show that H3(T) lies in the kernel of a. The Kiinneth formula implies that H3(T) is generated by the subgroups H3(Ti j k), where Tijk is the subgroup consisting of diagonal matrices with 1's everywhere except for possibly in positions i,j, k. Note, however, that Tijk is conjugate to T3 by a permutation matrix and hence the image of each H3(Tijk) coincides with that of H3(T3)' But H3(T3) lies in the kernel of a. Now let m, n be positive integers. For a pair G I , G 2 of groups, denote by Pi the projection G I x G 2 --> G i . Denote by qi the composition

GLm(F) x GLn(F) ~ GL*(F)

-----t

GL(F).

Also, denote by j the embedding G Lm (F) x G Ln (F) --> G Lm+n (F) --> G L( F). Note that for any n, the embedding Tn --> GLn(F) induces a surjection on Hl and H2 (for HI this is clear; for H2 this follows from Theorem 2.3.1 since we have a surjection K~(F) -----t H 2 (GL 2 (F» induced by the homology product Hl (Tl ) ® Hl(Tl ) --> H 2(GL 2(F» and the domain of the latter map lies inside H 2(T2». Thus, since j(Tm x Tn) C T, we Hee that

j.(H2(GL m(F» ® Hl (GLn(F»)

C

H3(T).

3. Low-dimensional Results

78

Now let n be sufficiently large and consider the commutative diagram 1 - T 2 -GM2(F) x En-2-E2 x En- 2 - 1

111

1-

Tn

---~>

GMn(F)

------;>~

En

--->~

1.

Denote the Hochschild-Serrc spectral sequence of the top sequence by E' and of the lower sequence by E. We have E;q

= Hp(En, Hq(Tn)).

Denote the ith summand of Tn = (Fx)n by Ft. Then

i<j

i=l

The action of En is transitive on both terms of the summand with stabilizers /\

2

x

(FI h~n

_

-

El X E n - I

and

I

j Thus we have 2

E;2 = Hp(En-l,/\ FX) fBHp(E2

X

I

E n _ 2,F x ®FX).

,I

Similarly, we have

'2 = Hp(En-2' /\2 F X) fB Hp(E2 Ep2

X

En- 2, F X ® F X).

The homology of En stabilizes for large n, and hence we see that for p ~ 5, the two spectral sequences coincide. The same is true for the q = 1 rows of the sequences. It follows that the (1,2) and (2,1) terms of the limits coincide and that H8(GMn(F)) = H3(Tn)

+ ker{H3(GM2(F)

x E n- 2) - + H3(E2

X

E n-2)}.

Passing to the limit, we see that the image of Hg(GMn(F)) is contained in H3(T) +H3(GM2(F)), which is precisely the kernel of 8. The reverse inclusion is clear. 0 We now have the following result. THEOREM

3.2.7. The Hurewicz map induces an isomorphism

K 3(F)/1rg(BGM(F)+) ~ H 3 (GL(F))/Hg(GM(F)),

where 1r~(BGM(F)+) is the kernel of the projection 1r3(BGM(F)+) -t1r3(E+).

3.2. The Bloch Group

79

PROOF. This is based on a number of lemmas whose proofs may be found in Section 5 of [128]. Let h : S3 ~ S2 be the Hopf bundle. This induces a map h* : 1I"2(X) ~ 1I"3(X) for any space X. The map h* is not a homomorphism in general, but if X is an H-space, then it is a homomorphism. One checks that if X is a simply-connected H -space, then there is an exact sequence .

1I"2(X) Taking X = E(A)

h*

~

1I"3(X)

~

H3(X)

~

O.

c GL(A), we obtain an exact sequence .

K 2(A)

h'

~

.

~

K3(A)

H3(E(A))

~

O.

Now consider the group GM(F) and its commutator subgroup M. The group M is the subgroup of monomial matrices having determinant 1 and an even permutation. This group is perfect. Consider the spaces BSL(F)+ and BM+. We have a commutative diagram

7r2(BM+)

-------+

1 K 2(F) ~ow,

7r3(BM+)

-------+

1 -------+

H3(M)

-------+

0

-------+

O.

1

K3(F)

-------+

H3(SL(F))

we have homotopy equivalences

B([G, G])+ x BG8b ~ BG+

(3.4)

for the groups G = GL(F), GM(F) ([128], Lemma 5.3). Thus, for i ~ 2, 7ri(BM+) = 1I"i(BGM(F)+). Moreover, for i ::; 2, the map 7ri(BGM(F)+) ~ Ki(F) is surjective. Thus,

K3(F)/1I"3(BGM(F)+) = H3(SL(F))/H3(M). The same is true if we divide out by 1I"8(BGM(F)+) and by

Hg(M) = ker(H3(M) ~ H3(E+)). Denote by SM(F) the group of monomial matrices in SL(F). The projection SM(F) ~ SM(F)8b S:!. 71../2 has a section. Using the Kiinneth formula and the equivalence (3.4) for SM(F), we see that

H3(SL(F))/H3(M)

=

H3(SL(F))/H3(SM(F))

and similarly if we divide by H3. Now, applying the equivalence (3.4) to G = GL(F), we find

H3(GL(F)) = H3(SL(F)) EB (H2(SL(F)) ® FX) EB H3(FX). Since the last two terms lie in Ha(GM(F)), the map

H3(SL(F))/ H3(SM(F))

-+

H3(GL(F))/ Ha(GM(F))

80

3. Low-dimensional Results

is surjective. We define a left inverse by a 1---+ a EB det( a-I). (Note: the fact that H2(SL(F)) 0 F X C H3(GM(F)) follows from the more general fact that H 2 (GL m (F)) 0 HI (GLn{F)) C H3(T) for any m, n.) The same is evidently true if we divide out by Hg. D It is possible to give a description ofthe image of 7l"~{BGM(F)+) in K3(F). There is a ring structure on 7l".(BGM(F)+) such that the maps

7l".(BGM(F)+) ----7 K.(F) tlnci

7l".(BGM(F)+) ----77l".(E+) are compatible with the multiplication. We have

7l"l(BGM(F)+) = H 1 (GM(F)) = F X EB7/.,/2, with 7l"~(BGM(F)+) = F X and 7l"l(BE+) = 7/.,/2. The map

7l"l(BGM(F)+)

--+

Kl(F)

= F

X

is the identity on F X and takes the generator of 7/.,/2 to -1 E FX. Thus, the image of 7l"~(BGM(F)+) in K 3(P) contains the decomposable part

im(Ktt (F)

--+

K3(F)).

Since the image of the decomposable part of 7l".(BE+) is contained in the decomposable part of K.(F), the image of 7l"~(BGM(F)+) in the indecomposable group K 3(F)ind = K 3(F)/Ktt(F) coincides with the image of 7l"~(BGM(F)+)ind

= ker(7l"3(BGM(F)+)ind ----77l"3(BE+)ind).

Vilt Home technical arguments using the Atiyah-Hirzebruch spectral sehomotopy theory, and Chern class maps on K-theory, we obtain the following mmlts. t. There are natural maps

qlltllW<1,

Tor(J.L(F),J.L{F)) ----7 7l"~(BGM(F)+)ind ----7 Tor(/l,(F),J.L(F)), and the composition is multiplication by 2. 2. The composition Tor(/l,(F), Jl(F)) ----7 7l"~(BGM(F)+)ind ----7 K3(F)ind is injective. 3. There is an exact sequence

0----7 J.L2(F) ----7 7r~(BGM(F)+)ind ----7 Tor(J.L(F), J.L(F)) ----7 O. If char(F) i- 2, the extension is nontrivial. Denote this extension by Tor (J.L (F), J.L( F))~. LE:vIMA

3.2.8. The map 7l"~(BGM(F)+)itld -; K:1(F)lnd is injective..

3.2. The Bloch Group

81

PROOF. This follows from 2 above by passing to the algebraic closure of F (see [128], Lemma 5.8). 0 Summarizing, we obtain the following. THEOREM 3.2.9. For any infinite field F, there is an exact sequence

o ~ Tor(J.t(F),J.t(F))'"

~

K3(F)ind

~

B(F)

~

O.

COROLLARY 3.2.10. If F is algebraically closed, then B(F) is uniquely divisible. PROOF. By Suslin's solution to the Lichtenbaum-Quillen conjecture for algebraically closed fields [126, 127], the group K3(F) is divisible with torsion subgroup ffil!#char(F) Qi/ZI!. The same is true for K 3(F)ind since Kf (F) is known to be uniquely divisible by the work of Bass and Tate [5]. Now, Tor(J.t(F), J.t(F))'"

E9

=

Qi/ZI!.

Nchar(F)

Since every injective map of Qi/ZI! into itself is bijective, we see that the group Tor (J.t(F) , J.t(F))'" is the torsion subgroup of K 3(F)indi that is, the quotient B(F) is uniquely divisible. 0 This provides another proof of the unique divisibility of the scissors congruence group P(Ji3). Since p(Ji3) ~ p(C)-, it suffices to show that p(C) is uniquely divisible. But this follows from the above corollary since we have an exact sequence o ~ B(F) ~ p(F) ~ (F X 0FX)u and the group (FX 0FX)u is uniquely divisible when F is algebraically closed. If F is not algebraically closed, then B(F) need not be uniquely divisible. Let x, y E FX. Consider the elements [x] + [1 - xl, [y] + [1 - y] E B(F). LEMMA

3.2.11. In B(F), [x]

+ [1- x] = [y] + [1 - yl.

PROOF. We may assume x =I- y. The elements

[x]- [y] [1- y]- [1 - x]

+ [y/xl- [(1- x-l)/(l- y-l)] + [(1- x)/(1 - y)l + [(1 - x)/(1 - y)]- [(1 - x- l )/(1 - y-l)] + [y/x]

are zero in p(F). Subtracting one from the other, we obtain the result. Denote the element [x] LEMMA

3.2.12.

6CF

+ [1 - x]

= O.

by cF.

0

82

3. Low-dimensional Results

PROOF. If x E F

X -

{I}, denote by (x) the element [x]

+ [X-I]. We have

the relations

+ [y/x]- [(1- x- 1 )/(I- y-l)] + [(1- x)/(I- y)] = 0 + [x/y]- [(1- x)/(I- y)] + [(1- x- I )/(I- y-l)] = O. Adding these, we see that (x) - (y) + (y/x) = O. Interchanging x and y, we see that 2(y/x} = 0 (since (y/x) = (x/y)). Thus, for any z, 2(z} = O. Now, [x]- [y] [x-l]_ [y-l]

3CF

[x] + [1- x] + [x-I] + [1- x-I] + [(1- x)-l] + [1- (1- x)-l] = (x)+(I-x)+(I-x- l ) = (x) + (1- x) + (-(1 - x)/x) = (x) + (1- x) + (-(1 - x)} - (x) = (1-x}+(-(I-x)) = (1 - x) + (-(1- x)2/(1 - x)) = (-(1 - x)2).

o = O. So 6CF = O. Now, if the equations x 2 + 1 = 0 and x 2 - x + 1 = 0 are solvable in F, one

But for any z, 2(z}

sees that CF = 0 ([128], Lemma 1.5). In particular, if F is algebraically closed, then CF = O. However, Suslin shows that the element CF is nontrivial and has order exactly 6 for any F ~ R Thus B(F) is not uniquely divisible since it has torsion. REMARK 3.2.13. The group B(Q) is cyclic of order 6, generated by CQ (see [128], Corollary 5.3).

3.3. Extensions and Generalizations S.S.l. Extensions. Many of the results in the previous section hold for rings other than fields. In particular, if A is a local ring with infinite residue field F, we can define the Bloch group B(A) exactly as we did for F. It was shown by the author in [70] that the proof of Theorem 3.2.2 goes through without modification for A. Thus we have an exact sequence

H3 (GM2(A))

--t

H3(GL2(A))

--t

B(A)

~

O.

The proof of Theorem 3.2.7 requires a bit of modification, but we still have the same result; namely, there is an exact sequence

1rg(BGM(A)+)

--t

K3(A)

--t

B(A) ~ O.

The only problem with the proof of Theorem 3.2.7 in this case is that it makes IlHe of a projection IP2(F) -+ IPl(F) which is not well-defined over A. However, it iH pOARible to COIlHtruct a map on the approprintfl ehniu complexes.

3.3. Extensions and Generalizations

83

Unfortunately, the proof of Theorem 3.2.9 does not generalize to A. Thus, we cannot deduce the exact sequence

o ~ Tor(J.t(A),J.t(A))'"

~

Ka(A)ind ~ B(A) ~ O.

(3.5)

However, if we assume that A is a Hensel local F-algebra, then we can say the following. Let p be a prime distinct from char(F). Since tensor product is right exact we obtain a commutative diagram 7rg(BGM(F)f) ® Zip

t 7rg(BGM(A)+) ® Zip

~

Ka(F) ® Zip

~

Ka(A) ® Zip

t

~

B(F) ® Zip

~

B(A) ® Zip

t

~

0

~

0

with exact rows. The map B(F)®Zlp ~ B(A)®Zlpis injective (since B(F) ~ B(A) splits) and the map Ka(F) ® Zip ~ Ka(A) ® Zip is an isomorphism [127]. A simple diagram chase gives the following result. PROPOSITION 3.3.1. Let F be an infinite field and let A be a Hensel local F -algebra. Then the natural map

B(F) ® Zip ~ B(A) ® Zip

is an isomorphism for P =1= char (F) .

o

Since K~ (A) ® Zip ~ K~ (F) ® Zip (this follows from the isomorphism A x ® Zip ~ F X ® Zip), and since J.t(A) = J.t(F), we see that we do obtain the exact sequence (3.5) for A after tensoring with Zip. The group Ka(R)ind can be described explicitly for a large class ofrings. A ring R is an HI-ring [50] if the following holds for alll ~ 2, k ~ 1: For any family of k surjective linear maps Ji : RI ~ R, there exists vERI such that Ji(v) E RX for i = 1, ... k. For example, semilocal rings with infinite residue fields are HI. Also, HI-rings have many units and have stable rank 1. We have the following result due to P. Elbaz-Vincent [36]. THEOREM

3.3.2. Let R be a commutative HI-ring. Then there is an iso-

morphism

This relies on the following stability result proved in [36]: The map GL2(R) ~ GLa(R) induces an injection Ha(GL2(R),Q) ~ Ha(GLa(R),Q).

The proof of this result is much like the proof of Theorem 3.1.8. Indeed, the proof follows that of Sah's proof of the following result [107].

3. Low-dimensional Results

84

THEOREM 3.3.3. Let D be one of the three real division algebms or any infinite field with F X = (FX)6. Then the map

H 3 (SL2(D), Z)

-+

H3(SL3(D), Z)

o

is injective.

As a corollary, Sah proves that for any infinite field F, we have K 3 (F)ind = Ho(FX, H 3 (SL2(F)))

modulo the class of abelian groups annihilated by powers of 6. 3.3.2. Generalizations. The derivation of the exact sequence

in Section 3.2 required a nontrivial amount of computation. One expects, however, that a similar approach to the study of Hn+l(GLn(F)) might prove fruit- _ ful. Unfortunately, the resulting spectral sequence is very complicated. In his thesis [133] S. Yagunov constructed a new resolution of GLn(F)modules and studied the resulting simpler spectral sequence. This leads to a definition of higher Bloch groups which are related to Hn+l (G Ln(F), A). One (minor) drawback is that we must assume that n! is invertible in A. For each n ~ 2, define a group Pn(F) as follows. Let P:(F) be the acyclic complex with PJ:(F) the free abelian group with basis the (k + I)-tuples of rational points of pn-l(F) which are in general position. This is clearly a complex of GLn(F)-modules. Define Pn(F) = Hn+l(P:(F)GL n (F))'

=

When n 2, this is precisely the group denoted p(F) in Section 3.2. It is clear that there is a surjective map Hn+l(GLn(F),Z)

-+

Bn(F)

for some subgroup Bn(F) C Pn(F). Determining this group Bn(F) seems rather difficult in general (certainly the complexity grows as n increases) since it involves the calculation of higher differentials in a spectral sequence. Yagunov defines A-modules Pn(F) in the following way. Let C: be the complex with Ci: the free A-module with basis (k + I)-tuples of vectors in Fn in general position. This complex is clearly acyclic. Define a sub complex C:,P by taking C~'P to be generated by the elements

ifk>paild by

.

J

.

3.3. Extensions and Generalizations

85

otherwise. Here, the 0i E FX. Then the quotient complex e~ = C~ /e~,n-l is easily seen to be acyclic. The corresponding spectral sequence has

El = H (GL (F) en) = {Hq((FX)P+! x GLn-p-l(F)) p,q q n 'P 0

p
Here, GLo(F) = {I}. The bottom row is the complex (C~)GL",(F)' One checks easily that

E2 _ p,o -

{A0

p= 0

0

< p S n.

We say that a vector v E Fn is monomial if it has exactly one nonzero entry and affine if all its entries are nonzero. Define D~ to be the free Amodule on the following basis. Consider all n x (k + 1) matrices of the form (vo, . .. , Vj, Wj+!, ... , Wk) where the Vi are independent monomial vectors and the submatrix (Wj+!, ... ,Wk) is such that any minor is non-zero. The positions of the entries in (vo, ... , Vj) uniquely define an ordered subset of {I, ... , n}. Two such matrices are projectively equivalent if they have the same affine parts and the same ordered subsets of {I, ... , n}. Now take the set of projective equivalence classes of such matrices as the basis of D'k. With the usual differential, the complex D~ is acyclic and there is a natural action of the monomial group G Mn (F) on D~. There is an isomorphism

H.((Dn) ,

)_

• GM",(F)

and a surjection A ® (F X )[n/2] following result.

-#

-

{A0

i = 0 0
+1

Hn+l ((D~)GM",(F»)' Moreover, we have the

PROPOSITION

3.3.4. If P > 0 and n! is invertible in A, then

Hp(GMn(F),

D~)

=

{HOp((Fx)q+l x GMn- q+1 (F))

0 S q S n-1 q 2: n.

Now consider the complex

Indg;;S;')D~ = AGLn(F) ®AGMn(F) D~. There is a natural surjective map I

defined by a ® x

I-t

dGL",(F)

n GM",(F)

Dn

• ----+

ax. Denote the kernel by

en•

K~.

86

3. Low-dimensional Results

Here, H.(GLn(F), GMn(F); A) are the relative homology groups (see Appendix A). On the other hand, we have the spectral sequence associated to the action of GLn(F) on K;:. It has the following form. PROPOSITION 3.3.6.

E~,q

If q > 0, then

Hq(GLn(F), K;) Hq+1(GL n- p-l(F) x (FX)P+1, GMn- p-l(F) x (FX)P+1; A).

= =

We also have an exact sequence of complexes

o ~ Ho(GLn(F), K;:) ~ (D~)GM,,(F)

----+

(C;:)GLn(F) ~ O.

This follows from the fact that for each p, the map

H 1(GMn- p(F) x (Fx)P)

~

HI (GLn_p(F) x (FX)P)

is surjective so that we have an exact sequence

o ~ Ho(GLn(F),K;)

~ (D;)GM,,(F) ~ (C;)GL,,(F) ~ 0

(this is the tail end of the long exact sequence associated to

o ~ K;

~ Indgt;3;'~)D;

----+

C;; ~ 0).

Thus, we obtain the following result. LEMMA 3.3.7.

Ho(GLn(F), K;) =

{~ker(Hn+'((D~)CM'(F») ~ HnH((C!')CL.(F)))

DEFINlTlON 3.3.8.

p
The A-module Pn(F) is defined as

There is an obvious map

Yagunov makes the following conjecture. CONJECTURE 3.3.9. Let F be an infinite field, let n 2: 2 and assume that n! is invertible in A. Then there is a natural isomorphism

Pn(F) =

E8 Hn+1-2i(GLn- 2i (F), {GMni~O

2i (F),

GLn-2i-l (F)}; A)ind

3.4. Invariants of hyperbolic manifolds

87

where ind denotes the indecomposable part of the homology. The prod'u,ct structure on homology is defined via the compositions

Hi(GLn(F),GMn(F)) @Hj(FX)

~

~

Hi+j(GLn x FX,GMn x FX) Hi+j(GLn+1(F), GMn+l(F)).

Yagunov verifies this conjecture for n :S 4 (the case n = 4 requires that F be algebraically closed). It is natural to ask the relationship between Pn(F) and Pn(F)@A. Yagunov constructs a natural map Pn(F) @ A ----4 Pn(F) for all n ;::: 2. 3.3.10. Suppose 2 is invertible in A. Then the map

PROPOSITION

o

is an isomorphism.

For further generalizations of Bloch groups, the reader is referred to Zagier's paper [134] where a conjectural description ofthe groups Bm(F) is given for m odd. Zagier focuses on the case where F is a number field and relates these groups with the Borel regulator map on K2m-dF) and values of the zeta function (F (s).

3.4. Invariants of hyperbolic manifolds The connection between the Bloch group and scissors congruences in hyperbolic 3-space suggests that it may be possible to define invariants of manifolds in B(C). W. Neumann and J. Yang [93] showed that this is indeed the case. We briefly discuss this here. Let IHI3 be hyperbolic 3-space with boundary Cpl. An 'ideal simplex ~ in IHI3 = IHI3 U Cpl is one whose vertices Zj, Z2, Z3, Z4 lie in Cpl. Such a ~ is determined up to congruence by the cross ratio Z = [Zl : Z2 : Z3 : ZIJ] =

(z:~ - Z2)(Z4 - zd ( )( ). Z3 -

Zl

Z4 -

Z2

Take finitely many geometric 3-simplices and glue them together along 2-faces to obtain a cellular complex Y which is a manifold except possibly at isolated points. If the complement of these points is oriented, we call Y II geometric 3-cycle; the complement Y - y(O) of the vertices is an oriented tnanifold. Suppose 11;1 = IHI 3 /r is a hyperbolic manifold of finite volume, where I' CPS L2 (C) is discrete. A degree orw ideal triangulation of M consists of a g(~ornctric 3-cyc:k Y with H lllap f : Y - y(O) __._, M sntisfying

3. Low-dimensional Results

88

1.

f is degree one almost everywhere in

M;

2. For each 3-simplex 8 in Y, there is a map fs to an ideal simplex in JH[3, bijective on vertices, such that fls-s(o) : 8 - 8(0) --) M is the composition 'IT 0 (fsls-s(o»), where 'IT : JH[3 --) M is the projection. Thurston proved that compact hyperbolic 3-manifolds have degree one ideal triangulations with IYI ':::' jIll. Most noncom pact j1lf also have such triangulations as well. DEFINITION 3.4.1. Let f : Y - y(O) --) M be a degree one ideal triangulation of j1lf. Each 3-sim plex of Y maps to an ideal simplex in JH[3: denote by Zl, . .. , Zn the cross ratio parameters of these ideal simplices. The Bloch invariant f3(M) is the element 2:7=1 [Zj] E p(C). If the Zj all lie in a subficld K ~ C, we may consider f3(M) as an element of p(K). A priori, the invariant f3(M) depends on the triangulation. This is not the case, however.

PROPOSITION 3.4.2 ([93], Proposition 4.3). The invariant f3( M) is independent of the triangulation. D This is proved by defining a fundamental class

[M] E H 3 (r, Cpl)

= Tor~r (JCpl,

Z),

where JCpl is the kernel of the augmentation map Z[Cpl] --) Z. There is homomorphism H 3 (r,Cpl) --) p(C) which maps [M] to f3(M), and this implioH the result.

It

TIIEOR~;M

3.4.3. The invariant f3(M) Ues in B(C).

D

The proof of this is fairly straightforward, but lengthy and technical. The ()HHcntial point is that the relation

L

Zi 1\

(1 -

Zi)

=0

holds in CX 1\ C X; that is, 2: [Zj] E B(C). The invariant f3(M) is related to other invariants of M. Denote by vol(M) the volume of M. For any hyperbolic 3-manifold A1, there is a well-defined invariant C8(M), called the Chern-Simons invariant, which lies in lRj'IT2Z. It is defined as the integral of a certain 3-form built from the connection and curvature forms over a nonsingular section s(M) of the positively oriented orthonormal frame bundle. Define the Bloch regulator map f! : B(C)

----->

CjQ

J

89

3.4. Invariants of hyperbolic manifolds

as follows [11]. For

E

Z

CX

-

{I}, define

10g(l- z) ( ) _ logz 2. A 2· 11"~ 11"2

pz -

+

'R(z)

= ! 10gzlog(l- z) _

I

where 2

'R(z)

A 22 E 11"

r

Jo

C

C A ,

10g(l- t) dt. t

This map p vanishes on the defining relation of p(C) and hence induces a map

We have a commutative diagram

B(C) -----+

1

1

C/Q -----+ CAC ~ CX ACx

where € = 2(e A e) with e(z) B(C) ~C/Q.

= exp(211"iz).

So p restricts to give a map p :

THEOREM 3.4.4 ([93], Theorem 1.3). The Bloch invariant satisfies p(f3(M))

=

2!2 (vol(M)

+ iCS(M)) E C/Q.

It is possible to extend this idea to higher dimensions. Let Sq(8lHn be the abelian group generated by (q+ I)-tuples of distinct points in BiEr modulo the relations

(zo, ... , Zq) = sgn(T)(zT(O),""

ZT(n))

for any permutation T E ~q+l' With the usual boundary map this defines an acyclic complex S.(BiEr). The group Isom+(IHIn) acts on this complex. Define

Pn = Hn (S.(8iEr)Isom+(nnn)). If n = 3, then we recover the group p(C). These groups are similar to the groups denoted by Pn(F) in Section 3.3. Let M be a complete hyperbolic n-manifold of finite volume and let f : 1I"1(M) ~ Isom+(IHIn) be a homomorphism. Then one may define an invariant fJ(f) E P n . If n = 3, then if f is the homomorphism corresponding to some "Dehn filling" M' of M, then f3(f) = f3(M'). Thus, this f3(f) is a generalization of f3(M).

90

3. Low-dimensional Results

Exercises 1. Prove that the inclusion Simo(n)

~

Sim(n) induces an isomorphism

H.(Simo(n),zt) __ H.(Sim(n),Zt). 2. Prove that Sim(n)lRj

~

Sim(j) x O(n -

j,~).

3. Prove the equality in Equation 3.1. 4. Verify the formula in Equation 3.2. 5. Verify that the maps (defined in the proof of Theorem 3.2.2)

dt,q : Hq(T2) - - Hq(B2) and

dL : Hq(FX) __ Hq(F X E9 FX)

are u - id and ~*' respectively. 6. Prove that the map

tP : p(F) - - Eg,2 is defined as = x 1\ (1- x) - x® (1- x) (see the proof of Theorem 3.2.2). 7. Consider the extension d 3 ([x])

8. 9. 10. 11.

1 - - T2 - - GM2(F) - - 1::2 - - 1. Use the Hochschild-Serre spectral sequence to verify the assertions made about H3 (GM2 (F)) in the proof of Theorem 3.2.2. Compute HI (1:: 2, H 2(T2)). What is the answer for F algebraically closed? Verify the various fonnulas and assertions of acyclicity of the complexes ; defined in the discussion prior to Proposition 3.2.5. Verify that the diagram (3.3) commutes. Prove that a (semi)local ring with infinite residue field(s) is an HI-ring.

Chapter 4

Rank One Groups The homology of rank one linear groups (SL 2, PGL 2 ) often can be computed via the action of the group on a suitable simplicial complex. There is the well-known tiling of the hyperbolic plane by SL2(Z)-translates of an ideal triangle (see, e.g. [21], p. 215) and there is the BruhatTits tree associated to a field with discrete valuation. Often, the action implies something about the structure of the group such as the existence of an amalgamated free product decomposition. In this chapter we study the groups SL 2 and PGL2 over the rings Z[l/p], p a prime, and k[C], where C is a smooth affine curve over the field k. In these cases we are able to obtain a great deal of information about the homology (often, we can compute it completely). Applications of these results will be given in Chapter 5. We also discuss the generalization of these results to higher rank groups.

4.1. SL 2 (Z[1/p]) We begIn with the following classical fact ([110], p. 11): SL2(Z) ~ Zj4 *Zj2 Zj6,

where the subgroups Zj2,Zj4,Z/6 are generated by

(-~ -~), (~1 ~), (~ -~) respectively. There is a map SL2(Z) ~ Zj12 defined by sending the generator of Z/4 to 3 mod 12 and the generator of Zj6 to 2 mod 12. 4.1.1. The map SL2(Z) integral homology: THEOREM

----+

Z/12 induces an isomorphism in

4. Rank One Groups

92

PROOF. Associated to any decomposition G Vietor is sequence

=

G} *AG 2 , there is a Mayer-

The homology of a cyclic group 7l/n is given by (sec Appendix A)

Hi (7l/n, 7l)

=

7l { 7l/n

o

i = 0 i odd i even.

Moreover, it is easy to see that if 7l/n --'> 7l/(nm) is the inclusion 1 f---+ m, then the map H 2i - 1 (7l/n) ~ H2i-d7l/(nm)) is the same inclusion. It follows that the sequence for computing H.(SL 2 (7l)) has the following form:

o ~ H2i(SL2(7l))

~

7l/2

~

7l/4 (jj 7l/6

.-+

H 2i - 1 (SL 2(7l))

~

0,

since H 2i (7l/n) = 0 for all n, i ~ 1. Note that the map 7l/2 ~ 7l/4 (jj 7l/6 is injective so that H2i(SL2(7l)) = 0 for i ~ 1 and H 2i - 1 (SL 2(7l)) has order 12. One checks easily that H 2i - 1 (SL2(7l)) is in fact cyclic. D The groups SL2(7l[1/p]), p prime, also admit a decomposition:

SL2(7l[1/p]) ~ SL2(7l)

*ro(p)

SL 2(7l),

where ro(p) is the subgroup of SL2(7l) consisting of matrices which are upper t.riangular modulo p. The two copies of SL2(7l) are embedded as the standard

mpy and as the set of matrices of the form

(p!!:le P:) where ad - be = 1.

TlllH cic!cornposition is obtained in [110] using the action of SL 2(7l[1/p]) on a cOl'tnin tree; this is a special case of the construction in Section 4.2. The cohomology of SL 2(7l[1/p]) was completely calculated by A. Adem and M. Naff'ah in [1]. We shall not give full details here, but we will indicate the line of proof. Since the cohomology of SL2(7l) is the same as that of 7l/12, one need only compute He(ro(p)) and then use the Mayer-Vietoris sequence. The proof is divided into cases: p = 2, p = 3, p ~ 5. The first computation is the following. PROPOSITION

4.1.2. If P 2:: 5, then Hl(ro(p),7l) ~ 7lN(p) , where

(P-7)/6

p===1

mod12

N(p) = { (p + 1)/6 p === 5 mod 12 (p-l)/6 (p + 7)/6

p===7 mod12 p === 11 mod 12.

93 PROOF.

Let r(p) be the level p congruence subgroup. We have an exten-

sion 1 ~ r(p) ~ ro(p) ~ B ~ 1, where B is the upper triangular subgroup of SL 2(7L.). The group SL 2(7L.) acts on a tree T with finite stabilizers and quotient an edge (this is one way to prove the amalgamated free product decomposition). Since r(p) is torsionfree, it acts freely on T and thus G = SL2(lFp) acts on T/r(p) with isotropy groups '1./4 and '1./6 for the vertices and 7L./2 for the edge. Let EB be the universal cover of BB. Via the projection ro(p) -+ B, ro(p) acts diagonally on EB x Tj the isotropy groups are trivial. Since EB x T is contractible, we have Bro(p) !:::! EB XB T/r(p). Let C· be the cellular cochain complex of T/r(p). Then we have CO ~ Z/[G/(Z/4)]lB E9 Z[G/(7L./6)]lB' and C 1 ~ 7L.[G/(7L./2YIIB, where IB means restriction of modules (this follows from the restrictioninduction formulas for modules). The associated spectral sequence has Efq

= Hq(B, CP) ===> HP+q(ro(P), 'I.).

This collapses to give a long exact sequence ... -+

JP(ro(P))

-+

Hi(B, CO)

-+

Hi(B, C 1 )

-+ ....

o

The result follows easily.

One then uses the fact that the cohomology of ro(p) is periodic to reduce the computation to that of H2 and H3. The final result is the following. THEOREM

4.1.3. If i

~ 1,

then

Z/12 EB 7L./6 H2i(ro(P) , 'I.) ~ { '1./4 E9 '1./2 '1./3 E9 7L./6 '1./2

=1 p =5 p=7 p = 11 p

mod 12 mod 12 mod 12 mod 12,

and H 2i + 1 (r o(p), 'I.) ~ (Z/2)N(p).

This is proved by first passing to the projective group pro(p) =

r o(p)/{±I}

and computing the cohomology of pro(p) via its action on the tree defined above. One then uses the Hochschild-Serre spectral sequence associated to the extension 1 ~ {±1} - + ro(p) - + pro(p)-+1.

4. Rank One Groups

94

To complete the calculation, one uses the Mayer-Vietoris sequence along with some facts about the co homological dimension of the group SL2(Z[I/p]). There is an extension problem to be solved, but it presents no real difficulty. THEOREM

4.1.4. Let p ~ 5. Then Hl(SL2(Z[I/pJ), Z) =

a and

p= 1

mod 12 p= 5 mod 12 p= 7 mod 12 p = 11 mod 12.

For i

~

2, we have

(Z/2)(p-7)/6 EB Z/12 ' (Z/2)(P+l)/6 E9 Z/12 E9 Z/3 H 2~ SL Z 1 Z ~ { ( 2( [/p]), ) - (Z/2)(p-7)/6 E9 Z/12 E9 Z/4

=1

mod 12 p= 5 mod 12 p= 7 mod 12 (Z/2)(P+l)/6 E9 Z/12 EB Z/12 p = 11 mod 12, p

and

=1 p=5 p =7

Z/6 p

H2i-l(SL2(Z[I/p]),Z) ~ { Z/2 Z/3

a

mod 12

mod 12 mod 12 p = 11 mod 12.

The case p = 3 is completely analogous and yields i odd

i=2 i = 2j,j > 1.

The case p = 2 is complicated by the fact that the principal congruence Hubgroup r(2) C ro(2) is not torsion-free. Still, it is possible to work around this and obtain i odd

i=2 i = 2j,j > 1.

This last computation is interesting since SL2(Z[I/2]) has no subgroups of order eight. The question of computing He (SL2(Z[I/n]), Z) for an arbitrary integer n remains open. The rational cohomology was computed by K. Moss [86J, and Naffah [87J has computed the 3-primary part.

4.2. The Bruhat-Tits Tree

95

4.2. The Bruhat-Tits Tree The amalgamated free product decomposition of SL 2 (Z[1/pJ) is a consequence of a general construction due to F. Bruhat and J. Tits [23]. In this section we describe the tree associated to a rank one group over a field with discrete valuation. Let K be a field with valuation v : KX --+ Z. Denote by 0 the valuation ring, by m the maximal ideal of 0, and by k the residue field O/m. By a lattice we mean a free, rank two O-submodule of K2. Two lattices L, L' are equivalent if there is a nonzero x in K with L' = xL. We denote the equivalence class of a lattice L by [L]. Let 7r be an element with v(7r) = 1. Define a simplicial complex X as follows. The vertices of X are classes of lattices in K2. Two vertices are adjacent if there exist representatives L, L' with 7r L c L' c L. This defines a graph and one has the following result (see Serre [no], p. 70). THEOREM 4.2.1. The graph X is a tree.

o

There is an obvious action of GL2(K) on X: if g E GL 2(K) and if [L] is a vertex of X, then g. [L] = [gL]. This action is obviously transitive and the center of GL2(K) acts trivially. If one considers the induced action of SL 2(K), then one checks easily that there are two orbits of vertices with representatives OEBO and OEB7I"0, and one edge orbit. The stabilizers of the vertices in SL 2 (K) are SL2(0) and diag(l, 7I")SL2(0)diag(1, 71"-1). The edge stabilizer r consists of those elements of SL 2 (0) which are upper triangular modulo 71". COROLLARY 4.2.2. There is an amalgamated free product decomposition

SL 2(K)

~

SL 2(0) *r SL2(0).

PROOF. This type of decomposition exists for any group G which acts on a tree with an edge as fundamental domain (see [nO], p. 32). 0 One also has the following result. PROPOSITION 4.2.3. Suppose that R is a dense subring of K (in the topology defined by v). Denote by OR the intersection 0 n R and by r(R) the subgroup r n SL 2 (R). Then there is a decomposition

SL 2(R) PROOF. See [110], p. 78.

~

SL 2(OR) *r(R) SL 2(OR).

o

EXAMPLE 4.2.4. Consider the field Q of rational numbers with the p-adic valuation. Since the ring Z[l/p] is dense in this topology we obtain the decomposition SL 2(Z[1/pJ) ~ SL 2(Z) *r(Z(l/pj) SL 2('1.) mentioned in the previous section.

96

4. Rank One Groups

EXAMPLE 4.2.5. Denote by £, the field of formal Laurent series over a field k with valuation

V( L

antn) = no,

ano

=1=

O.

n~no

The ring k[t, elJ is dense in £, and k[t, elJ nO = k[tJ. Thus we have SL 2(k[t, t- l ]) ~ SL 2(k[tJ) *r SL2(k[t]).

4.3. SL2(k[tJ) In 1959, H. Nagao [88] published an elementary proof of the following fact. If k is any field, then there is an amalgamated free product decomposition

SL 2 (k[t])

~

SL 2(k)

*B(k)

B(klt]),

where B(R) denotes the subgroup of upper triangular matrices over R. Using the tree X associated to K = k(t), v(f Ig) = deg(g) - deg(f), Serre provided a new proof of this fact by finding a fundamental domain for the action of SL 2 (kl:tD on X. It turns out that such a subtree is an infinite path vo, VI, V2, •.. , where Vi = [tiO EB OJ [110], p. 87. This decomposition allows one to compute the homology of SL2(k[t]). THEOREM 4.3.1. Suppose the field k is infinite. Then the natural inclusion

SL 2(k)

-+

SL2(k[t]) induces an isomorphism H.(SL 2(k),Z)

---+

H.(SL 2(k[tJ),Z).

REMARK 4.3.2. This result is clearly false for finite fields. For example, one NeeN eMily that HI (S L2 (IF2 [t]), Z) contains a copy of an infinite-dimensional

F2 -vector space while HI (SL2(IF2), Z) is finite. However, if one uses ZIp-coefficients, where p ~

an blomorphism H.(SL 2(k[t]),Zjp)

=1=

char k, then we do have

H.(SL 2(k),Zlp)·

PROOF OF THEOREM 4.3.1. The decomposition yields a Mayer-Vietoris Moquonce: ... -+

Hi(B(k))

-+

Hi(B(k[t])) EB Hi(SL 2(k))

-+

Hi (SL 2 (k[tJ))

-+ . . . .

Since the map B(k) -+ B(k[t]) is split by evaluation at t = 0, the long exact sequence breaks into short exact sequences

0-+ Hi(B(k))

-+

Hi (B(k[tJ)) EB Hi(SL 2(k))

-+

Hi (SL 2(k[tJ))

-+

O.

We now need the following result. If the field k is infinite, then

H.(B(k),Z)

~

H.(B(kl"tJ),Z).

This is trivial if char k = 0 since one can use the Hochschild-Serre spectral Hcquence associated to the extension () -+

R

-+

B(R)

-+

P

-+

1

97

for R = k,k[tJ. A center kills argument shows that Hi(kX,Hj(R)) = 0 for j ~ 1 so that H.(B(R), Z) ~ H.(P, Z). The positive characteristic case is more delicate, but it follows from a simple modification of the argument used to prove Theorem 2.2.2 (note also that this isomorphism was observed modulo p-torsion (p = char k) by Alperin [2]). Given this fact, the theorem now follows easily by considering the above exact sequence. 0

4.4. SL2(k[t, rID The isomorphism H.(SL 2(k[tJ),Z) ~ H.(SL 2(k),Z) reduces the computation of H.(SL 2(k[t, rID, Z) to that of H.(f, Z), where r is the subgroup of matrices in SL 2(k[t]) which are upper triangular modulo t. Indeed the decomposition of SL 2 (k[t, t-IJ) given in Example 4.2.5 yields a Mayer-Vietoris sequence -4

Hi(f)

-+

H i (SL 2(k[tj)) EEl Hi (SL2(k[tJ)) ~ Hi(SL 2(k[t, rl]))

-+ .

Consider the split exact sequence 1 --t K

--t

f

t-Q --=-+ B(k)

~

1.

Here K consists of those matrices which are congruent to the identity modulo

t. THEOREM 4.4.1. If the field k is infinite, then the map r -4 B(k) induces an isomorphism in integral homology. Consequently, there is a natural isomorphism H.(f,Z) ~ H.(P,Z). REMARK 4.4.2. As in the previous section, one can show that if k is finite, then the mapT -4 B(k) induces a mod p homology isomorphism (p =F char k). PROOF OF THEOREM 4.4.1. An elementary proof (which works only in characteristic zero) was given in [66J. The idea is as follows. IT T is the fundamental domain for the action of SL 2(k[tJ) on the tree X associated to k(t), then the subtree DC X defined as

D=

u

sT

8ESL2(k)/ B(k)

is a fundamental domain for the action of K on X. This is proved by first showing that D is a tree. To do this it suffices to show that D is connected (since D is a subgraph of a tree). A set of coset representatives of SL 2(k)/ B(k) is

Each of these fixes the initial vertex [() EEl OJ of T; the connectivity of D follows immediately. One then showH that the vcrtkoH of D are inequivalent modulo

4. Rank One Groups

98

the action of K and that every edge of X is equivalent to an edge of D modulo K. This is left to the reader. If G denotes the subgroup of matrices of the form

{

(~ p~t)): pet) E tk[t j },

then we have a free product decomposition

K ~ liSESL2(k)/B(k)sGs-1. This implies that for i ;?: 1, Hi(K) = ffis ES L2(k)/B(k) Hi(sGs-I). Now, if char k = 0, C is a torsion-free abelian group and Hi(sGs-l) = 1\~(sGs-l). A straightforward (if tedious) calculation then shows that Hp(B(k), Hq(K)) vanishes for q ;?: 1. Consequently, Hp(r) = Hp(B(k)) (via the Hochschild-Serre spectral sequence ). A more streamlined proof (which works in any characteristic) is as follows (see [67]). One uses the action of B(k) on D to show that a fundamental domain for the action of r on X is the path D' consisting of the vertices ... ,U2,UI,VO,VI,V2, ... , where Vi is as in Section 4.3 and Ui = [0 EEl tiOj. The stabilizer in r of each vertex and edge is homologically equivalent to k x . Since X is contractible, we have a spectral sequence converging to H. (r) with El-term

whertl (7(p) is a p-simplex in D' and r 0" is the stabilizer of (7 in r. The qth row of this spectral sequence is the chain complex G.(D', Hq), where Hq is the (loofficient Hystem (7 I-t Hq(r 0")' In this case, the system Hq is constant: H,,(I',,)::::::: Hq(kX) for all (T. Since D' is contractible, we have E2

p,q

= H p(D' , H q) = {Hq(P) 0

p p

= > 00.

o

It followH that Hq(r) = Hq(P) for all q ;?: O.

Even though the pieces are now in place, the final computation of the homology of SL2(k[t,e l ]) remains elusive. Clearly, H.(SL 2(k),7I.,) lies in H.(SL 2(k[t, t- I l), 71.,) as a direct summand, but other than this, we only have the long exact sequence

...

~

Hi(k X )

~

H i (SL 2(k)) EEl Hi (SL 2(k))

~

Hi(SL 2(k[t, ell))

~

....

At present, little is known about the map Hi (k X) ~ Hi (S L2 (k)). A description of this homomorphism would simplify the calculation of the homology of SL2(k[t, ell) drastieally.

4.5. Curves of Higher Genus

99

Still, some things can be said. Suppose k is a number field. Denote by r the number of real embeddings and by s the number of conjugate pairs of complex embeddings. THEOREM

4.4.3. The map

is injective for i

~

2r + 3s + 1 and bijective for i

PROOF. By Theorem 1.5.1, H i (SL 2 (k), Q) = REMARK

~

2r + 3s + 2.

a for i

~

2r + 3s + 1.

0

4.4.4. With a bit more work (see [66]), one can show that

4.5. Curves of Higher Genus The rin~ k[tJ and k[t, t-IJ are the coordinate rings of Al and Al- {a}, respectively. Suppose S is a smooth projective curve over k and let q E S be a closed point. Denote by C the affine curve S - {q} and by A the coordinate algebra of C. The ring A is a Dedekind domain and its field of fractions, F, is the function field of S. We have AX = P. For example, if S = Pl, we have C

= Ai:,

A

= k[tJ,

F

= k(t).

The point q gives rise to a discrete valuation, v q , on F. Let X be the Bruhat-Tits tree associated to (F,vq ). The group GL 2 (A) acts on X as it is a subgroup of GL 2 (F). Serre [UO], p. 106, provided a partial description of the quotient GL 2 (A)\Xj it consists of a graph Y with cusps attached. (We will define a cusp below.) He also showed that the cusps are in one-to-one correspondence with the elements of PicO(S). In the case S = Pl, we saw above that the quotient graph GL 2 (kl:t])\X consists of an edge Y with vertices [0 EB OJ and [to EB OJ and a single cusp VI, V2, ••. , where Vi = [tiO EB OJ. Note that Pic°(JP>l) = a. Recently, Suslin [129J gave a more complete description of the quotient graph along with the stabilizers of the vertices and edges. We discuss this below. 4.5.1. Elliptic Curves. In [130J, S. Takahashi provided a complete description of GL 2 (A)\X when S is an elliptic curve. In fact, he exhibited a subtree D of X such that D ~ GL 2 (A)\X. Since the center of GL 2 (A) acts trivially on X, we have D ~ PGL 2 (A)\X. This allows us to compute the integral homology of PGL 2 (A) provided the field k is infinite.

100

4. Rank One Groups

We shall describe D only informally. Assume the affine elliptic curve S {oo} is defined via a Weierstrass equation F(x, y) = 0, where

If 1 E k and if F(l, y) = 0 has no k-rational solution, we denote by k(w) the quadratic extension of k in which F(l,w) = O. The tree D consists of the following. There is a distinguished vertex o. For each 1 E k U {oo}, there is a vertex vel) adjacent to o. Denote by D(l) the connected component of D - {a} which contains vel). The D(l) fall into three types. (1) Suppose F(x, y) = 0 has no rational solution with x = l. Then D(l) consists only of v(l) (sec Figure 4.1). 0 · - - - - -.... •

v(l)

FIGURE 4.1. F(l, y)

= 0 has no rational solutions

(2) Suppose 1 = 00 or F(x, y) = 0 has a unique rational solution with x = 1. Let p be the point at infinity of E or the rational point corresponding to the solution. Note that p is a point of order 2. Then D(l) consists of an infinite path c(p, 1), c(p, 2), ... and an extra vertex e(p) (see Figure 4.2).

O~.-------V~~)~----CW~,,\\\~I)----e-w-)~C~_'2)~-----

FIGURE 4.2. F(l, y) = 0 has a unique rational solution (3) Suppose F(x, y) = 0 has two different solutions such that x = 1. Let p, q be the corresponding points on E. Then D(l) consists of two infinite paths

c(p, 1), c(p, 2), ... and c(q, 1), c(q, 2), ... (see Figure 4.3). The infinite path c(p, 1), c(p, 2), ... is called a cusp. Note that there is a one-to-one correspondence between cusps and the rational points of S. The stabilizers of the various vertices are given by the next result (see [130], Theorem 5).

4.5. Curves of Higher Genus

101

c(p.2)

c(p.l)

o

v(1

c(q,1)

c(q,2)

FIGURE

4.3. F(l, y) = 0 has two distinct solutions

Up to isomorphism, the stabilizers in PGL 2 (A) of the vertices of D are as follows: PROPOSITION 4.5.1.

ro

rv(l)

= {I} ~

fW)X/k' P

rc(p,n)

~

{( ~

in case (1) in case (2) in case (3)

; ) :p,qE

e,v E kn }/kX

PGL 2 (k). The stabilizer of an edge is the intersection of its vertex stabilizers (one of which is contained in the other). D re(p)

~

The action of PGL 2 (A) on X yields a spectral sequence converging to H.(PGL 2 (A),Z) with El-term

E~,q

=

EB

Hq(rl7'Z),

l1(p)CD

As pointed out in Section 4.3, we have H.(rc(p,n),Z) ~ H.(P,Z), provided k is infinite. This leads to the following computation.

4. Rank One Groups

102

THEOREM 4.5.2. If the field k is infinite, then for all i

~

1,

lEkU{oo} F(l,y)=Ohas unique sol.

lEk

F(l,y)=Ohas two

801.

lEk

F(l,y)=Ohas no

801.

PROOF. If 1tq denotes the coefficient system U f---+ Hq(ru), then E!,q = C.(D,1t q ). Since the stabilizers of 0 and the edges adjacent to it are trivial, we see that

C.(D,1t q ) =

EB

C.(D(l),1t q ).

lEkU{oo}

= 0 has no rational solution then C.(D(l), 1tq ) consists of the group Hq(k(w)X IP) sitting in degree zero, whence the factor of Hq(k(w) X IP) in Hq(PGL 2(A)) If F(l, y) = 0 has two solutions, then the stabilizer of each edge and vertex in D(l) is homologically equivalent to P. Thus, since D(l) is contractible (Figure 4.3), Hi (D(l), 1tq) vanishes for i > 0 and equals Hq(P) for i = O. If F(l, y) = 0 has a unique solution, then the complex D(l) is as in Figure 4.2. The vertex v(l) and the edge e joining it to c(p, 1) have stabilizers iso-

If F(l, y)

morphic to k. The map p;roup r r; has the form

re

-+

{

r v(l)

is an isomorphism (this is clear). Since the

(~ ~): v E k},

til(! map r" -+ r c(p,l) factors through the inclusion k -+ r c(p,l) and hence InduceH the ~cro map on homology Thus, if D(l)' denotes the complex obtained by deleting v(l) and e, we have Hi (D(l), 1tq) = Hi (D(l)' , 1tq) for all i. A Htraightforward computation shows that the latter group vanishes for i > 0 0 and equals Hq(PGL2(k)) for i = O. REMARK 4.5.3. Theorem 4.5.2 is valid for finite fields for i ~ 2 with integral coefficients, and in all degrees with ZIp-coefficients (p i- char k). 4.5.2. Application: The K-theory of Elliptic Curves. Theorem 4.5.2 can be used to derive information about the K-groups of elliptic curves. There are many deep conjectures about the structure of these groups, especially if the ground field k is a number field. Perhaps the most famous, due to A. BeHinson [7] (and later modified by Bloch and Grayson [12]), asserts that if E is an elliptic curve over It number field k, then K 2 (E) ® Q has dimension equal to

1

4.5. Curves of Higher Genus

103

the number of infinite places of k plus the number of primes p C Ok where E has split multiplicative reduction modulo p. At this time, it is not known if dim(Kz(E) ® Q) is finite (even for a single curve). Let E be an elliptic curve, q E E a closed point, and denote by A the affine coordinate algebra of E-{q}, as in the previous section. Denote by F the function field of E and consider the obvious embedding i : GLz(A) --+ GLz(F). We have the following result. THEOREM 4.5.4. The image of the map

coincides with the image of

PROOF. This is the main theorem of [71]. It suffices to show that j PGLz(A) --+ PGL 2 (F) satisfies

im(j*) = im((j/ PGL 2(k»)*)' One then uses the Hochschild-Serre spectral sequences associated to the extensions and 1 ---t F X

---t

GL 2 (F)

----'>

PGLz(F)

---t

1

along with the Universal Coefficient Theorem to finish the proof. The isomorphism of Theorem 4.5.2 is induced by the various inclusions r vel) --+ PGL 2 (A) and r e(p) --+ PGLz(A). The subgroup r eeoc) is the usual copy of PGLz(k). All the other r's are not subgroups of PGLz(k). However, one can show (via a lengthy calculation) that each r vel) and r e(p) is conjugate by an element of PGL 2 (F) to a subgroup of PGL 2 (k). Since conjugation induces the identity map in homology (Appendix A), it follows that im(j*) = im((j/PGL2(k»)*)' 0 Recall the rank filtration of the rational K-groups of a ring R: Ki(R)Q := Ki(R) ® Q. This is an increasing filtration defined by

Theorem 4.5.4 has the following consequence. COROLLARY 4.5.5. The image of the map TZKn(A)Q - TZKn(F)Q coin0 cides with the image of T2Kn(k)Q -+ T2Kn(F)Q.

4. Rank One Groups

104

Now suppose k is a number field. The localization sequences for A and E give rise to a commutative diagram

where all three maps are injections. COROLLARY

4.5.6. If k is a number field, then r2K2(A)Q = O.

PROOF. The image of r2K2(A)Q in r2K2(F)Q coincides with the image of r2K2(k)Q = O. But the map K 2(A)Q --+ K 2(F)Q is injective. 0

Theorem 2.2.18 implies that r3K2(A)Q = K 2(A)Q. Thus, to show that K 2(A)Q (and hence K 2(E)Q) is finite dimensional, it would suffice to show that the image of H 2(GL 3(A),Q) in H2(GL 3(F),Q) is finite dimensional. 4.5.3. Genus Greater than One. A complete description of the structure of PGL2(k[C])\X for curves C = S - {q} of genus 9 ~ 2 remains elusive. However, A. Suslin computed H.(PGL 2(k[C]), Z/p) , where pi-char k and k is algebraically closed. As mentioned above, Serre showed that PGL 2(k[C])\X consists of a graph Y with cusps attached (one for each element of the Jacobian of S) by interpreting the vertices of X as rank two vector bundles on S. Suslin showed (using .ome results of Seshadri [9:1.]) that the vertices and edges of PGL2(k[C])\X {eullp,,} are "trivial" in the following sense. If v E Y - {Y n {cusps} }, then the .tablHzer of v in PGL 2(k[C]) is a k-vector space. Thus, with Zip-coefficients, Hi(rv,Z/p) = 0 for i ~ 1. It follows that for i ~ 2, the groups Hi(PGL 2(k[C]),Z/p) depend only on the cusps. Note that Hl(Y,Z/P) will contribute to the group Hl(PGL2(k[CJ),Z/p) (this is not an ilnme for elliptic curves since in this case Y is contractible). The "nontrivial" part of PGL 2(k[C])\X is therefore a disjoint union of trees indexed by the Jacobian, PicO(S). If L E Pic°(S) satisfies 2L = 0, the tree associated with L is a path VO, Vb ... where r Vo ~ PG L2 (k) and r Vi is the semidirect product of P with k i (this is the same as the group r c(p,i) of Proposition 4.5.1). By the same arguments as above, this branch contributes a copy of Hi(PGL 2(k), Zip) to Hi (PGL 2(k[C]) , Zip) for i ~ 2. If L E PicO(S) is not of order 2, then the branch associated to L consists of a path Vb V2, .... Each stabilizer is the semidirect product of k X with a k-vector space. Homologically, this is equivalent to kX. Moreover, the branch is adjacent to a vertex VL and one has VL = V-L' Thus, we have a tree Himilar to that in Figure 4.3.

4.6. Groups of Higher Rank

105

This contributes a copy of Hi(P, Zip) to H i (PGL 2(k[C]), Zip) for i > 2. Thus, we have the following result. THEOREM

i

~

4.5.7. Suppose the field k is algebraically closed. Then for all

2,

EB

H i (PGL 2(k),Zlp) EB

LEPic°(S) 2L=0

EB

Hi(k X , ZIp)·

LEPico(S) 2L-I-0,L=-L

Note that this agrees with Theorem 4.5.2. Indeed, an elliptic curve E and its Jacobian coincide. The points on E corresponding to the I E k U {oo} where F(l, y) = 0 has a unique solution are precisely those of order 2, and the identification of a point with its negative is built into the indexing of the Hi(P,Zlp) factors in Theorem 4.5.2. These homology decompositions have important corollaries; we discuss this in Chapter 5.

4.6. Groups of Higher Rank The impressive array of results for rank one groups is something of an anomaly; once one passes to 8Ln , PGLn , n ~ 3, things become much more complicated. In this section we discuss what is known for groups of higher rank. 4.6.1. SLn(Z). It is possible, in principle, to compute the homology of SLn(Z) in a manner similar to that of 8L2(Z), There is an obvious contractible space X upon which 8L n (Z) acts, namely the space of positive definite quadratic forms on ]Rn (up to positive scalars). When n = 2, this space can be identified with the hyperbolic plane 1i c C with the usual action of 8L 2 (Z) by linear fractional transformations. Unfortunately, the quotient SLn(Z)\X is not compact. Compactifications exist, but it is still difficult (if not virtually impossible) to compute the cohomology in this way. C. Soule [113] discovered a way to get around this in the case n = 3. The basic idea is to replace X by a contractible subspace X, such that SL 3 (Z)\X ' is compact (it is also contractible). The space X, is defined as follows. The space X consists of symmetric positive definite 3 x 3 matrices (modulo scalars). Write [h] for the class of a matrix h = (h ij ). The group 8L3 (Z) acts on X via [h].g = [gThg]. A fundamental domain for this action is the set D of "reduced" points. These are the matrices which are minimal in their own orbit with respect to the ordering [h] < [h'] if and only if the sequence of diagonal coefficients of II. iH less than

4. Rank One Groups

106

that of h' in the lexicographic order on R3. Let D' be the set of reduced points [hJ such that hll = h22 = h 33 . Then X' = D'.SL 3 (Z). SouIe described the cells of SL3(Z)\X' and completed the (formidable!) spectral sequence calculation to arrive at the following result. THEOREM 4.6.1. The integral cohomology of SL 3 (Z) is given by the following table (where (a)G means a direct sum of a copies of G).

n 12m+1 12m+2 12m+3 12m+4 12m+5 12m+6 12m+7 12m+8 12m+9 12m+ 10 12m + 11 12m+ 12

(6m)Zj2 (6m)Zj2 (6m+ 2)Zj2 (2)Zj3 EB (2)Zj4 EEl (6m)Zj2 (6m+ 1)Zj2 (6m+ 4)Zj2 (6m+ 3)Zj2 (2)Zj2 EB (2)Zj4 E9 (6m + 1)Zj2 (6m+ 5)Zj2 (6m+ 5)Zj2 (6m+ 4)Zj2 (2)Zj3 EB (2)Zj4 @ (6m + 5)Zj2

No such computation exists for n ~ 4; this technique is just too complicated. Armed with Theorem 4.6.1, it should be possible to compute the cohomology of SL3(Z[1jpJ) using a higher dimensional analogue of the technique of SectiOIl 4.1. One now needs to compute the cohomology of subgroups r c SL 3 (Z) " which aro pttI'ubolic modulo p. This has been carried out by Henn [56J in the' CMe p = 2.

4.8.2. SLn(kl:t]) and SLn(k[t, t- 1 ]). There is a generalization to higher rank of the Bruhat-Tits tree. If K is a field with valuation v, local ring 0, and residue field k, one defines lattices in Kn as before. Define a simplicial complex X as follows. The vertices are equivalence classes of lattices. A collection Ao, Al , ... ,Ai forms an i-simplex if there exist representatives L o, L 1 , •.. , Li such that 7r Li

C

Lo

c ...

C

Li.

One checks that the maximal dimension of a cell is n - 1. The complex X is contractible (see Brown's book [221 for a nice discussion of this). In the case K = £, = field of Laurent series over k, a fundamental domuin for the action of 8Ln(£) on X is the (11. - I)-simplex D with vertices

107

4.6. Groups of Higher Rank

Vi

= [0 E9 ... E9 0 E9 !O E9 . ~. E9 tOJ. i

The stabilizers of the Vi are conjugates of SLn{O) by diagonal matrices with n - iI's and it's. Since k[t, t- I ] is dense in £, the simplex D is also a fundamental domain for the action of SLn{k[t, rl]) on X. The vertex stabilizers are conjugates of SLn(k[t]). The stabilizer of a simplex is the intersection of its vertex stabilizers. Thus, the computation of H.(SLn{k[t, rl]), Z) reduces to the computation of the homology of groups rp ~ SLn{k[t]) of the form 1 ---+ K ---+ f

P

·t-O

--=-+ P

---+

1

where P ~ SLn(k) is a standard parabolic subgroup containing the upper triangular matrices and K is the subgroup of SLn(k[t]) consisting of matrices congruent to the identity modulo t (compare with Section 4.4). All simplex stabilizers are conjugates of the various f p . In Section 4.4, we showed that the map f ---+ B{k) induces an isomorphism in integral homology. It should come as no surprise that this generalizes to higher dimensions. THEOREM 4.6.2. If the field k is infinite, then the split surjection f p induces an isomorphism H.{fp, Z) ---+ H.(P, Z). COROLLARY 4.6.3. The inclusion SLn(k) morphism H.(SLn(k),Z) ---+ H.(SLn{k[t]),Z). PROOF. This is the special case P

---+

P

---+

SLn(k[t]) induces an iso-

= SLn(k).

o

PROOF OF THEOREM 4.6.2. The first step is to compute a fundamental domain for the action of SLn(k[t]) on the Bruhat-Tits building Y associated to k(t); this was done by Soule [1141. It is a subcomplex T C Y which looks like an infinite ''wedge''; see Figure 4.4 for the case n = 3. The vertices of T are the classes [trlel,tr2e2, ... ,trn-len_I,en], where el,e2, ... ,en is the standard basis of k(t)n and TI ~ T2 ~ ... ~ Tn-l ~ O. " The stabilizers have the form LI

V12

a

L2

0 0

0 0

VIm

Vm- 1 ,1TI 0

Lm

4. Rank One Groups

108

(1) E {I}

FIGURE 4.4. The fundamental domain T for n

Li ~ GLn; (k) and Vij lunt, t.o t.he reductive group

whor~

OIl(1

C

=3

Mni,nj (k[tJ). Homologically, these are equiva-

then uses the spectral sequence

E~,q

=

EB

Hq(f a)

====?

Hp+q(SLn(k[t]))

a(p)CT

to prove Corollary 4.6.3 as follows. Denote by Va the vertex [el,"" en] and by Vi the vertex [tel, ... , tei, ei+l, ... ,en] for i = 1, ... ,n - 1. For a k element subset I = {i l , ... , id of {I, 2, ... , n - I}, define E}k) to be the sub complex of T which is the union of all rays with origin Va passing through the (k - 1)simplex (ViI' ... ,Vik)' If I = {I, ... ,n - I}, then Ejn-l) = T. When we write E}l), the superscript denotes the cardinality of the set J.

4.6. Groups of Higher Rank

Define a filtration V· of T by setting

V(k) =

UE}k) ,

V(O) = Vo

109

and

1 ~ k ~ n - 1,

I

where I ranges over the k element subsets of {1, 2, ... ,n - 1}. One checks easily that if 1 ~ k - 1 and if (J' is an l-simplex in E}k), where I = {i l , ... , ik}, not lying entirely in any E}k-l) , where J c I, then r a has the block form of the intersection r Vil n··· n r Vik' Thus, the coefficient system (J' r-t Hq(r a) is "locally constant" in the sense of Proposition A.2.7 and hence we see that the inclusion Vo ---+ T induces an isomorphism

H.(vo, Hq)

---->

H.(T, Hq)

for all q ~ O. Thus, the E2-term of the spectral sequence is E2 = {Hq(SLn(k)) pq

0

P= 0 p> O.

The general case is proved by showing that a fundamental domain for the action of r p on Y is the sub complex

U

Dp =

sT,

sE'En/H

where En C SLn(k) is the symmetric group on n letters, and H = EnnP. This is done in stages as follows. Let K C SLn(k[tJ) be the subgroup of matrices congruent to the identity modulo t. Then as in the case n = 2, the sUbcomplex

U

D'=

sT

sESLn (k)/ B(k)

is a fundamental domain for the action of K on Y. Then using the action of B(k) on D', one shows that

DB =

U sT sEEn

is a fundamental domain for the action of r B on Y. Finally, since r p contains a subgroup of permutation matrices, the action of P on DB shows that Dp is the required fundamental domain for r p. Thanslating the filtration V· of T gives a filtration W;' of Dp. Another application of Proposition A.2.7 shows that the map

H.(vo, Hq)

---->

H.(Dp, Hq)

is an isomorphism. Here, the coefficient system iH (J' r-t Hq(r a) where r a is the stabilizer of (J' in r p. The proof is finished by noting that the Htabilizer r Vo is P. D

4. Rank One Groups

110

Theorem 4.6.2 provides a complete computation of the El-term of the spectral sequence for computing H.(SLn(k[t, rl]), Z):

E;,q

=

E9 Hq(rp). qeD

The differential, d l , is difficult to compute. However, one has the following results. THEOREM

4.6.4. If n ~ 3, then the E2-term of the spectral sequence is

°

Z

a E~,q= a kX H 2 (SL n (k), Z)

P = O,q = p> O,q = a p # l,q = 1 p = l,q = 1 p = O,q = 2.

PROOF. See [67]. The row q = a of El is the simplicial chain complex of the (n-l)-simplex D. Since D is contractible, the computation of E;,Q follows easily. The row q = 1 of El is similar to the bar construction; this makes it possible to write down a contracting homotopy. D COROLLARY

4.6.5. If n

~

3, then

H2(SL n (k[t, rl]), Z) = =

H 2 (SL n (k), Z) EB k X K2(k) EB Kl(k) K 2 (k[t, rl]).

PRom'. This follows directly from the spectral sequence calculation. The ;' flld tlutt H2 (SL n (k) , Z) = K 2 (k) for n ~ 3 is standard (see, e.g. [80]). The hUlJt oqultlity is the Fundamental Theorem of Algebraic K-theory. D Tho other differentials, d l , remain a mystery. One needs a description of t.ho lIlaps H.(GLn(k)) ----+ H.(GLm(k)) for n < m in order to carry out the (~alculation.

4.6.3. Unstable Homotopy Invariance. Corollary 4.6.3 may be viewed as an unstable version of homotopy invariance in algebraic K-theory:

K.(R[t])

~

K.(R)

for R regular. A natural question to ask is if the hypotheses in Corollary 4.6.3 can be weakened; that is, do we have H.(SLn(R[tj), Z) ~ H.(SLn(R), Z) for all regular R? The answer is certainly no: consider R = IFp • Still, one might hopo for a result in this direction provided the ring R has enough units. We discllss such a result now.

4.6. Groups of Higher Rank

111

Suppose R is an integral domain with field of fractions Q. Denote by E2(R[tJ) the subgroup of SL 2(R[t]) generated by elementary matrices. In general this is a proper subgroup. Indeed, Krstic and McCool [74] have shown that if R is an integral domain which is not a field, then there is a surjective homomorphism

SL 2(R[t])!U2(R[t])

----->

F,

where U2 (R[t]) is the normal subgroup generated by all unipotent matrices (we have E2 ~ U2) and F is a free group of countable rank. Nagao's theorem asserts that there is a decomposition

SL 2(Q[t])

~

SL 2(Q)

*B(Q)

B(Q[t]).

By applying a result of Serre [110], p. 6, we obtain the following. THEOREM 4.6.6. If R is an integral domain, then

E2(R[t])

~

E 2(R)

*B(R)

B(R[t]).

This theorem has the following application. By a modification of Theorem 2.2.2, if A has many units (see Section 2.2), then H.(B(A)) ~ H.(B(A[t])). Thus, using the same argument as in Theorem 4.3.1 we obtain the following result. THEOREM 4.6.7. If R is an integral domain with many units, then the

inclusion E 2(R)

---->

E2(R[tJ) induces an isomorphism H.(E2(R), Z) -----> H.(E2(R[t]), Z).

Theorem 4.6.7 cannot hold in general for all n, however. Consider the ring - x 2 ))(x,y)' If H.(En(R), Z) ~ H.(En(R[t]), Z) for all n, then we would have

R = (k[x, y]!(y2 - x 3

H2(E(R[t]), Z)

~

H2(E(R), Z);

i.e., K2(R[t]) ~ K2(R). But Weibel [132] has shown that K2(R[t]) i- K2(R). Perhaps the added assumption that R be regular would suffice to prove the result for all n 2: 3. Theorem 4.6.7 does hold for n = 3, however. THEOREM 4.6.8. If R is an integral domain with many units, then the

inclusion E3(R)

---->

E3(R[t]) induces an isomorphism H.(E3 (R), Z)

~

H.(E3 (R[tj), Z).

PROOF. Let Q be the field of fractions of R and denote by T the fundamental domain for the action of SL3 (Q[tj) on the Bruhat-Tits building Y associated to Q(t). Define a sub complex U of Y by

U=

U ,r.E R:I(R.[I.])

xT.

4. Rank One Groups

112

Clearly, T is a fundamental domain for the action of E3{R[tJ) on U. Using the argument in the proof of Theorem 4.6.2, we are finished provided the complex U is acyclic. To prove this we need the following result of SouIe [112]. Suppose G is a group which acts on a simplicial complex Z with fundamental domain Z', Denote by G the amalgam of the stabilizers of the simplices of Z (Le" view the complex Z' as a partially ordered set, and take the colimit of the stabilizers Uldng this poset as the indexing category). Then there is an exact sequence

7r1 (Z)

---+

G ----+ G ----+ 7ro(Z)

---+

O.

Since E3(R[t]) is generated by the vertex stabilizers in T, the map E3{R[t]) ---+ E3{R[t]) is surjective; that is, U is connected. We claim that U is also simply connected. This finishes the proof in light of the long exact sequence o ----+ H2{U) ----+ H 2(Y) ------t H2{Y' U) ----+ • , • and the fact that H 2 {Y) = O. The simple connectivity of U is proved as follows. The amalgam E3(R[t]) is the colimit of the stabilizers of the simplices in T. Since Y is contractible, the group SL3{Q[t]) is the amalgam of its vertex stabilizers. Thus we have an injective map of systems

{E3(R[t])u}uET and consequently an inclusion

<-t

{SL 3(Q[t])U}UET,

E3{R[t])

<-t

SL3(Q[tJ),

Moreover, the image of E3(R[t]) in SL3{QI:tD is clearly the subgroup generated by the Ea(RI:t])u; that is, E3{R[tJ) ~ E3{R[tJ). Thus, the map 7rl{U) ---+ t3(R[t]) is the zero map. Sou16's theorem also asserts that the map 7rl{Z) ---+ G is injective if (1) Z'is ..imply connected, and (2) for each 9 E G, the complex gZ'nZ' is connected (or empty). Since this is clearly the case for E3{R[t]), the map 7rl{U) ---+ E3{R[t]) Iii Injectivo. Thus, 7r1 (U) = O. 0 REMARK 4.6.9. The above argument works for any n to show that the complex U is connected and simply connected. Moreover, we always have H 1I - 1 (U) = O. Thus, for larger n and the ring R mentioned above, there must be some nontrivial 2-cycle in U.

Unstable homotopy invariance is an important step in attacking the Friedlander--Milnor Conjecture. We discuss this in Chapter 5 In [731, we used Theorem 4.6.6 to study the homology of SL2 {Z[t]). Note that Theorem 4.6.7 does not apply to E2{Z[t]). However, since B{Z) = Z/2 x Z and B{Z[t]) = Z/2 x Z[tl, the Kiinneth formula implies that Hi{B{Z[t])) contains a. copy of Hi(tZ[t]) = I\~ tZ[t] as fl. SUlllUHUld. Since H.i(Z) = 0 for

4.6. Groups of Higher Rank

113

i> 1 and since the map H 1 ('Z) --+ H 1 (Z[t]) is the usual inclusion, we see that Hi (E2(Z[t]), Z) also contains !\~ tZ[t] as a direct summand. Now consider the group SL2(lFp[t]) for p = 2,3. Nagao's Theorem applies here and we have B(lFp[tJ) = IF; x lFp[tJ. Again using the Kiinneth formula and results about the homology of abelian groups [21J, p. 123, we see that Hi (SL2(lFp[tj) , Z) contains a copy of !\~ tlFp[t]. Moreover, it is clear that under the obvious map

(tpp)* : Hi (E2 (Z[tJ),Z)

----+

Hi (SL 2(lFp[tJ), Z),

the element t h /\ ... /\ t li E !\~ tZ[t] maps to t h /\ ... /\ t li E !\~ tlFp[tJ. Now consider the map j : E 2(Z[t]) --+ SL 2(Z[t]) and the elements xl = j*(th /\ ... /\ tli). Via the commutative diagram

E2(Z[t]) -

SL2(Z[t])

~! SL2(lFp[t]) we see that xl i= 0 and if Xm is another such element, then xl. and xm are independent. Thus we have the following. THEOREM 4.6.10. For all i

~

1, the group Hi (SL 2(Z[tj) , Z) is not finitely

generated.

0

Moreover, if we consider the group Hl(SL2(ZI:tj), Z), we see that the situation is worse than was known previously. Write

Hl(SL2(Z[t]), Z)

= H 1 (SL 2(Z), Z) ffi H 1 (F, Z) ffi X

where X is some abelian group. Since E 2(Z[tJ) maps to 1 in F, we have H l (E2(Z[tj),Z) 1--+ X. Thus, the group X is also not finitely generated. 4.6.4. Twisted Coefficients. Another natural question about unstable homotopy invariance is the following. Suppose char k = 0 and V is a rational SLn(k)-module. The group SLn(k[tJ) acts on V via evaluation at t = O. What can be said about the cohomology H-(SLn(k[tJ) , V)? In [68J, we studied the group Hl. Denote by B the upper triangular subgroup of SLn(k), by K the subgroup of SLn(k[t]) consisting of matrices congruent to I modulo t, and by C the upper triangular subgroup of K. THEOREM 4.6.11. Suppose char k = 0 and let V be a finite dimensional rational SLn(k)-module. Then

Hl(SLn(k), V) Hl(SLn(k[tJ), V)

= { Hl(SLn(k), V) $

F oo Hl(SLn(k), V) ffi F

V = Ad, n V = Ad,n

=2 ~

3.

4. Rank One Groups

114

The case n = 2 may be proved using the long exact sequence associated to the amalgamated free product decomposition of SL2(k[t]): ~ HI (SL2(k[t]) , V) ~ H 1 (SL 2(k), V) Eli HI (B(k[tJ) , V) ~ Hl(B, V).

To compute HI (B(kl:t]) , V) we use the Hochschild-Serre spectral sequence associated to the split extension

1~C

-------+

B( k[t])

-------+

B ~ 1.

This yields a split exact sequence

o ~ HO(B, Hl(C, V))

~ Hl(B(k[t]), V) ~ Hl(B, V) -------+ O.

Since C acts trivially on V, Hl(C, V) = Hom(H1 (C), V). Thus, we have HO(B, Hl(C, V)) = HomB(C, V) (since C is abelian). The action of B on C is given by

(~ 1~a): (~

tPi t ))

~ (~

a2tf(t)).

If V is an irreducible SL 2 -module, then B acts on V with weight n for some integer n. Hence if n =I 2, HomB(C, V) = O. If n = 2 (Le., V = Ad), then HomB(C, V) = Homk(tk[t], k). If n ~ 3, then one uses the spectral sequence associated to the action of SLn(k[t]) on the Bruhat-Tits building associated to k(t)n. The bottom row of the spectral sequence satisfies E;,o = 0 for p > 0 and hence HI (SLn(k[tj), V) = E~,l' A careful analysis of the structure of various groups of the form HOIIlp(Hl(Cp), V), where P is a parabolic subgroup of SLn(k) and Cp is fLll aHHociatcd subgroup of K, yields the final result that

ES,

1

~ HomB (HI (C), V).

1'hl" group is easily seen to vanish if V =I Ad. It is a one-dimensional k-vector I!Ipooe 1f V = Ad. Nothing is known about the higher cohomology groups.

Exercises 1. Prove that if Z/n ~ Z/(mn) is the inclusion 1 ~ m, then the map H 2i - 1 (Z/n) ~ H 2i - 1 (Z/(mn)) is the same inclusion. 2. Show that H 2i - 1 (SL 2 (Z)) is cyclic of order 12. 3. Complete the proof of Proposition 4.1.2. 4. Show that there are two orbits for the action of SL 2 (K) on the BruhatTi ts tree X; the representatives are 0 Eli 0 and 0 Eli 1l'O. Show also that there is a single edge orbit. 5. Show that HI (SL 2 (lF 2 [t]) , Z) contains a copy of an infinite dimensional lF 2 -vector space, while HI (SL 2 (lF2), Z) is finite.

Exercises

115

6. Let k be a field and let p be a prime distinct from the characteristic of k. Prove that the inclusion SL 2(k) -+ SL2(k[tJ) induces an isomorphism in mod p homology. 7. Let r be the subgroup of SL 2 (k[t]) consisting of matrices which are upper triangular modulo t. Prove that the map r -+ B(k) induces an isomorphism in mod p homology for p not equal to the characteristic of k.

8. Consider the complex D defined in the proof of Theorem 4.4.1. Show that the vertices of D are inequivalent modulo the action of K. Show also that every edge of X is equivalent to an edge of D modulo K. 9. Prove that if k is of characteristic zero, then

Hp(B(k), Hq(K))

= 0,

q~1

(see the proof of Theorem 4.4.1). 10. Use the action of B(k) on D to show that the path D' defined in the proof of Theorem 4.4.1 is a fundamental domain for the action of ron X. 11. Prove the assertions in Remark 4.5.3. 12. Prove that the maximal dimension of a simplex in the Bruhat-Tits building associated to Kn is n - 1. 13. Prove that the stabilizers in SLn{k[t]) ofthe simplices in the fundamental domain T (described in Theorem 4.6.2) have the form indicated. 14. Prove that the inclusion of the reductive part of the stabilizers in the previous exercise into the full stabilizer induces a homology isomorphism. (Hint: use induction on m = the number of diagonal blocks.) 15. Calculate the differential d 1 in the row q = 1 of the spectral sequence of Theorem 4.6.4. (Hint: it looks similar to the bar construction in Appendix A). 16. Prove that for all i ~ 1, the group Hi(B(Z[t])) contains a copy of Hi(tZ[tJ) as a summand. 17. Prove that for all i ~ 1, the group Hi (SL 2 (lFp[tj) , Z) contains a copy of tlFpl:t] (provided p = 2,3). Show that under the obvious map

Ai

Hi (E2 (Z[t]) , Z)

-+

Hi (SL2(lF p[t]) , Z)

the element t h 1\ ... 1\ t l • maps to t iI 1\ ... 1\ t li . 18. Show that HomB{C, Ad) = Homk(tk[tJ, k). 19. Prove that if n ~ 3, then the group HomD(H 1 (C), V) vanishes if V-# Ad and is one-dimensional if V = Ad,

Chapter 5

The Friedlander-Milnor Conjecture Aside from Quillen's conjecture (Section 1.2), the most important unsolved problem in the study of the homology of linear groups is E. Friedlander's generalized isomorphism conjecture [41]. This is usually called the FriedlanderMilnor conjecture because of Milnor's study of the special case of Lie groups

[81]. Let G be a reductive algebraic group over an algebraically closed field k and let p be a prime distinct from char k. Consider the simplicial classifying scheme BGk and the simplicial set BG(k) (G(k) denotes the discrete group of k-rational points of G). The cohomology of BG(k) is simply the cohomology of the discrete group G(k). The conjecture asserts that there is a natural map HZt(BGk,Z/P)

-+

H-(BG(k),Z/p)

which IS an isomorphism, where H;t denotes the etale cohomology of the simplicial scheme BGk (see Appendix C). In the case k = C, we have HZt(BGc,Z/p) ~ H-(BG,Z/p), where BG is the classifying space of the complex lie group G defined in Appendix B. Since a great deal is known about HZt(BGk,Z/P) (see, for example, [42]), this conjecture would give detailed information about H-(G(k),Z/p). In this chapter we describe the conjecture in detail along with all known results. In contrast with Quillen's conjecture, there are no known counterexamples.

5.1. Lie Groups While the generalized isomorphism conjecture assumes that the ground field is algebraically closed, we may say something about the case k = lR as well. Let G be a Lie group and let GO be the group G with the discrete topology. 5.1.1. The natural map BGo -+ BG defined in Appendix B induces isomorphisms on homology and cohomology with Zip coefficients. CONJECTURE

To begin our attack on this conjecture, we first prove a few lemmas. If G is a topological group, denote by G the homotopy fiber of the map GO -+ G; that is, if P(G) denotes the space of paths in G, then

G = {(g,1) E GO x P(G) : 1(0)

= e E Gli , 1(1) = g}.

5. The Friedlander-Milnor Conjecture

118

If G is locally contractible, so that the identity component Go has a universal covering group U, then the homomorphisms U ~ Go ~ G induce isomorphisms U ~ Go ~ G. So the homology groups of BG depend only on the universal covering group of G. If G is a Lie group, it follows that H.(BG) depends only upon the Lie algebra g. LEMMA 5.1.2. Conjecture 5.1.1 is true for a connected lie group G if and only if the space BG has the mod p homology of a point for every prime p. Moreover, if H is locally isomorphic to G, then if Conjecture 5.1.1 holds for G, it also holds for H. PROOF.

Consider the homology spectral sequence 2

-

If Hq(BG, Zip)

= 0 for q > 0, then

6

Ep,q = Hp(BG, Hq(BG, Zip)) ==> Hp+q(BG ,Z/p). Hp(BG 6,Z/p) = E~o = E~,o = Hp(BG,Z/p). Conversely, suppose H.(BG6,Z/p) 9:! H.(BG,Z/p). Note that since = {I}, the space BG is simply connected. We proceed by induction on q to show that Hq(BG, Zip) = o. Suppose q = 1 and consider the map 1l'0

(G)

d2 : H2(BG,Z/p)

-+

Ho(BG,Hl(BG,Z/p)).

Since H2(BG6,Z/p) ~ H2(BG,Z/p) by hypothesis, we see that tP is the zero . map. It follows that we have a short exact sequence -

0- Ho(BG,H 1 (BG,Z/p))

---->

6

H 1 (BG ,Zip)

~

~

H 1 (BG,Z/p)

---->

O.

Sillet! DC iH simply connected, we have

Ho(BG,H1 (BG,Z/p)) Now tUolHllme that Hi(BG,Z/p)

dq+1 : Hq+1(BG,Z/p)

= Hl(BG,Z/p) = o.

= 0 for i < q. -+

Consider the map

Ho(BG, Hq(BG,Z/p)).

By hypothesis, this map is trivial and by the inductive hypothesis we have an exact sequence Ii o ~ Ho(BG, Hq(BG,Z/p)) ----> Hq(BG ,Z/p) ~ Hq(BG, Zip) ----> o. It follows that Hq(BG, Zip) = o. ~

The second assertion is proved by considering the spectral sequence assodated to the fibration BG - + BHIi - + BH. 0 LEMMA 5.1.3. Let r be a discrete uniqu.ely divi,9ible grou.p. Then the mod p homolo,qy of a point.

Br has

5.1. Lie Groups PROOF. By definition, have

r

119

is a rational vector space. Thus, for i

> 0, we

o THEOREM 5.1.4. If the component of the identity of G is solvable, then Conjecture 5.1.1 holds for G. PROOF. Proceed by induction on the dimension of G. Lemma 5.1.2 allows us to assume that G is simply connected. If dimIR G = 1, then G ~ ffi.. Since Bffi. is contractible, and Hi (Bffi.D, Zip) = 0 for i > 0 by Lemma 5.1.3, we see that the conjecture holds for G. Now if G has real dimension greater than 1, choose a homomorphism G -+ ffi. with kernel N. Then we have a fibration

B N ------; B G ------; B"i. By induction, we may assume that BN has the mod p homology of a point. The associated spectral sequence may then be used to show that Hi(BG, Zip) = 0 for i > O. 0 Theorem 5.1.4 allows us to reduce Conjecture 5.1.1 to the case of simple groups. Indeed, if G is a Lie group, the Lie algebra g has a maximal solvable ideal n. The quotient gin splits as a direct product of simple Lie algebras .ai. Let Si be the corresponding simple Lie groups. Then we have a fibration

B N ------; B G ------;

II B S

i.

Since BN has the mod p homology of a point by Theorem 5.1.4, we see that BG does if and only if each BS i does. For an arbitrary Lie group G, not much more can be said. We do have the following result. THEOREM 5.1.5. Assume G has finitely many components. Then the canonical map "h : Hi(BGD,Zlp) ------; Hi(BG,Zlp) is a split surjection (and hence the map '1]* on cohomology is a split injection). PROOF. We use the notion of Becker Gottlieb transfer [6]; we shall not define it explicitly. Let 7r : E -+ B be a smooth fiber bundle with a closed manifold F as fiber. Then there is a transfer map tr : Hi(B) -+ Hi(E) such that the composition

Hi(B) ~ Hi(E) ~ Hi(B) is multiplication by the Euler characteristic X(F). ~ow let G be a Lie group and choose a maximal compact subgroup K. It is a standard fact that G I K hi contractihle so that t.ho lllap i : BK -> BG is a

120

5. The Friedlander-Milnor Conjecture

homotopy equivalence. Let N be the normalizer of a maximal torus in K. The manifold KIN has Euler characteristic 1 [58J. Considering the transfer map associated to the fibration KIN - BN ~ BK, we see that the map

is a split surjection. By Theorem 5.1.4, the map H.(BN 6,7llp)

H.(BN, 7llp)

----+

is an isomorphism. The proof is finished by noting that the diagram Hi (BN 6, 7llp)

-------t

Hi (BG6 , 7llp)

~1

1~

Hi(BN,7llp) ~ H i (BG,7llp)

o -

commutes.

COROLLARY 5.1.6. Every element of order n in H i (BG,7l) lifts to an element of ordern in H i (BG 6,7l). PROOF. Consider the exact sequence

o ----+ 7l ~ 7l ----+ 7l I n

----+

0

and the commutative diagram of associated long exact sequences Hi+l(BG6,7lln)

-------t

H i (BG6,7l) ~ H i (BG 6,7l)

Hi+l(BG,7lln)

-------t

H i (BG,7l) ~ H i (BG,7l).

Slnc(J r/. : Hi+l(BG 6,7lln) - H i + 1 (BG,7lln) is surjective, the result follows.

o

Rational Coefficients. A natural question to study is the behavior of the map H i (BG 6,Q)

----+

Hi(BG,Q).

In this case, a great deal can be said. THEOREM 5.1.7. If G is compact, then the canonical map H i (BG 6,Q) i,~

zero for i > O.

-+

Hi(BG,Q)

5.2. Groups over Algebraically Closed Fields PROOF.

121

The proof makes use of the Chern-Weil homomorphism () : InvG(JR[g']) ~ H-(BG, 1R)

associated to the Lie group G and its Lie algebra g. The algebra InvG(IR[g']) is the graded algebra of real-valued polynomial functions on the vector space g which are invariant under the adjoint action of G. The map () is defined as follows. Let f : g --+ IR be an invariant polynomial which is homogeneous of degree n. Let M be a smooth manifold with a smooth principal G-bundle (see Appendix B) E --+ M and let w be a smooth G-invariant connection on E (Le., w is a I-form with values in g). Let 0 be the curvature 2-form of w. Then is a 2n-form on E and it gives rise to a closed 2n-form f(O) whose cohomology class lies in H2n(M, 1R). This class corresponds to the required class 0(1) E H2n(BG,IR) under the canonical map

on

H 2n (BG,IR) ~ H2n(M,IR) (for further details, sec [116]). The Chern-Weil theorem [18J asserts that if

G is compact, then () is an isomorphism. In particular, BG has only evendimensional cohomology with real (and hence with rational) coefficients. Now, any homology class x in H 2n (BGIi,Q) can be realized as the image of a homology class from some smooth open manifold M which is mapped into BG Ii . To prove that its image in H 2n (BG,Q) is zero, we evaluate on an arbitrary real cohomology class in H-(BG, 1R). If n > 0, choose a homogeneous polynomial f E InvG(IR[g'J) of degree n and consider the class (1(0)) of the induced bundle over M. This induced bundle has curvature zero and hence the value ofx , when evaluated on (1(0)), is zero. 0 There is a complex version of the Chern-Weil theorem. If G is a semisimple complex Lie group, then the analogous map

is an isomorphism [81J. A similar argument now proves the following. THEOREM 5.1.8. If G is a complex semi-simple group with finitely many components, then the natural map

Hi(BGIi,Q) ~ Hi(BG,Q) is zero for i >

o.

o

5.2. Groups over Algebraically Closed Fields There is a natural generalization of Conjecture 5.1.1 for algebraic groups over fields. This generalized hmmorphism conjecture if! due to E. Friedlander [41].

122

5. The Friedlander-Milnor Conjecture

Let S be a simplicial set and let k be an algebraically closed field. Define a simplicial scheme S 0 spec( k) by (Sl8>spec(k))n =

II spec(k) Sn

with simplicial structure induced from S. This scheme has the property that

Now let r be a discrete group and let Gk be an algebraic group over k. If we have a homomorphism p: r - + G(k) from r to the discrete group G(k) of k-rational points of G k , then we obtain an induced map p : rk = r 18> spec(k) - + G k of group schemes over k. In turn, this induces a map

of simplicial schemes. Thus, we have an induced map j

Bp* : H;t(BGk,Zjn)

-+

He(Brk,Zjn) ~ H e ( B r , Z j n ) . 1

j

5.2.1. Let k be an algebraically closed field and let p be a1 prime distinct from char k. Let G k be an algebraic group over k. Then the natuml map of group schemes G(k)k - + G k induces an isomorphism CONJECTURE

AH In tho ease for Lie groups, we can reduce to the case of reductive groups.

5.2.2. Let Gk be a connected algebraic group over k and let 5.2.1 holds for G k if and only if it holds for the reductive group GkjUk. PROPOSITION

Uk dt!fl.ote the unipotent radical of G k· Then Conjecture

The discrctc group U(k) is a successive extension of ko-vector spaces (where ko denotes the prime subfield of k). It follows that U(k) is acyclic for cohomology with Zjp-coefficients, p i= char k. Thus, the natural map He(BGjU(k), Zjp) - + He(BG(k), Zjp) PROOF.

is an isomorphism. On the scheme side, since Gk - + GkjUk is an affine bundle, the map BGk - + BGkjUk induces an isomorphism in etale cohomology. Since the map H;t (BGk, Zjp) - + He(BG(k), Zjp) iH Batural, the wmlt follows. 0

5.2. Groups over Algebraically Closed Fields

123

PROPOSITION 5.2.3. Let Gc be an algebraic group over C. Then there is a natural commutative diagram H;t(BGc, Zip)

---------~

~

H·(BG(C)to p, Zip)

~

where the two diagonal maps are induced by the maps G(C)c -+ Gc and id : G(C)6 -> G(C)toP. Thus, Conjecture 5.2.1 holds for Gc if and only if Conjecture 5.Ll holds for the complex Lie group G(C)top. PROOF. This is obvious from the definitions.

o

In the proof of Proposition 5.2.2, we showed that Conjecture 5.2.1 holds for unipotent groups. This is also true for tori and normalizers of maximal tori Uust as in the complex case). We will prove this in Section 5.3, but we will use this fact before then. We now prove Conjecture 5.2.1 for the case k = lFp • We first need the following result. THEOREM 5.2.4. Let G be a reductive complex Lie group and let 1, p be primes with I i= p. Then there exists a map BG(lFp) ~ BG which induces an isomorphism H.(BG(lFp), ZII) ----+ H.(BG, Z11) where G(lFp ) is the discrete group oflFp-rational points of a Chevalley integral group scheme associated to G. PROOF., The proof is rather technical and relies on some facts from etale homotopy theory [40J. Let Gz be an integral Chevalley group scheme associated to G and let GFp = G z 0lFp. Choose an embedding of the Witt vectors of lFp into C. Then for any pth power q = pd we have a homotopy commutative square ([40], Theorem 12.2) BG(JF q ) Dq

!

(ZIl)oo(BG)

J

(ZIl)oo(BG)

(5.1)

!~

~ (ZII)00(BGx2)

such that some map on the homotopy fibers fib(D q ) -+ fib(~) ~ (ZI1)00(BG) induces a homology isomorphism with Zil-coefficients. Here, (ZII)oo(BG) is the Bousfield-Kan I-completion of the singular complex of BG [20J (the relevant property of this completion is that in this ca.'1e the map BG-+ (ZIl) 00 (BG) induces a mod I homology isomorphism); ¢9 is associated to the geometric

5. The Friedlander-Milnor Conjecture

124

F'robenius map ¢q : G'Fp ~ GFp j and ~ is induced by the diagonal map G ~ GX2. The proof of this fact uses the Lang isomorphism, which asserts the following. Let'IjJ : G k ~ G k be a surjective endomorphism such that the group of 'IjJ-invariant k-rational points, H = Gk(k)'I/J, is finite. Then the map 1/'IjJ : G k

----4

Gk

defined by sending a k-rational point 9 to g'IjJ(g)-l is a principal H-fibrationj that is, 1/'IjJ : Gk/H ~ Gk.

The above diagram is obtained by interpreting the Lang isomorphism in this context. Note that the F'robenius map satisfies the condition of the theorem. Denote by i : BG(lFq) ~ BG(lFql) the map induced by an inclusion G(lFq ) ~ G(lFql). Then as a corollary to the above statement (see Corollary 12.4 of [40]), we have, for any q' = qe = pde, a natural map of fibration sequences

fib(Dq) -

BG(lFq) -

ji

ji

Dq

D"

(5.2)

(Z/l)oo(BG)

jid

fib (Dql ) - - BG(lFql) - - (Z/l)oo(BG) such that j'" : He (fib{Dql ), Z/l) ~ He (fib (Dq) , Z/l) can be identified with the map ()* : HZt(GFp ,Z/l) ----4 HZt{G-Fp ,Z/l) induced by () = J.to(1 x ¢q x··· x ¢q'/q): GFp ~ G'Fp ,where J.t: (G'Fp )e ~ GFp i. tht! product map. Associated to each sequence in (5.2), we have a spectral lIequClncCl. Taking direct limits we obtain a spectral sequence with 2 ·

-

Eli,t = Hs{BG, !!!!hHt(fib{Dq), Z/l)) ~ Hs+t(BG{lFp), Z/l). We now show that E;,t = 0 for t =I- O. LEMMA 5.2.5. For any q = pd there exists some q' = pe such that the endomorphism on the Z/l-dual Hopf algebra of HZt(G'Fp ,Z/l)

j. : HZt{G'Fp ,Z/l)# ~ He(fib(Dq),Z/l) ~ He(fib{Dql),Z/l)

induced by j : fib(Dq)

~

fib(Dql) satisfies

1. ifxEHZt(GFp ,Z/l)# is primitive, thenj.(x) =0

2. if x € HZt{GFp ,Z/l)# is such that j.{x) =I- 0 while j.{y) = 0 for all y with homological degree less than the degree of x, then j. (x) is primitive.

5.2. Groups over Algebraically Closed Fields

125

PROOF. Identify i. with the dual of the map ()'" defined above. Then if x is primitive, we have

i.(x) = x + q)~(x) + ... + q)~

0

q)~

0'"

0

q)~(x)

where q)~ is the dual of q)q*. If the order of q)~ as an automorphism of the finite-dimensional Hopf algebra HZt(GFp ' Z/l)# is m, then i.(x) = 0 provided that q' = qe is such that e = 1m. To prove the second assertion, assume that x satisfies the stated conditions and let e = 1m. Denote by the comultiplication in the Hopf algebra HZt(GFp ,Z/l). We have Ll.(x) = 1 ®x +x® 1 + L:Xi ®Xj in HZt(GFp x GFp ,Z/l)#. Then

a

a.(j.(x»

=

i*(Ll.(x»

+ j*(x) ® 1 + Li",(Xi) ®j",(Xj) = 1 ®i.(x) + i*(x) ® 1

=

1 ®j.(x)

o

so that i*(x) is primitive. Lemma 5.2.5 implies that for each q = pd there is some q'

is the zero map.

= qe such that

i. : iI.(fib(Dq ), Z/l) ---? iI.(fib(Dql), Z/l) Hence, we have E;,t = a for t i= a and therefore the map

(~Dq). : H.(BG(IFp),Z/l)

=

~H.(BG(lFq),Z/l)

-

H.((Z/l)oo(BG), Zll) H.(BG,Z/l)

=

is an isomorphism. Since iI.(BG(IFp), Q) a unique lift [120]

= iI.(BG(IFp), Z/p) = 0, we can find

which gives the required isomorphism H.(BG(IFp), Z/l)

---?

H.(BG, Z/l).

This completes the proof of Theorem 5.2.4.

o

5.2.6. Let p be a prime and let Gfjip be an algebraic group over IFp. Then Conjecture 5.2.1 holds/or Gfji,.' COROLLARY

5. The Friedlander-Milnor Conjecture

126

PROOF. We may assume that Gjjip is reductive and that G'Fp = Gz The map ~Dq

QS)

iFp.

: BG(iFp) ~ (Z/l)oo(BG(C)toP)

is induced by the map G(iFp)'Fp ~ G'Fp and a choice of embedding of the Witt vectors of iFp into C induces an isomorphism

HZt(BG(iFp),Z/l) ~ HZt(BGc,Z/l) ~ H-(BG(C)toP,Z/l). (The first isomorphism is implicit in the statement of Theorem 5.2.4 and is due to Friedlander and Parshall [42]. They proved that if k is a separably closed field with ring of Witt vectors R, then the base change maps

Gk

~

GR

GK ~ Gc K +- C (K algebraically closed) induce f--

associated with embeddings R - t isomorphisms in mod 1 etale cohomology.) Consider the commutative diagram

H:t(BG'Fp,Z/l) -

H-(BG(C)to p, Z/l)

~1 H-(BG(iFp), Z/l). By Theorem 5.2.4, the right arrow is an isomorphism. The result follows.

0

Finite fields revisited. Diagram (5.1) allows one to compute the cohomology of '1 the finite group G(lFq) via the associated Eilenberg-Moore spectral sequence. Thill was noted by Quillen in his ICM address [97], and discovered independent.iy by Friedlander [39]. This spectral sequence has the form E~'·

-= Tor~:«Z/I)oo(BG)X(Z/I)oo(BG» (H-((Z/l)oo(BG)), H-((Z/l)oo(BG))) ~

H-(BG(lFq),Z/l).

Nottl that this holds for the unstable group BG(lFq). The computation given in Chapter 1 involved this spectral sequence for the stable space BU followed by n difficult restriction argument to the unstable case. For example, if G = GL n , then we have

H-((Z/l)oo(BG),Z/l)

= H-(BG,Z/l)

H- (BUn, Z/l) = H-(G(n,oo),Z/l) Z/l[Cl, C2,.'" en], The spectral sequence then has the form

degci = 2i.

Tor~l[cl, ... ,c.. ,c;, ... ,c~l (Zjl[Cl,"" en], Z/l[dt, ... , c~&]) ~ H-(BG(lF q)). The E 2 -term is easily calculated (see Chapter 1), giving the result.

5.2. Groups over Algebraically Closed Fields

127

We now return to our study of Conjecture 5.2.1. If H ~ G is an inclusion of discrete groups and if A is a G-module, then x E He(BH,A) is G-invariant if the images of x under the maps He(BH, A) ~ He(B(H

n gHg- 1 ), A)

and He(BH,A) ~ He(B(gHg-l),A) ~ He(B(H

n gHg-1),A)

coincide, where c(g)* is the map induced by conjugation (see Appendix A). Denote the set of G-invariant elements by He(BH,A)s. For example, if G is finite with p-Sylow subgroup H, then the p-primary part of He(BG, A) is detected on H; that is, Hn(BG, A)(p)

--=-. Hn(BH, A)B

for n > O. 5.2.7. Let Gjiip be a connected linear algebraic group over Gjiip be the normalizer of a maximal torus in Gjiip . Then the

PROPOSITION

IFp and let Njiip

C

restriction map is injective with image equal to the set of G-invariant elements. PROOF. Let G z be an associated reductive group scheme over Z. The group N(lF q ) oflFq-rational points of N z contains an l-Sylow subgroup of G(lFq) [117]. Thus the restriction maps

He(BG(lF q ), Zll)

are isomorphisms for any q

-'>

He (BN(lF q ) , 7lll)s

= pd. Now pass to the limit.

o

THEOREM 5.2.8. Let G k be a connected linear algebraic group over k and let Nk C Gk be the normalizer of a maximal torus. Then the composition

HZt(BGk, Zll)

-'>

HZt (BNk' Zll) ~ He(BN(k), Zll)

is an injection with image the stable elements with respect to the embedding N(k) C G(k). PROOF. The proof that Conjecture 5.2.1 holds for Nk will be postponed until Section 5.3. Let R C C be the strict henselization of the local ring 7l(p). Then R has residue field IFp and field of fractions contained in Q. By Hensel's is uniquely I-divisible, as are the lemma, the kernel of the surjection R X --cokernels of the injections R X ___ QX and QX ___ C x . As a consequence, we have isomorphisms (with Z I l-coefficients)

W;

He(BN(fp)) ~ He(BN(R))

J!-

He(BN(Q))

:=. H e (BN(C)6).

5. The Friedlander-Milnor Conjecture

128

Hence the map

He(BN(iFp),z/l)S ~ He(BN(R),Z/l)s is injective. To see that it is surjective, let x E He(BN(R), Z/I) be stable for 9 E G(R). Denote by 9 the image of gin G(iFp ). Note that

He(BN(iFp)

n N(iFp)7i,z/l) ~ He(BN(R) n N(R)9, Z/I)

HO that the element in He(BN(iFp),Z/l) corresponding to x is stable for g. Since R is Hensclian, the map G(R) - t G(iFp ) is surjective, so we have an isomorphism

He(BN(iFp),Z/l)S ~ He(BN(R),Z/l)s. Now consider the commutative diagram

He(BN(Fp),Z/l)S ~ He(BN(R),Z/l)s _

r

He(BN(IQ),Z/l)s

r

r

The lower horizontal maps are induced by base change and hence are isomorphisms. By Corollary 5.2.6 and Proposition 5.2.7, we have an isomorphism

HZt(BG Fp ,Z/I) ~ He(BN(Fp),Z/l)s and hence all the vertical arrows are isomorphisms. In particular,

HZt(BGQ,Z/l) ~ He(BN(IQ),Z/l)s.

Now if k is any algebraically closed field, let ko

~ k be the algebraic the prime field. Then since k X /k~ is uniquely I-divisible, we have "'II injection He(BN(k),Z/l)S ~ He(BN(ko),Z/l)s.

(!IUMUrC uf

CouHider the commutative diagram

HZt(BGko, Z/l)

"" ---=---+

1~ HZt(BN(ko),Zjl)S -

HZt(BGk,Z/l)

1 He(BN(k),Z/l)s.

The top map is an isomorphism by base change. Since the bottom map is injective, we obtain the desired isomorphism

for any k.

D

5.2. Groups over Algebraica.lly Closed Fields

129

COROLLARY 5.2.9. The map HZt(BGk,Z/l) ~ He(BG(k),Z/l) is split injective and the map He(BG(k),Z/l) ~ He(BN(k),Z/l)S is split surjective.

PROOF. Consider the commutative diagram

o

The left map is an isomorphism.

COROLLARY 5.2.10. Conjecture 5.2.1 holds for Gk if and only if the map He(BG(k),Z/l)

~

He(BN(k),Z/l)

o

is injective.

Conjecture 5.2.1 admits an equivalent formulation in terms of finite subgroups. CONJECTURE 5.2.11. Let k be an algebraically closed field, let Gk be an algebraic group over k, and let 1 be a·prime distinct from the characteristic of k. Then for any nonzero x E He (BG( k), Z/l), there exists a finite subgroup He G(k) such that x restricts nontrivially to He(BH,Z/l). THEOREM 5.2.12. Conjecture 5.2.1 holds for Gk if and only if Conjecture 5.2.11 does. PROOF. Suppose Gk satisfies 5.2.11. We show that the map He(BG(k), Z/l)

-+

He(BN(k), Z/l)

is injective. Let x E He(BG(k),Z/l) be nonzero and choose a finite subgroup He G(k) such that x restricts nontrivially to He(BH,Z/l). We may replace H by an I-Sylow subgroup and hence assume that H is an I-group. Such a group consists entirely of semi-simple elements and hence is conjugate to a subgroup of N(k). The restriction of x to He(BN(k), Z/l) is therefore nontrivial. Conversely, if Gk satisfies 5.2.1, then the map

5. The Friedlander-Milnor Conjecture

130

is injective. If char k = p > 0, choose an embedding isomorphisms

iFp C

k. This induces

H-(BN(k),Z/l) ~ H-(BN(iFp),Z/l) ~ ~H-(BN(lFq),Z/l) and hence the mod 1 cohomology of G(k) is detected on the finite subgroups N(lFq) C G(k). If k has characteristic zero, choose a prime p which does not divide the order of the Weyl group W of G k . Choose an embedding of the strict henselization R of Z(p) into k. This gives rise to maps iFp ...... R -> k" TheMe induce isomorphisms

H-(BN(iFp),Z/l) ~ H-(BN(R),Z/l) ~ H-(BN(k),Z/l).

IF;

Since the map R X has a splitting with uniquely IWI-divisible cokernel, we get a W-equivariant map T(Fp) - T(R) inducing the inverse to the reduction map "" H-(BW, T(lF H-(BW, T(R)) --=-. )). p

In particular, the reduction map N(R) -> N(Fp) (which is a map of extensions of W) admits a splitting which induces an isomorphism

H-(BN(R),Z/l) ~ H-(BN(Fp),Z/l). The composite map N(IFp) -> N(R) - N(k) - G(k) detects the mod 1 cohomology of G(k). Since the mod 1 cohomology of N(Fp) is detected by its finite Mubgroups N(lFq), the result follows. D COROLLARY 5.2.13. Let k = U ka, where each ka is algebraically closed. Then Gk Hatisjies Conjecture 5.2.1 if and only if each Gkc. does.

Pn.OOF. Suppose G k satisfies 5.2.1. For each

Oi,

k=

U A{1,

where A{1 is a -> ka. It

f\nlt,tJiy g(lfICwtcd ka-algebra. Each A{1 admits a ka-algebra map A{1 fUUUWM thut t.Iw map

H_(BG(ko,), Z/O

------->

H_(BG(k), Z/l)

1M ill.ioetiv(1, and hence

iM

~mrjcctive.

Consider the diagram

H-(BG(k), Z/l)

~

H-(BN(k),Z/l)

1~

1 H-(BG(k a ), Z/l)

~

H-(BN(koJ,Z/l).

It f'ollowR that the bottom map is injective. The converso is easy and is left to the reader.

D

5.2. Groups over Algebraically Closed Fields

131

As a consequence we see that we need only find a "large enough" field of each characteristic to prove Conjecture 5.2.1. Indeed, let k be an algebraically closed field of infinite transcendence degree over the prime sub field and let L be algebraically closed with char(L) = char(k). Assume that 5.2.1 holds for Gk. Write L = ULOI. where each LOI. is algebraically closed and of finite transcendence degree over the prime subfield. Then each LOI. admits an embedding into k and hence each GLa. satisfies Conjecture 5.2.1. Since L is the union of the LOI.' we see that GL does also. There is another reformulation of Conjecture 5.2.1, due to Jardine [62], which involves ultraproducts of simplicial sets. Recall that an ultrafilter U on a set I is a collection of subsets of I such that (1) U does not contain the empty set; (2) if X c Y and X E U, then Y E U; (3) U is closed under finite intersections; and (4) if XuY E U, then X E U or Y E U. Given an ultrafilter U, the corresponding ultraproduct of a collection.of sets {FihEI is the set

Fi) / '" (II iEI where the equivalence relation is defined by (ai) = (b i ) if there is a Z E U with ai = bi for all i E Z. Now assume that the sets Fi are fields and let R be their ordinary product. Let p be a prime ideal of R. If X c I, define an element ex = (ex,i) E R by IV

o

eX,i

= {1

ifi E X if i f/. X.

Let U(p) be the set of subsets X C I such that ex E p. One checks easily that U(p) is an ultrafilter. Conversely, every ultrafilter U corresponds to some prime ideal p(U) of R. In particular, the kernel mi of the projection pri : R

-----+

Fi

is a maximal ideal of R which corresponds to the ultrafilter Ui = U(mi) that is generated by the element e{ i}. Denote by j the canonical morphism of schemes j :

II sPec(Fi) -----+ spec(R).

iEI If Y is a collection of sets indexed by I, then there is a sheaf j*Y on spec(R) such that the ultraproduct of the sets Yi corresponding to U is the stalk j. Yjl(U) of j.Y at the point p(U). We will denote the ultraproduct by j.(Y)u. CONJECTURE 5.2.14. Let I be the set of prime numbers and let P : J -----+ I be a fu.nction, where .J is some infinite .~et. Let B S Ln (F p) denote the sheaf on the di.'lc'f'r,tf. .'1pacc 'P('/) of all .'11J,b.~et.~ of .J corrr."ITJOnding to the family of

132

5. The Friedlander-Milnor Conjecture

simplicial sets {B S Ln (iFP(j»)}, and let B S Ln (iF p ) t be the sheaf associated to the family oil-completions {BSLn(iFp(j))t}. Let U be an ultrafilter on the set J such that the set {j E J : P(j) = l} is not a member of U. Then the induced map j*(BSLn(iFp))u ~ j*(BSLn(JFp)t)u is a mod I homology isomorphism. According to Jardine ([62], Theorem 17), this conjecture is equivalent to Conjecture 5.2.1 for SL n . Moreover, one expects that this construction works equally well for any simple simply connected algebraic group. Also, this can be reduced to a statement about ultraproducts of classifying spaces for finite special linear groups. The reader is referred to [62] for further details.

5.3. Rigidity The study of Conjecture 5.2.1 is closely related to the concept of rigidity, which _ we now explain. Let V be a contravariant functor on an appropriate category of schemes over a field k (e.g., the category of smooth affine schemes) with values in the category of torsion abelian groups. DEFINITION 5.3.1. The functor V is rigid if for every connected variety X over k and closed points x,Y: spec(k) ~ X, the corresponding maps

x*,y*: V(X)

~

V(spec(k)) = V(k)

coincide.

To produce examples of rigid functors, we need the following notions. We lIay that V is homotopy invariant if V(X x A1J = V(X) for any scheme X. The functor V admits transfers if for any finite flat morphism X ~ Y, there 'N Ito homomorphism trx/y : V(X) ~ V(Y) satisfying the usual properties (e,g,. tho (~omposition V(Y) ~ V(X) ~ V(Y) is Ulultiplication by the degree of the map). THEOREM 5.3.2. Suppose V is a homotopy invariant functor with transfers. Then V is rigid. PROOF. It suffices to consider the case where X is a smooth affine curve. Let Div(X) be the group of divisors on X and let DivO(X) be the subgroup of principal divisors. Consider the pairing Div(X) x V(X)

~

V(k)

defined by (x, u) 1-+ x*(u). We show that this vanishes on DivO(X) x V(X). Let X be a smooth projective model for X and set X 00 = X - X. Let f be

5.3. Rigidity

133

a rational function on X which is equal to 1 on Xoo. Consider the principal divisor (1). Let Xo be the curve obtained from X by deleting the points where f equals 1. Then f defines a covering X o ---+ Af, = lP'f, - {I}. The properties of transfer imply that the image of ((1),u) in V(k) coincides with the image of ((0 - (0), trxO/Al (ulxo»j the latter is zero by homotopy invariance. Thus, the pairing factors through PieO(X, Xoo) (8) VeX). Since Pic°(X, Xoo) is divisible and VeX) is torsion, we see that the pairing vanishes on DivO(X) x VeX); that is, x*(u) = y*(u) for any x, y : spec(k) ---+ X. 0 As an easy corollary, one sees that if V commutes with limits: V (spec(lim Ai»

= lim V(spec(Ai»

,

then V(k) = V(ko) for any extension ko ---+ k of algebraically closed fields [125]. To see this, note that if Xo is a connected variety over ko and if x, y : spec(k) ~ X o are any two ko-points, then x* = y*. In turn, this implies that the image of the map V(Xo) ~ V(k) corresponding to x : spec(k) -+ X o lies in the image of V(ko): choose a rational point spec(ko) ~ Xo and apply the above remark Lo x and y: spec(k) ~ spec(ko) ~ Xc. Now write k as the limit of the finitely generated ko-subalgebras of k. Then V(k) = li!!l V(spec(A)). The homomorphism V(spec(A)) ~ V(k) is induced by a ko-point spec(k) ~ spec(A) corresponding to the embedding of A in k. The image of this map is contained in the image of V(k o) and hence the map V(ko) -+ V(k) is surjective. The injectivity of this map is clear since each algebra A is split over ko. Since the K -groups are homotopy invariant, admit transfers and commute with limits, we obtain the following result. COROLLARY 5.3.3. Let ko ---+ k be an extension of algebrairolly closed fields. Then for any n, the natural maps Ki(ko)/n ---+ Ki(k)/n, nKi(ko) ---+ nKi(k), and Ki(ko,Z/n) ---+ Ki(k,Z/n) are isomorphisms. 0 COROLLARY 5.3.4. Let k be an algebraically closed field of positive characteristic p. Then Ki(k) is divisible for i ~ 1. The torsion subgroup is trivial for i even and is isomorphic to ESI:;!:p QdZI if i is odd.

PROOF. The follows from the computation 1.1.12 for ko

= iFp •

0

Corollary 5.3.3 begs the question: are the homology functors

Hi(G( -), Z/n), where G is an affine group scheme over k, rigid on the category of smooth k-algebras? This question has not been settled. CONJECTURE 5.3.5. Let G be a group over an algebraically closed field k and let p be a prime distinct from the characteristic of k. Let X be. a smooth

5. The Friedlander-Milnor Conjecture

134

affine curve over k and let x, y be closed points on X. Then the corresponding specialization homomorphisms Sx,Sy: H.(G(k[X]),Zjp) ~ H.(G(k),Zjp) coincide.

Note that 5.3.5 holds for the affine line when G = GLn,SLn,PGLn : By Corollary 4.6.3, the inclusion G(k) ~ G(k[t]) induces an isomorphism on integral homology and this map is split by evaluation at any x E AI . More generally, it holds for G = PGL 2 , X = C-{x}, where C is a smooth projective curve over k and x is a closed point on C. This follows from Theorem 4.5.7 by an argument similar to the proof of the rigidity theorem in [126]. Consider the functors Hi (E2 (-), Z) on the category of smooth algebras, where k is an infinite field. In Section 4.6.3 we showed that these satisfy

H i (E2 (R[tj), Z)

~

H i (E2 (R), Z)

whenever R has many units. If R is a smooth k-algebra, then

k[spec(R) x Al] = R[t]; thus, the functors H i (E2 ( -), Z) are homotopy invariant. If they also admit transfers, then Theorem 5.3.2 would imply that the functors H i (E2 ( -), Zjn) are rigid for (n, char (k)) = 1. Unfortunately, there appears to be no way to equip these functors with transfer maps. Conjecture 5.3.5 is implied by the following stronger statement. CONJECTURE 5.3.6. Let G be an algebraic group over k and let p be a prime distinct from the characteristic of k. Let X be a smooth affine curve over k and denote by O~ the henselization of the local ring Ox, x EX. Then the naturol map H.(G(k),Zjp) ~ H.(G(O~),Zjp) t, an iSOTn01phism.

conjecture (and hence Conjecture 5.3.5) holds in a few special cases. this in Section 5.5. AH promised, we have the following result.

ThiH

We

CliHCUHH

PROPOSITION 5.3.7. Conjecture 5.3.6 implies Conjecture PROOF.

5.3.5.

Let x, y be closed points on the smooth affine curve X. Note that H.(G(O~),Zjp)

= ~y/x etaleH.(G(k[Y]),Zjp).

Let a E H.(G(k[Xj),Zjp) and denote by a(x) the image of a under the specialization map Sx' Since H.(G(k), Zjp) is a summand of H.(G(k[X]), Zjp) , we can view a(x) in the latter. Consider the class a - a(x). This specializes to 0 under Sx and by assumption maps to 0 in H.(G(O!:,), Zjp). Thus there

5.3. Rigidity

135

exists a curve Y which is etale over X such that 0: - o:(x) maps to 0 in H.(G(k[Y]),Z/p). Denote by O:y the image of 0: in H.(G(k[Y]),Z/p). Then o:y = o:(x), which is a constant class. This implies that the specialization of o:y is the same at all points of Y and hence there is an affine open neighborhood of x E X on which the specialization of 0: is constant. Similarly, we can find an open neighborhood of y on which the specialization of 0: is constant. Since these two neighborhoods intersect nontrivially, we have o:(x) = o:(y). 0 On the surface, rigidity has no apparent connection to Conjecture 5.2.l. The next result provides the link. PROPOSITION 5.3.8. Let k be an algebraically closed field and denote by K the algebraic closure of k(T). Then if rigidity holds, the natural map

i* : H.(G(k), Zip)

---7

H.(G(K), Zip)

is an isomorphism for p =1= char(k).

Injectivity does not require rigidity. We have K = C ranges over the smooth affine curves over k, and each map PROOF.

H.(G(k),Z/p)

---7

~ k[C],

where

H.(G(k[C]),Z/p)

is split injective. It follows that i* is injective. To prove surjectivity, note that each class in H.(G(K),Z/p) comes from some 0: E H.(G(k[C]), Zip), where C is a smooth affine curve over k. If x, y : spec(K) ---7 C are two points of the k-variety C, then the corresponding specialization maps Sx,Sy:

H.(G(k[C]),Z/p)

---7

H.(G(K),Z/p)

coincide (see the discussion following Theorem 5.3.2). It follows that the image of Sx is contained in the image of H.(G(k),Z/p). Indeed, choose a rational point spec(k) ---t C and let y : spec(K) -+ C be the composition

spec(K) Then the image of homomorphism

Sy

---7

spec(k)

---7

C.

is clearly contained in the image of H.(G(k),Z/p). The

H.(G(k[C]),Z/p)

---7

H.(G(K),Z/p)

is induced by a map spec(K) ---t C of schemes over k. In view of the above statements, the image of this map is contained in the image of H.(G(k), Zip). It follows that i", is surjective. 0 REMARK 5.3.9. Consider the functors Hi(E2(-)'Z/p). If we knew these were rigid, then the coneiusion of Proposition 5.3.8 would hold for G = 8L 2 •

5. The Friedlander-Milnor Conjecture

136

Indeed, we would have an isomorphism

H.(E2 (k),Z/p) and

E2

=

----+

H.(E2 (K), Z/p) ,

SL 2 for fields.

COROLLARY 5.3.10. If rigidity holds and k has positive characteristic, then Conjecture 5.2.1 holds for G. PROOF. Note that if k is an algebraically closed field of characteristic p> 0, then k is a union of transcendental extensions of iFp • It follows that the natural map H.(G(iFp), Z/l) ----+ H.(G(k), Z/l) is an isomorphism. Consider the commutative diagram HZt(BGk, Z/l) - - - - - t H·(BG(k), Z/l)

1

r HZt(BGFp,Z/l)

-----t

H·(BG(iFp),Z/I).

The left vertical arrow is an isomorphism by base change and the bottom arrow is an isomorphism by Theorem 5.2.6. 0 REMARK 5.3.11. If we could prove 5.2.1 for a field of characteristic zero (e.g., Q), then Corollary 5.3.10 would apply in that case as well. To summarize, we have the following. If, for each smooth affine curve X over k and closed points x, y E X, the map

H.(G(k), Zip) -----+ H.(G(O~), Zip) 1", an b:lomorphiHm, then the specialization maps .9 x ,Sy :

H.(G(k[XD,Z/p)

----+

H.(G(k),Z/p)

coincide, and hence the canonical map

HZt(BGk,Z/p)

-----+

H·(BG(k),Z/p)

II! an isomorphism. This allows us to prove more cases of Conjecture 5.2.l. THEOREM 5.3.12 ([61.D. Suppose G is one of the following groups over k: a unipotent group, a torus, a solvable group, or the normalizer of a maximal torus in a connected algebraiC group. Then Conjecture 5.2.1 holds for G. PROOF. Let X be a smooth affine curve over k and let x be a closed point on X. If G is the additive group, then G(O~) is the abelian group underlying O~j since p is invertible in k, this group is uniquely p-divisible. Thus,

Hi(G(k),Z/p) = Hi(G(O~),Z/p)

= 0,

i > O.

137

5.3. Rigidity

An arbitrary connected unipotent group U has a normal connected codimension one unipotent subgroup U'; the quotient is the additive group. Any map spec(O~) ----+ U/U'lifts to U. Arguing by induction and using the short exact sequence we see that Hi(U(O;),Z/p) = 0, i > O. The same is true for U(k); thus, 5.2.1 holds for any unipotent group. If G is the multiplicative group, then G(O~) = (O~)X is p-divisible and contains the group Z/poo of p-primary roots of unity as its p-torsion subgroup. It follows that the induCed map H.(kX,Z/p)

----+

H.((O;)X,Z/p)

is an isomorphism. Using the Kiinneth formula, we obtain the result for any torus. Now, if G is solvable, we have a split exact sequence 1

~

U

----+

G

----+

T

~

1,

where T is a torus and U is unipotent. The associated spectral sequence calculation shows that H.(G(k),Z/p) ~ H.(G(O;),Z/p). Finally, let G be the normalizer of a maximal torus T in a connected algebraic group. The quotient W = G /T is a finite discrete group; hence W(k) = W(O~). We have a commutative diagram 1

----+

T(k)

1

----+

T(O~)

----+

G(k)

----+

G(O~)

!

----+

W(k)

----+

W(O~)

----+

1

----+

l.

II

!

Comparing the spectral sequences, we see that the map H.(G(k),Z/p)

----+

H.(G(O;),Z/p)

o

is an isomorphism.

There is one more important case which we now mention. Consider the infinite general linear group GL. We have the following result of Gabber [43] and Gillet-Thomason [47]. THEOREM 5.3.13. Let X be a smooth variety over afield k (not necessarily algebraically closed) and let x E X be a closed point. Then if p is prime to char (k), the natural map

K.(O;,Z/p) is an isomorphism.

----+

K.(k,Z/p)

o

138

5. The Friedlander-Milnor Conjecture

COROLLARY 5.3.14. Under the same hypotheses as 5.3.13, the natural map H.(GL(O~),Z/p) ~ H.(GL(k),Z/p)

is an isomorphism. PROOF. This is true more generally. Suppose f : X ~ Y is a map of connected H-spaces and m is an integer such that both the kernel and cokernel of the map 7l'1(X) ~ 7l'1(Y) are uniquely m-divisible. Then 7l'.(X,Z/m) ~ 7l'.(Y,Z/m) is an isomorphism if and only if H.(X,Z/m) ~ H.(Y,Z/m) is an isomorphism ([127], Proposition 1.5). In this case, taking X = BGL(O~)+ and Y = BGL(k)+, Theorem 5.3.13 implies the result. 0 REMARK 5.3.15. 1. The analogue of Corollary 5.3.14 holds for the group SL since BE(R)+ is the universal cover of BGL(R)+. 2. The analogue of Corollary 5.3.14 is known for the infinite orthogonal group 0 and the infinite symplectic group Sp by the work of M. Karoubi [65]. COROLLARY 5.3.16. If k has positive characteristic p, then the natural map

is an isomorphism. PROOF. By the previous corollary, the map Ir(BGL(k),Z/I) ~ H·(BGL(Fp),Z/l)

ill an isomorphism. To extend etale cohomology to direct limits of schemes,

we view a scheme as a sheaf and work in the sheaf category (see [61]). Then DGLAl = I.!.m BGLn,k and there is an isomorphism HZt(BGLk,Z/I) ~ fuEHZt(BGLn,k,Z/l)

([81]. Stldiol1 4). Consider the commutative diagram HZt(BGL k , Z/l) - - - - - t H·(BGL(k), Z/l)

r~ HZt(BGL,p ,Z/l)

l~ -----t

H·(BGL(Fp),Z/I).

The bottom arrow is an isomorphism by Theorem 5.2.6.

o

REMARK 5.3.17. Jardine's proof ofthis result ([61], Theorem 4.1) works in any characteristic. It is based upon the theory of simplicial etale sheaves on the category of smooth schemes over k. A thorough discussion of this would take IlH too far afield, but we can say what is going on in general. Corollary 5.3.14 implies that the homology sheaves H.(BGL, Zip) are constant. Comparing the universal coefficient spectral sequences for discrete cohomology and etale cohomology yields the isomorphism H6t.(BGLA~'Z/P) ~ H·(BGL(k),Z/p). The

5.4. Stable Results

139

same argument works for any algebraic group G, so the analogue of Corollary 5.3.14 for G implies the Friedlander-Milnor conjecture for G. (See [61] for further details.) The proof of Corollary 5.3.16 would work in characteristic zero if we knew the result for C (or Q). In the next section, we show that there is an isomorphism H·(BGL(C)top,Z/p) ~ H·(BGL(C),Z/p).

5.4. Stable Results The results in this section are due to A. Suslin [127]. Let k be a field and ~onsider the simplicial scheme BG Ln,k' The ith space Xn,i is the i-fold product

Xn , i

= ,GL n' k X .••• x GL n ' k,, v i

and the jth face map p; : Xn,i

-----+

X n,i-1 is defined by

(92, ... , 9i) j = 0 { P~(91'''' ,9i) = (91, ... ,9~9H1'''' ,9i) ~ =- ~ S i - I J - ~.

(91, ... ,9,-1)

Consider the k-rational point spec(k) ~ Xn,i given by x f---t (1,1, ... , 1), where 1 E GLn,k is the unit element. Let X!,i be the henselization of Xn,i at this closed point and denote its coordinate algebra by O~,i' The ring O~,i has maximal ideal m~ ,i' Since the maps preserve the unit section, we have induced morphisms X!,i -----+ X!,i-1' Denote by (p~)'10 the maps

p;

O~,i-1

-----+

GL(O~,i_ll m~,i-1)

O~,i'

-----+

GL(O~,i' m~,;),

-----+

C.(GL(On,i' mn,i)' Z/p) ,

and -

h

h

C.(GL(On,i_l, mn,i-1), Zip)

-

h

h

where C(G,Z/p) denotes the reduced standard complex of G with coefficients in Zip (i.e., Co(G,Z/p) = 0). We have morphisms of schemes over spec(k):

X!,i

-----+

Xn,i ~ GLn

preserving the unit section. These give rise to canonical matrices

am E GLn(O~,i' m~,J (i.e., O!,cvalued points of GLn). Denote the chain [a] ,... ,n·d ® 1 E Ci(GL(O!:,i,m":,i)'Z/P) by Un,i (set ?Ln,o

= 0).

5. The Friedlander-Milnor Conjecture

140

LEMMA 5.4.1. There exist chains Cn,i E Ci+l(GL(O~,i,m~,i),Zlp) such

that Un,i - I:(-l)j(p;)*(cn ,i-t). j=O PROOF. Let cn,o = 0 and assume that cn,o, ... , Cn,i-l have been constructed . .:--rote that d(un,d = L~=o(-l)j(p~)*(un,i-d. Then d(c n,;}

=

d(1Ln'i - I)-l)j(P;)*(Cn J=O

t(

-1 F (p;)*

j=O i

(I:

,i-d) =

(_l)k (pj-l)* (C n ,i-2))

k=O

i-I

j=Ok=O

O. This shows that Un,i - L~=o(-l)j(P;)*(Cn,i-l) is a cycle and hence a boundary since iIi(GL(O~i,m~i),Zlp) = 0 by Corollary 5.3.14. This gives us the required chain Cn,i" , 0 Lemma 5.4.1 leads to a computation of the K-theory of any algebraically closed field. The idea is to prove that if k is an algebraically closed field of positive characteristic l, and if E is the algebraic closure of the quotient field of the ring of Witt vectors of k, then there is an isomorphism K.(k, Zip) ~ K.(E, ZIp). Thus the ~roups K.(F, Z/m) for (m, char(F)) = 1 do not even dClPfllld on the characteristic of the algebraically closed field F. Thus, the (!omputnt.ioll of Corollary 5.3.4 is valid for any F. LOllllIl!l 5.4.1 also plays a role in the computation of the Zip homology of JIG L( k) top for k = JR, C We have the map

H.(BGL(k), Zip)

--->

H.(BGL(k)to p , Zip)

obt.ained by taking the limit of the maps

H.(BGLn(k), Zip)

--->

H.(BGL n (k)iO P , Zip).

The space BGL(C)to p has the homotopy type of BU and BGL(JRfop has that of BO (see Appendix B). The mod p (co)homology of these spaces is well llnderstood. THEOREM 5.4.2. Let k

= JR

OT

H.(BGL(k), Zip) is an isomorphi8m.

C The canonical map --->

H.(BGL(k)to p , Zip)

5.4. Stable Results

141

We first prove a few lemmas. Let G be a discrete group. The space BG is the geometric realization of a certain simplicial set (see Appendix B). If G is a Lie group with finitely many components, fix a left invariant metric on G and denote by Go; the c-ball with center at 1 E G. Let BGo; be the geometric realization of the simplicial set whose p-simplices are p-tuples [g1,"" gp] of elements of G such that

Go;

n gl Go; n ... n gl ... gpG i= 0 0;

with the usual face and degeneracy maps. 5.4.3. If c > 0 is small enough, then .the sequence BGo; ---? BGtop is a fibmtion up to homotopy (i.e., BGo; is homotopy equivalent to the space BG of Section 5.1). LEMMA

BGo

-->

Let Gtop ---? Eat° P ---? Bat op be the universal principal Gtop_ bundle. For a space X, denote by Sin(X) its singular simplicial set (the homology of the associated simplicial abelian group is the singular homology of X). We obtain a commutative diagram of fibrations PROOF.

ISin(GtoP)1

1

Gtop

~

ISin(EGtoP)I

----;..~

1

EGtop

~

ISin(BGtOP)1

----;..~

1

BGtop

where the vertical arrows are homotopy equivalences. The discrete group G acts on ISin(Eat°P)I freely and hence we obtain a new fibration ISin(GtoP)I/G

---?

ISin(EGtoP)I/G

-->

ISin(BctoP)I·

Since ISin(EatoP )I is contractible, the space ISin(EGtoP)I/G is homotopy equivalent to BG. Thus, we have a fibration up to homotopy ISin(GtoP)I/G

---?

BG

---?

BGtoP.

Suppose that E is small. Then Go; is geodesically convex and hence every nonempty intersection goGo; n ... n gpGo; is contractible. Denote by X. the simplicial space with

Xp =

II

goGo; n··· n gpGo;.

go, .... gp

Let Y•• be the bisimplicial set with Ypq = Sinp(Xq) and denote by (Sin(GtoP))o; the subobject of Sin(GtoP) cOI1.'listing of singular simplices lying in some gGe;. Let Ee; be the simplicial set with p-simplices the (p+ 1)-tupiOH (gO, ... 'flp) with

142

5. The Friedlander-Milnor Conjecture

gaGe

n··· n gpG e =1= 0.

Consider the evident maps of bisimplicial sets

where we view (Sin(at°P))e (resp. Ee) as a bisimplicial set trivial in the q (resp. p) direction. One checks easily that r.p and 1/J are homotopy equivalences. Moreover, the embedding (Sin(GtoP»e ---+ Sin(GtoP) is also a homotopy equivalence. Thus, we have homotopy equivalences ISin(GtoP)I ~

IY•• I ~ IEel·

The discrete group G acts freely on all of these and the above maps are Gequivariant. Thus, we get a homotopy equivalence

and it is clear that the diagram

eommut.cs up to homotopy.

o

LEMMA 5.4.4. Let k = R or C. If c is small enough, then the map IJGLn(k)~ ---+ BGL(k) induces the zero map on iI.( -, ZIp). PROOF. We use the construction of Lemma 5.4.1. Denote by o~ort , the ring of gtlnus of continuous functions GLn(k)i ----... k defined in a neighborhood of the identity. The group GLr(O~ort) may be identified with the group of gorms of continuous maps GLn(k/ ---+ GLr(k) defined in a neighborhood of 1. Thus, every chain c E Cq(GLr(()~rt),Zlp) defines a continuous map from some neighborhood of 1 E GLn(k)i to Cq(GLr(k), ZIp). Now, the ring t t OC()!1 nt" is henselian so we obtain a unique map Oh n,t. ---+ OC0!1 n,t. . Denote by (:~;~rt E Ci+l(GL(()~~it), Zip) the image of the chain Cn,i constructed in 5.4.1. Let N > 0 and find c > 0 so that the chains c~~it are defined on GLn(k)e x ... x GLn(k)e for 0 ~ i ~ N. Then we get maps ·'1i :

Ci(BGLn(k)t:, Zip) ----... Ci+l(GL(k), Zip) = Ci+l(BGL(k), Zip)

5.4. Stable Results

for 0 :s; i :s; N. It is clear from 5.4.1 that the embedding C.(BGLn(k)c:, Zip)

------>

Si

143

is a null-homotopy (for i :s; N) for

C.(BGL(k), Zip)

=

C.(GL(k), ZIp)·

o

The result follows. COROLLARY 5.4.5. If E

> 0 is small, then the embedding

BSLn(k)c: induces the zero map on

BSL(k)

------>

iI.( -, Zip).

PROOF. This follows since for any field k, the map H.(SL(k))

------>

H.(GL(k))

o

is split injective. COROLLARY 5.4.6. If E

> 0 is small, then

Hi(BSLn(k)c:, Zip)

=

0

for 0 :s; i :s; (n - 1)/2.

PROOF. Consider the spectral sequence associated to the fibration BSLn(k)c:

------>

BSLn(k)

------>

BSLn(k)to P •

By Theorem 5.1.5, the map H.(BSLn(k), Zip)

------>

H. (BSLn(k)to p , Zip)

is (split) surjective. Let io be the least positive integer for which the group Hio(BSLn(k)c:,Zlp) -=J O. Then the spectral sequence shows that we have a short exact sequence (with ZIp-coefficients)

o ------> Hio (BSLn( k)c:)

------>

Hio (BSLn( k))

Now, the homology Hi(SL n ) stabilizes at i i :s; (n - 1)/2, we see that Hi(BSLn(k)c:, Zip)

is the zero map. Thus, io

------>

=

------>

Hio (BSLn(k )toP)

------>

O.

(n-1)/2 (Theorem 2.1.7). So if

Hi(BSLn(k), Zip)

> (n - 1)/2.

o

PROOF OF THEOREM 5.4.2. By Lemma 5.4.6, we see that the map BSL(k) -----; BSL(k)to p

induces an isomorphism on homology with Zlp--codnciellt.s. Since hoth spaces arc simply connected, the map ext.ends t.o BSL(A:) I ~. BSL(k)to p Itnd t.his

144

5. The Friedlander-Milnor Conjecture

map is an isomorphism on homotopy with Zip-coefficicnts. Consider the commutative diagram of fibrations BSL(k)+ -

BGL(k)+

1

1

BSL(k)to p _

-

BP

1

BGL(k)top _

B(P)top.

The outside arrows induce isomorphisms on homology and homotopy with finite coefficients. Hence, so does the middle arrow. D COROLLARY 5.4.7. The map Hi (BGL n (k), Zip)

---+

Hi (BGL n (k)top , Zip)

is an isomorphism for i :S n. D

PROOF. This follows from Theorem 2.3.1.

Aside from this last low-dimensional result, we can say a few more things.

5.5. HI, H 2 , and H3 Recall from Section 5.1 that if G is a Lie group, then Conjecture 5.1.1 holds for G if and only if the space BG has the mod p homology of a point. This is true if and only if Hi(BG,Z) is uniquely divisible for all i > O. PROPOSITION 5.5.1. Let G be a connected semi-simple Lie group. Then HI (BO, Z) = 0 and there is a short exact sequence -

0----+ H 2(BG,Z)

----+

6

H 2(BG ,Z)

---+

H 2(BG,Z)

----+

O.

PROOF. For the computation of HI (BG), we may assume that G is simply r.onner.t(lci. This implies that H2(BG, Z) = o. Since G is perfect, HI (BG 6, Z) = o. Tho spectral sequence associated to the fibration BG_BG6_BG shows that HI(BG,Z) = o. For any connected Lie group, results of Borel [13] imply that the group Ha(BG, Z) is finite (since the rational cohomology of BG is a polynomial algebra on even dimensional generators). By Corollary 5.1.6, we see that Ha(BG 6, Z) surjects onto H 3 (BG, Z). If in addition G is semi-simple, then HI (BG, Z) = 0 and the spectral sequence gives us the required exact sequence. D THEOREM 5.5.2. If G is a Chevalley group over the real or complex number·8, then Conjectu,re 5.1.1 holds for H2.

145

5.5. Ht, H'J, and H3

PROOF. By the work of Sah and Wagoner [108], the group H2(BG,Z) is isomorphic to K 2(C) in the complex case and to K 2(C)- in the real case. Since K 2(C) is uniquely divisible, we see that H2(BG, Zip) = o. 0

The results of Section 3.2 can be used to prove Conjecture 5.1.1 for

H3(SL2 (C), Zip) and H3(GL 2(C), Zip) (the former implies the latter). Let F be an algebraically closed field of characteristic zero. We have an exact sequence

o ~ Q/Z ~ H3(SL2(F), Z) ~ p(F) (here, Q/Z is the group of roots of unity in F). Since p(F) is uniquely divisible, H3 (SL 2 (F), Z) is the sum of a uniquely divisible group and Q/Z. If F = C, then H3(SL2(C)6,Zlp) = O. Since this group surjects onto the group H3 (SL 2(C)top , Zip), this implies the result. With a bit more work, one can prove the same result for H3(SL2(~)' Zip) [107]. In positive characteristic, we can prove Conjecture 5.2.1 for Hi(GL n ) for i ~ 3. AP, we saw in Section 5.3, it suffices to show that the natural map

is an isomorphism when X is a smooth affine curve over k. Theorems 2.3.1 and 5.3.13 show that this holds for i ~ n. In Section 3.3, we showed that if F is an infinite field and if A is a hensel local F-algebra, then the natural map of Bloch groups

B(F) ® Zip is an isomorphism for p

~

B(A) ® Zip

=I char(F). Moreover, there is an exact sequence

H3(GM2(R))

~

H3 (GL2 (R))

~

B(R)

~

0

for any local F-algebra R.

Let O~ be the henselization of the local ring Ox, where x is a closed point on the curve X. Then the natural map PROPOSITION 5.5.3.

is an isomorphism. PROOF. Consider the commutative diagram

H3(GM2(k)) ® Zip

1

-----+

H3(GL2(k)) ® Zip

1

-----+

B(k) ®Zlp

1

5. The Friedlander-Milnor Conjecture

146

The left vertical map is an isomorphism since we have an isomorphism

H.(GM2(k),Z/p) ~ H.(GM2(O;),Z/p) (see the proof of Theorem 5.3.12). The result now follows from the Five Lemma. D

COROLLARY 5.5.4. The natural map i,~

H3(GL2(k),Z/p) -- H3(GL2(O;),Z/p)) an isomorphism.

PROOF. Consider the commutative diagram of universal coefficient sequences

H3 (GL 2 (k)) ® Zip

~

H3(GL 2(k), Zip)

1

1

H3(GL2(O~)) ®Z/p ~ H3(GL2(O~),Z/p) ~ pH2(GL2(O~)).

The last map is an isomorphism by stability and the corresponding statement for H 2 (GL). The result now follows from the Five Lemma. D COROLLARY 5.5.5. If k has positive characteristic 1, then the map

H2t(BGL n,k,Z/P) __ H3(BGLn(k), Zip) is an isomorphism. PROOF. Stability implies the result for n

~

3. The case n

= 2 now follows

from the previous corollary.

D

Exercises 1. Let G be a locally contractible topological group, Go its identity compommt, and U its universal covering group. Prove that the homomorphisms U --+ Go --+ G induce isomorphisms U --+ Go --+ G. 2. Prove the second assertion of Lemma 5.1.2: if H is locally isomorphic to G, then if Conjecture 5.1.1 holds for G, it also holds for H. 3. Prove that if N is the normalizer of a maximal torus T in a Lie group G, then the identity component of N is T. 4. Prove that if R is the strict henselization of the local ring Z(p), then there is an isomorphism

H·(BN(iFp) n N(IFp)9, Z/l)

9:!

H·(BN(R)

n N(R)9, Z/l).

5. Complete the proof of Corollary 5.2.13. 6. Show that U(p) is an ultrafilter and that every ultrafilter on I corresponds to some prime ideal of R = TIiU Fi .

Exercises

147

7. Let k be an algebraically closed field and let p be a prime not equal to the characteristic of k. Let X be a smooth affine curve over k and let x be closed point on X. Let O~ be the henselization of the local ring Ox' Prove that the homomorphism H.(k\ Zip) ~ H.((O;) x • Zip)

8. 9. 10.

11.

12. 13.

is an isomorphism. Prove that Corollary 5.3.14 holds for the infinite special linear group SL. Prove that fIi(GL(O~ i. m,~ i)' Zip) = 0 (you may assume that the action of GL(k) on this homology is trivial). Prove that the map 'P in Lemma 5.4.3 is a homotopy equivalence as follows. It suffices to show that for each p, the map Yp ,. ---+ (Sin( atoP) )e,p is a homotopy equivalence. Use the definitions of these spaces and note that the fiber of this map over a point t is the simplicial set whose ~ simplices are (p+1)-tuples of elements ofGt = {g E G : t E Sinp(gGe)}. This set is contractible. Prove that the map 't/J in Lemma 5.4.3 is a homotopy equivalence by showing that the fiber of Y.,q ---+ (Ee)q over (go, . .. ,gq) is Sin(goGe n ... n gqG e). Show that the map Si defined in the proof of Lemma 5.4.4 is a nullhomotopy. Prove that for any field k, the map H.(SL(k),Z) __ H.(GL(k),Z)

is split injective.

Appendix A

Homology of Discrete Groups In this appendix we summarize basic facts about group (co)homology. We assume a basic familiarity with the homology of topological spaces and homological algebra. Good references for most of this material are the books of K. Drown [21] and 1. Evens [37].

A.I. Basic Concepts A.I.I. The definition. Let G be a group and choose a presentation of G: 1 ---+ R

---+

F

---+

G

---+

1

where R and F are free groups. Construct a CW-complex BG as follows. Take a point x and for each generator of G (i.e., for each element of F) attach a 1-cell to x. One now has a bouquet of circles, X(1), with 71"1 (x(1») = F. Each element of R is a word in the generators of F and hence corresponds to a path 'Y in X(1). For each such element, attach a 2-cell e2 via a map f : oe2 ---t 'Y. This yields a space X(2) with 71"1 (x(2») = Fj R = G (each path corresponding to an element of R is now nullhomotopic). Now, attach a 3-cell for each generator of 7I"2(X(2») to obtain a space X(3) with 7I"1(X(3») = G and 7I"2(X(3») = O. Continue this process, adding i-cells to obtain a space X(i) at each stage with 7I"} (X(i») = G and 7I"j(X(i») = 0 for 1 < j < i. Now define

BG= UX(i). i

Clearly, 7I"1(BG) = G and 7I"j(BG) = 0 for j > 1. This construction is covariantly functorial in G and one checks easily that BG is the unique space, up to homotopy equivalence, satisfying 71"1 = G and 7I"j = 0 for j > 1. EXAMPLE A.1.1. If G = Z, then a presentation for G is 'd o ---+ 0 ---+ Z ~ Z ---+ O.

Thus, to build BG, we take t1 point and attach a I-coH. This gives Since 7I"i(81) = 0 for i > 1, the prOCClHH stoPH and hence BG = 8'.

X(1)

= 8 1.

A. Homology of Discrete Groups

150 EXAMPLE

A.1.2. Consider G = Z/2. A presentation is

o -----+ Z ~ Z -----+ Z / 2 -----+ O.

oe

Thus, X(I) = 8 1 and we attach a 2-cell e 2 to 8 1 via the map 2 = 8 1 ~ 8 1 • The resulting space X(2) is the real projective plane RIP2. Now, 1f2(~IP2) = 1f2(82 ) = Z and the generator f : 8 2 ~ 00p2 is the double cover. Hence, X(3)

=~ Uje3 =~.

Continuing this process, we see that at each stage we attach an i-cell via the double covering map so that XCi) = OOPi . Hence, BZ/2 = OOPoo. DEFINITION A.l.3. The homology of the group G with coefficients in the trivial module A is H.(G,A) = H.(BG,A).

The cohomology is defined similarly as

The above computations of the homotopy types of BZ and BZ/2 give the following homology groups:

{z

o

i = 0, 1 i> 1

z { Z/2 o

i=O i odd i even.

Unfortunately, it is usually impossible to obtain such simple models for 8G. Moreover, we would like to be able to compute homology with coefficients In nontrivial G-modules M. We carry this out via the following device. Lot ZG be the group ring of Gover Z and let M be a (left) ZG-module. A pmjcctive resolution of Mover ZG is an exact sequence of ZG-modules ...

~

Pi

~

Pi -

1 ~ ••• ~

PI

~

Po

~

M

~

0

where each Pj is a projective G-module. Such resolutions exist for any M. Now, let G be a group and choose a resolution

p.

-----+

Z

of the trivial module Z. If M is a G-module, we define the homology groups of G with coefficients in M to be

A.I. Basic Concepts

151

(In the sequel, we shall abbreviate I8lZG by I8lG.) The cohomology groups are defined similarly: H·(G,M) = H·(HomzG(P.,M)). That this is well-defined is a consequence of the following. PROPOSITION A.1.4. Let p. ---+ Z and Q. ---+ Z be two projective resolutions over ZG. Then there is a ZG-linear chain map f. : p. ---+ Q. such that f. is a homotopy equivalence of chain complexes which is unique up to a unique 0 chain homotopy equivalence.

To see that our new definition agrees with the previous definition using BG, consider the (contractible!) universal cover X ---+ BG. The group G acts on X as the group of deck transformations and hence the cellular chain complex G.(X) is a chain complex of G-modules. Moreover, since X is contractible, the augmented complex G.(X) ---+ Z is a (free) resolution of Z over ZG. Each Gi(X) is a free G-module with one basis element for each G-orbit of i-cells. This new definition allows us to use ad hoc resolutions to compute homology. For example, let G = (t : t n = 1) be the cyclic group of order n with generator t (written multiplicatively). Denote by /:1 the endomorphism of ZG given by multiplication by 1 + t + ... + tn-I. Consider the sequence

... ~ ZG .!=..; ZG ~ ZG .!=..; ZG ~ Z This sequence is exact since /:1(t - 1) = t n result. PROPOSITION

-

= O.

O.

This gives the following

A.1.5. The integral homology of the cyclic group Z/n is

z { Hi (Z/n, Z) = Z/n o PROOF.

1

-+

i=O i odd i even.

Apply - I8lc Z to the above resolution:

... ~ Z .!=..; Z ~ Z .!=..; Z. Since t acts trivially on Z, this complex has the form

···~Z~Z~Z~Z. The homology of this complex is easily computed.

o

The standard resolution. This resolution is obtained from the "simplex" spanned by Gj i.e., we build a space X with vertices the elements of G and simplices the finite subsets of G. This space is clearly contractible. The corresponding free resolution F. = G.(X) is explicitly given ItS follows. The module Fn. is the free abelian group with hnsis all (n + I)-tuples (lIo, lJl, ... ,,I}II)' The

A. Homology of Discrete Groups

152

G-action is given by 9(90, . .. ,9n) = (ggo, ... , ggn) and the boundary map 8 is defined as n

8(90, ... ,9n) = ~)_I)i(go, ... ,gi, ... ,9n). i=O

A basis for the free ZG-module Fn consists ofthose (n + I)-tuples whose first element is 1. Write such a tuple as (l,gl,glg2, ... ,glg2··· 9n) and introduce the bar notation

[gllg21···lgn] = (l,gl,glg2, ... ,glg2···gn). (If n = 0, there is only one such element, denoted [ ].) In terms of this basis, the map 8 is given as 8 = E~=o(-I)idi' where

ddgll· .. Ign]

=

9d921 .. ·Ign] { [gIl· . ·lgi-llgigi+llgi+21· . ·Ign] [gIl· . ·Ign-l]

i

=0

0
In low dimensions, the bar resolution has the form

F2 ~ Fl ~ ZG where £(1) = 1, 8 1 ([g])

--=-. Z ---+ 0,

g[ ]- [ ] = 9 - 1, and 82([glh]) = g[h]- [gh]

+ [g].

PROPOSITION A.1.6. Let M be a G-module. Then Ho(G, M) is the module MG of coinvariants; that is,

MG =M/(gm-m: 9 E G,m EM). PRom'. Apply - 0G M to the standard resolution:

~ F10GM ~ZG0GM. W(l have Ho(G, M) = coker(81 ). The formula for 8 1 is given by

8 1 ([g] 0 m) = (g - 1) 0 m Il.nd under the isomorphism ZG0 GM

--t

M, we have (g-1)0m

t-->

gm-m.

0

REMARK A.1.7. Let M and N be G-modules. Then there is a canonical isomorphism M 0G N ~ (M 0 N)G, where G acts diagonally on M 0 N. This gives us a second definition of H.(G, M). If F. is a projective resolution of Z, then H.(G, M) = H.((F. 0 M)G).

PROPOSITION A.l.8.

tion of G.

For any group G, Hl(G,Z) = Gab, the abelianiza-

A.l. Basic Concepts

153

PROOF. Applying - i8lc Z to the standard resolution, we obtain

EeZ~EeZ~Z. [glhJ

[gJ

The map (h maps the element [g] i8l n to gn - n. But since Z is a trivial Gmodule, gn - n = n - n = O. Thus, Hl(G, Z) = coker(02) and the map 02 is

g[h] - [gh] + [g] [h]- [gh] + [g]. Therefore, Hl(G,Z)

= EeZ/([g] + [h] = [gh]), [g)

o

which is Gab, as claimed.

REMARK A.1.9. This also follows from the fact that the first homology group of the space BG is the abelianization of 7rl(BG) = G.

The definition of H. (G) is clearly functorial. Relative Homology. Let G be a group, MaG-module, and H ~ G a subgroup. We define the relative homology groups H. (G, H j M) as follows. Consider the sequence of complexes

o ~ C.(H) i8lH M

--+

C.(G) i8lc M

--+

C.(G,H;M)

~

0,

where C.(G, Hj M) = C.(G) i8lcM/C.(H) i8lH M. We claim that this is exact; that is, we claim that the map C.(H) i8lH M ~ C.(G) 0c M

is injective. To see this, note that this map is none other than the inclusion C.(BH; M)

--+

C.(BG; M),

where if is the coefficient system on BO given by M. A less topological proof is obtained by noting that the map in question is the composition C.(H) i8lH M

--+

C.(G) 0H M ~ C.(G) i8lc M.

It therefore suffices to treat the special case ZH0H M

~

ZG0 H M

--+

ZG0cM

where the composition is clearly an isomorphism. DEFINITION

A.1.lO. The relative homology groups H.(G, Hj M) are de-

fined as H.(G, H; M)

= H.(C.(G, Hi M)).

154

A. Homology of Discrete Groups

Note that there is along exact sequence

...

~

Hi(H; M)

~

Hi(Gj M)

~

Hi(G, H; M)

~

Hi-l(H; M)

~

Center kills. Let g E G and consider the map of pairs c(g) : (G, M) given by c(g) : (h ~ ghg-I, m t----7 gm). This induces a map c(g) ... : H.(G,M) ~ H.(G,M).

....

~

(G, M)

PROPOSITION' A.l.ll. The map c(g)* is the identity. PROOF. Consider the map hi: Ci(G,M) ~ CHl(G,M) given by i

hi([gll .. ·Igi] ® m) =

L( -l)j[gll· .. lgjlg-llg-lgj+1gl·· ·Ig-lgig] ® m. j=O

Then h. is a homotopy from c(g) ... to the identity.

o

COROLLARY A.1.12 ("center kills"). Let z be a centml element of G and let M be a G-module. Then the endomorphism of M defined by m ~ zm induces the identity map on H.(G,M). PROOF. Conjugation by z is the identity on G.

o

EXAMPLE A.l.l3. Center kills is very useful in computing homology. Consider the group GLn('Z) and let zn have its usual GLn(Z) action. Then - I acts as multiplication by -1 on zn and hence acts the same way on 1I.(GLn (Z), zn). But since -1 is central, the induced map on homology is thl.llclmltity. Thus, id* = -id*; that is, H.(GLn(Z), zn) is all 2-torsion.

A.l.2. Induced Modules. Let G be a group with subgroup H. The ring ZG is (',\.Idly HOCIl to be a free ZH-module with basis a set of coset representatives of 11\0. Let M be an H-module. We define the induced module as Ind~M

= ZG ®:m

M.

This is n G-module via g : g' ® m ~ gg' ® m. Clearly, we have a decomposition Ind5}M =

EB

gM.

gEG/H

This characterizes modules of the form Ind5}M. Let N be a G-module whose underlying abelian group is a sum EBiEI Mi' Suppose the G-action transitively permutes the summands. PROPOSITION A.1.l4. Let N be a G-module as above. Let M be one of the l~1.lmmd.nds Mi and let H c G be the stabilizer of i in G. Then M is an H-module and N ~ Ind5}M. 0

155

A.I. Basic Concepts

COROLLARY A.l.15. Let N = EBiEI Mi be a G-module. Assume that the G-action permutes the summands according to some action of G on I. Let G i be the stabilizer of i and let E be a set of representatives for I modulo G. Then Mi is a Gi-module and N ~ EBiEE IndgiMi . D EXAMPLE A.l.16.

Let X be a complex on which G acts. The G-module

Cn(X) is a direct sum of copies of Z, one for each n-cell of X, and G permutes the summands. Let ~n be a set of representatives for the G-orbits of n-cells, and let G a be the stabilizer of (j. Then

Cn(X) =

EB Indg".Za, aEL: n

where Za is the Ga-module associated to (j; that is, if g EGa, g acts on Za as +1 if g preserves the orientation of (j and by -1 if g reverses orientation. PROPOSITION A.l.17 (Shapiro's Lemma). If H module, then H.(H, M) ~ H.(G, Ind~M).

~

G and M is an H-

Let F. ---+ Z be a projective resolution over ZG. Since ZG is a free ZH-module, F. is also a resolution over ZH. Thus PROOF.

H.(H, M) = H.(F. ®H M).

But F. ®H M ~ F. ®G (ZG ®H M) ~ F. ®G (Indi}M).

D

A.1.3. Transfer. Let G be a group and let H be a subgroup of finite index. The inclusion i : H ---+ G induces a map i* : H.(H, M) ---+ H.(G, M). Since (G: H) < 00, we can construct a map

tr~ : H.(G, M)

-----+

H.(H, M)

as follows. Denote by X the space BG and by X the covering of X corresponding to the subgroup H. If (j is a cell of X, then there is a cell if of X lying above (j for each coset representative of G / H. Define a map

Cr,(X, M) where

M is the coefficient system on X (j

®m

Cn(X, M),

-----+

f----t

associated to M, by

L

if ® m.

GIH

The induced map on homology is tri}o Since the composition

Cn(X, M)

-----+

is the map (j

®m

f----t

Cn(X, M)

L GIH

we see that i ..

0

tr)~

= (G : H)id.

if ® m

f----t

-----+

L GIN

Cn(X, M) (j

® m,

A. Homology of Discrete Groups

156

PROPOSITION A.1.IS. If G is finite, then Hn( G, M) is annihilated by IGI for all n > O. If IGI is invertible in M, then Hn(G, M) = 0 for all n > O. PROOF. Let H = {I} C G. Then i. otr~ = IGlid. But Hn(H,M) = 0 for n > O. This proves the first assertion. The second follows from the first· since if IGI is invertible in M, then it is also invertible in H.(G, M). D A.1.4. Abelian groups. We state the following propositions about the homology of abelian groups. THEOREM A.l.I9. Assume that k is a principal ideal domain. 1. There is a map 'I/J : /\. (G ® k) ---t H. (G, k) which is injective for every abelian group G and split injective if G is finitely generated. 2. Suppose that every prime p such that G has p-torsion is invertible in k. Then 'I/J is an isomorphism. 3. If k has characteristic zero, then'I/J is an isomorphism in dimension 2. PROOF. This is all obvious if G is cyclic: /\i(G ® k) = 0 for i > 1 and Hi(G, k) = 0 for i > 1 if IGI is invertible in k. The case of G finitely generated now follows easily via the Kiinneth formula and induction since G is a finite direct product of cyclic groups. The general case follows from the fact that G = !!!QGa., where Ga. ranges over the finitely generated subgroups of G. D THEOREM A.1.20. Let G be an abelian group. Then there is a natural isomorphism (G ® Z/p) ® r(pG) ---t H.(G, Z/p)

1\.

where

r

i.9 a divided power algebra and pG denotes the p-torsion subgroup of

G. PROOF. See [21], p. 126.

D

A.2. Spectral Sequences A.2.1. Basic definitions. Spectral sequences are a generalization of the long exact homology sequence ••• ---t

Hi(C~)

---t

Hi(C.)

---t

Hi(C./C~)

---t

Hi-l(C~)

---t •••

H8/lociated to a short exact sequence

o ---t C~

---t

C.

---t

C./C~

---t

0

of chain complexes. Suppose we are given a complex C. and an increasing sequence of sub complexes {FpC.hEZ. Assume the filtration is dimensionwise finite; that is, {FpCnhEZ is a filtration of finite length for each n. There is an induced filtration on the homology H.(C.) given by

F1J H.(C.)

= im(H.(FpC.)

-+

H.(C.)).

A.2. Spectral Sequences

157

We have the associated graded module

grH.(C.) =

EB FpH.(C.)/ Fp_1H.(C.). p

The spectral sequence associated to the filtered complex C. is a sequence grH.(C.) with El consisting of the groups H.(FpC./ Fp-lC.). More precisely, Er is a bigraded module equipped with differentials {Er}r~o of "successive approximations" to

tt;;,q : E;,q ~ E;-r,q+r-l such that E7'+l is the homology of E7':

E;~l = ker(~,q)/im(~+r,q_r+1)' Note that if E;,q = 0 for some r, then for r' ~ r, E;:q = O. Since the filtration is assumed dimensionwise finite, the module E;,q' for fixed p, q, stabilizes at some point r = r(p, q). We define E'::q to be this stable module: E'::q = E;~,q). We say that the spectral sequence converges to H.(C.). This is best illustrated with an example. Suppose we have a first quadrant double complex of modules:

1

1

1

1

1

1

Co,o We have dh : Cp,q --+ Cp-1,q and dV : Cp,q --+ Cp,q-l, dhdv + dVd h = dhd h = dVdv = O. This is the EO-term of a spectral sequence with dO = dV • Thus E~,q = Hq(Cp,., dV ). Since dhd v + rfUd h = 0, the horizontal map dh induces a map d 1 -- (d h ) •.. Elp,q -----> Elp-l,q' Taking homology again, we obtain

E~,q = Hp(E;,q, d1 )

= H; H~(C.,.);

that is, the E2-term is obtained by first taking the vertical homology of the double complex and then taking the horizontal homology of the resulting complex. The map d2 is easily described (sec [77], Appendix D, for a good discussion).

158

q

A. Homology of Discrete Groups

Now, since Cp,q = 0 for p < 0 or q < 0, we have E;,q = 0 for p < 0 or < 0 for all r. AB a result, for fixed p, q, there exists r = r(p, q) such that

the differentials d! starting and ending at E;,q are zero. Thus, E;,q ... = Er;:q for r 2:: r(p, q). Consider the total complex TotG.,. defined by

TotnG.,. =

E9

= E;~l =

Gp,q'

p+q=n This complex has a differential induced by dh and rJV. There is a canonical filtration on TotG.,.:

FiTotnG.,.

= E9 Gp,n-p' p~i

In other words, we take only those modules in columns 0 through i. This induces a filtration on H.(TotG.,.):

oc THEOREM

Fo C Fl C ... C Fn = Hn(TotG.,.).

A.2.1. For all p, q 2:: 0,

E;:q = FpHp+q (TotG.,.) / Fp-1Hp+q(TotG.,.). In this case we see that the spectral sequence converges to the homology of TotG.,. and write E~,q ===} Hp+q(TotG.,.). Note that we could have filtered the complex by rows:

FiTotnG.,. =

E9 Gn-q,q. q~i

Tl1C1 tl.'lHociatod spectral sequence is then obtained by first taking horizontal homology and then taking vertical homology:

E~,q = H;H;(G.,.). This sequence also converges to H.(TotG.,.), but gives a different filtration and hence a different EOO-term. Example: Chain complexes of coefficients. Let G be a group and M a Gmodule. The homology H.(G, M) is defined as H.(F. ®G M), where F. is a projective resolution of Z. If G. is a nonnegative chain complex of G-modules, we set H.(G, G.) = H.(F. ®G C.). If C. consists of a single module M in dimension zero, then H.(G,G.)

H.(G,M).

=

A.2. Spectral Sequences

159

Now, F.&JeC. is the total complex of the double complex (Fp&JeCq). Thus, we have two spectral sequences converging to H.(G, C.). The first sequence has E~,q = Hq(Fp &Je C.) = Fp &Je Hq(C.) since Fp &Je - is an exact functor (Fp is projective). Computing the E 2-term, we have

E;,q PROPOSITIOK PROOF.

=

A.2.2. If C.

Since C. c::::

C~,

(A.I)

Hp(G,Hq(C.)). c:::: C~,

H.(C.)

then H.(G, C.)

= H.(C~). It

=

H.(G,

C~).

follows that

E;,q(C.) - Hp(G, Hq(C.)) = Hp(G, Hq(C~)) = E;,q(C~), and hence the E=-terms are isomorphic.

D

The second spectral sequence has E~,q

= Hq(F. &Je Cp) = Hq(G, Cp).

(A.2)

The E 2 -term is then the pth homology group of the complex Hq(G, C.). Suppose for example that each Cp is a free ZG-module, or more generally an H.-acyclic G-module (that is, Hq(G, Cp) = 0 for q > 0; for example Cp projective or induced). Then E~,q = 0 for q > 0 and E~,o = (Cp)e. Thus, in this case the second spectral sequence collapses to give an isomorphism

H.(G, C.) = H.((C.)e). Now, llsing the first spectral sequence A.I, we see that (A.3) This is a typical argument using spectral sequences. One uses one spectral sequence to identify a computable El_ or E2-term, and the other spectral sequence to identify the abutment. A.2.2. Two important examples. Let

I----->H----->G----->Q-----> I be a group extension. Then we have the following result, due to Hochschild and Serre. THEOREM

fOTm

A.2.3. FOT any G-module M, theTe is a spectml sequence of the

A. Homology of Discrete Groups

160

PROOF. Let F. ---+ Z be a projective resolution over ZG. Then F. ®c M can be computed by first factoring out the H -action, then factoring out the Q-action:

F. ®a M

= ((F.

® M)H)Q = (F. ®H M)Q.

Writing C. = F. ®H M, we have

H.(G, M) = H.((C.)Q)' Moreover, we have an isomorphism of Q-modules

H.(H, M) = H.(C.). Now, we must show that the Q-modules Cp = Fp ®H Mare H.- acyclic. Since we can take F. to be the standard resolution, it suffices to show that ZG ®H M is acyclic. But this latter module is an induced module ZQ ® A (see [2:1.], p. 69). Thus, l1',ing spectral sequence A.3, we have

E;,q = Hp(Q, Hq(C.))

=?

Hp+q((C.)Q)'

But the E2-terms are isomorphic to Hp(Q, Hq(H, M)).

o

EXAMPLE A.2.4. Let G be the group of 3 x 3 upper triangular matrices over Z with 1's on the diagonal (the Heisenberg group). Then we have an extension

o -----> Z -----> G -----> Z EB Z

----->

0

whm'c G ---+ Z e Z is the map (

1

o

a

b)

1 c 001

f---*

(a, c).

Tho kernel of this map is central; thus, Hq(Z) is a trivial Z EB Z-module. The Hochschild--Serre spectral sequence takpB the following form:

E;,q = Hp(Z EB Z, Hq(Z)) Since Hq(Z) = 0 for q q = 0 and q = 1:

=?

Hp+q(G).

> 1, the spectral sequence is concentrated on the lines

o E2

=

Z Z

0 Z EB Z ZEBZ

0 Z Z

0 0 0

The only nontrivial differential is d2 : E?,o ---+ Eg,l' We claim that this map is an isomorphism. Note that H 2 (Z EB Z) = 1\ 2(Z E9 Z) = Z generated by (1,0) 1\ (0,1). We claim that to compute d2 , we lift (1,0) and (0,1) to G and

A.2. Spectral Sequences

161

compute the commutator of the two elements. (The interested reader can check this.) The obvious lifts are

o~n

and

O!n

and their commutator is

o~n

so that

d2 :

(1,0)!\ (0,1)

f-t

1 E Hl(7L.). Thus,

7L. { Hi(G) = 7L. EB 7L.

°

i=0,3 i = 1,2 i

> 3.

A.2.5. One could also compute the homology of G by noting that BG is a circle bundle over the torus with Euler class 1. The Gysin sequence (which is really just a special case of a spectral sequence) then gives the same result. REMARK

Equivariant homology. Suppose C.(X) is the cellular chain complex of a Gcomplex X. The homology groups H.(G, C.(X)) are denoted Hf(X) and called the equivariant homology groups of (G, X). We can perform this construction with any G-module M:

Hf(X,M) =H.(G,C.(X)®M) where G acts diagonally on C.(X) ® M. Note that Hf(pt, M) = H.(G, M). Since any G-complex X admits a map to a point, there is a canonical map

Consider the two spectral sequences associated to H.(G, C.(X)®M). The first spectral sequence satisfies

E;,q = Hp(G, Hq(X, M)) ===> H;+q(X, M). PROPOSITION

A.2.6. If X is acyclic, then the canonical map

H;;(X,M) ----' H.(G,M) is an isomorphism.

A. Homology of Discrete Groups

162

PROOF. In this case, Hq(X, M) = 0 for all q > 0 so that the spectral sequence is concentrated on the line q = o. Thus

E~o = E;,o = Hp(G,M) = H;(X,M).

o The second spectral sequence provides an important computational tool. Let ~p be a set of representatives for the G-action on Cp(X). We have, for onch (J, the orientation module Zoo mentioned above. Let MO" = Zoo ® M. Then we have an isomorphism of G-modules

Cp(X, M)

=

ED Ind~tMO". O"EI:p

By Shapiro's Lemma, we have

Hq(G,Cp(X,M)) ~

ED Hq(GO",MO"). O"EI:p

The second spectral sequence A.2 then takes the form

ED Hq(GO",MO") ====> H;+q(X,M),

E~,q =

O"EI:p

and, if X is acyclic, then

E~,q =

ED Hq(GO",MO") ====> Hp+q(G,M).

(AA)

O"EI:p

Tho d l

lTutp

i!l easily seen to be induced by the boundary map in Xi that

bl,

if'! tlw direct Ilum of the maps induced by the inclusions GO" - G n where r rangel'! over the faces of (J. In more compact terms, we have

E;,q

= C.(X/G, 1tq)

where 1t q is the coefficient system

(J I-t

Hq(GO", MO"). The E2-term is then

E;,q = Hp(X/G, 1tq}. This point of view is used frequently in Chapter 4, where the following result if'! Ilsed repeateclly (see [113], Lemma 6, or [67], Lmmua 3.3).

A.2. Spectral Sequences

163

PROPOSITION A.2.7. Suppose p(O) c p(l) c ... C p(k) = X is a Jiltmtion of the simplicial complex X such that each p(i) and each component of p(i) _ p(i-l) is contmctible. Let M be a coefficient system on X such that the restriction of M to each component of p(i) - p(i-l) is constant. Then the inclusion p(O) ---t X induces an isomorphism

H.(P(O),M) -----) H.(X,M). PROOF. The filtration of X yields a filtration of C.(X,M). This gives a spectral sequence converging to H. (X, M) with E1-term having ith column

H. (F(i), F(i-l); M).

Consider the relative chain complex C.(F(i), F(i-l); M). By hypothesis, this chain complex is a direct sum of chain complexes with constant coefficients. Since each F(i) is contractible, it follows that H.(p(i),F(i-l);M)

= 0,

i ~ 1.

o

Thus, only the Oth column H. (F(O) ,M) is (potentially) nonzero.

The Solomon-Tits Theorem. In Chapter 3, we used the following fact. Let k be a field and let S be the partially ordered set of proper subspaces of k n , ordered by inclusion. Let T be the geometric realization of S. THEOREM

A.2.S. T is homotopy equivalent to a wedge of (n - 2)-spheres.

PROOF. We proceed by induction on n, beginning at n = 2. The only proper subspaces of k 2 are lines. It follows that T is simply a collection of points; that is, it is a wedge of O-spheres. Now assume that n ~ 3. Let l be a fixed line in kn. Let Y denote the set of hyperplanes in k n such that H + l = kn. Denote by So the complement S - Y and let To C T be the geometric realization of So. We claim that To is contractible. Define a poset map f : So ---t So by A 1--+ A + i. This map is well-defined since A +.e is a proper subspace for A E So. ~ote that f(A) ~ A for all A E So.

A.2.9. Let Pt and P2 be posets with Pt ~ P2' Let f : P l a map with f(s) ~ s for all s E Pl. Then IPll ~ If(Pdl· LEMMA

---t

P2 be

We need a homotopy F : IPll x [O,IJ ---t 1P21 with Fo = Iii and Pl = If I (here, i is the inclusion of Pl in P2). Triangulate !Pli x [0,1J in the usual way. The condition that f(s) ~ s implies that Iii x {OJ Illfl x {I} extends to all of !Pli x [O,IJ. 0 PROOF.

Now, the image of the map J defined above iH tho Het of HubHpaccH of k n which contain the liuc P. Tlw g()()tnotric realization of thiH iH dcmriy hOll1ot.opy

164

A. Homology of Discrete Groups

equivalent to the vertex £ E To (it is a minimal element of So). By the lemma, we have To ~ !(To) ~ {£}; that is, To is contractible. Now, if H is an element ofY, its link is, by definition, the set of subs paces of n k properly contained in H. The geometric realization of this link is isomorphic to the space T in dimension n-l (H is an (n-l)-dimcnsional k-vector space). By the induction hypothesis, this link has the homotopy type of a wedge of (n - 3 )-spheres. To complete the proof, note that T is obtained from To by attaching the links of all the hyperplanes in kn. Since To is contractible, we see that T is homotopy equivalent to a wedge of (n-2) spheres (i.e., when we contract To, we are getting the wedge of the suspensions of the links of the hyperplanes). 0

Appendix B

Classifying Spaces and K-theory This appendix gives a brief introduction to the general theory of classifying spaces and Quillen's definition of higher algebraic K-theory. Good references for this material are the books of Husemoller [59] and Srinivas [118].

B.l. Classifying Spaces Let F be a functor on some category of topological spaces (such as the category of finite CW-complexes). A classifying space for F is a space B together with an equivalence of functors F~[-,B];

that is, the set F(X) is the same as the set of homotopy classes of maps X - t B. In this section we find classifying spaces for principal G-bundles and for vector bundles of a given rank. This is a consequence of the more general notion of classifying space for small categories.

B.I.I. Principal Bundles. Let G be a group. A principal G-bundle consists of a locally trivial fibration p:E~B

with fiber G and a right G-action ExG~E.

Two such bundles are equivalent if there is a homeomorphism such that the diagram

f : El

-t

E2

commutes. DEFINITION

B.1.1. A classifying space for G is a space BG with a princi-

pal G-bundle p : EG

-----+

DG,

166

B. Classifying Spaces and K-theory

where EG is contractible, which is universal in the following sense: if q : E --+ B is any principal G- bundle, then there is a continuous map B --+ BG, unique up to homotopy, such that E is the fiber product E~EG

1

1

B~BG.

EXAMPLE B.1.2. Let U(n) be the group of unitary nxn complex matrices. Denote by S (n) the Stiefel manifold of unitary n- frames in Coo, and by G (n) the Grassmann manifold of n-planes in Coo. The space S(n) is contractible [119] and there is a fibration

U(n)

~S(n)

In G(n) where 7r sends a frame to the plane it spans. A proof that this U(n)-bundle is universal may be found in [59], p. 83. Thus, we have BU(n) = G(n). Moreover, since the inclusion U(n) --+ GLn(C) is a homotopy equivalence, we also have BGLn(C) = G(n). Similarly, if O(n) denotes the group of orthogonal n x n real matrices, then BO( n) is the Grassmannian of n-planes in lR. oo . REMARK B.1.3. Interestingly enough, the same space G(n) also classifies rank n complex vector bundles in the sense that if E .!!... X is such a bundle, there is a bundle map (unique up to homotopy)

where "In is the canonical bundle whose fiber over V E G(n) is V (that is, "In C G(n) x Coo is the set {(V, v) : v E V}). Similarly, rank n real vector bundles are classified by the real Grassmann manifold BO(n) (see [83], p. 61).

B.1.2. The Classifying Space of a Small Category. Let C be a small category (that is, the class of objects of C forms a set). Define a simplicial set NC, the nerve of C as follows. An n-simplex is a diagram in C of the form

hA 1 -----+ 12 A o -----+

in-l

• •• -----+

An -

in

1 -----+

An .

The ith face map applied to this simplex is

hA 1 -----+ 12 A o -----+

1;-1

. • . -----+

A i-I

fi+1 0 1; -----+

A 'i + 1

1;+2

in

-----+ • • • -----+

An ,

167

B.1. Classifying Spaces

and the ith degeneracy is hA 1 -----+ h A o -----+

Ii . • • -----+

A i --+ Id A i

in -----+ • • • -----+

ii+l

An .

DEFINITION B.1.4. The classifying space BC of C is the geometric realization of the simplicial set NC (see [118], Chapter 3). Let G be a topological group (perhaps with the discrete topology). Denote by G the category with a single object e with morphism set equal to G. Composition of morphisms is given by the group operation. The n-simplices of NG are n-tuples (gl, ... ,gn) of elements of G with ith face map (g1, ... , gn)

I-t

(g1, ... , gigi+ 1, ... , gn).

Consider the classifying space BG. THEOREM B.1.5 ([109]). If G is an absolute neighborhood retract (e.g., G discrete or G a Lie group), then BG is a classifying space for G on the category of paracompact spaces. CJ THEOREM B.1.6. Let G be a discrete group. Then the space BQ is an Eilenberg-MacLane space K(G,l), and hence is homotopy equivalent to the space BG defined in Appendix A. PROOF. Let G be the category with object set G and morphism set G x G; that is, if g1,g2 E-G, then HomQ(g1,g2) = (g1,g2)' There is a functor Q - Q defined by 9 I-t ej (g1,g2) I-t g2g]"1 E Homq(e,e). The group G acts on G by g.h = hg-I, g.(g1,g2) = (g1g-l,g2g- 1 ). This action is free and hence-G acts freely on BG. The induced map EO. -+ EO. is G-equivariant for the trivial action on BG-and so BG is a covering space of DG with fiber G. Since G has an initial object (any object is initial), the space BG is contractible. Thus, 7r1(BG) = G and all the higher homotopy groups vanish. Moreover, since BG -+ BG is a principal G-bundle, there is a map BG -+ BG which is an isomorphism on homotopy and hence is a homotopy equivalence. 0 REMARK B.1.7. The standard resolution of Z over ZG (see Appendix A) is the chain complex associated to the simplicial abelian group obtained by applying the free abelian group functor dimensionwise to the simpliCial set NG. The "simplex" X mentioned in Appendix A is the space EG. Now, if G is a topological group, denote by G li the group G viewed ~lS a discrete group. The identity map Gli -+ G is continuous (it is not continuous in the other direction) and hence induces a map BGli -+ BG of cla.'lsifying Spac:(lS. Moreover, any homomorphism G -+ H induces a map BG -+ DH (this WH.'l not obvious for t.he cO!lstmttjoll of nG !(iVell in Appondix A),

B. Classifying Spaces and K-theory

168

B.2. K-theory B.2.1. Topological K-theory. Topological K-theory is a generalized cohomology theory on the category of CW-complexes. Let F denote JR or C and denote by VectF(X) the set of isomorphism classes of F-vector bundles on X. This set is a semiring with addition given by Whitney sum of bundles and multiplication given by tensor product. The trivial bundle of rank zero is the additive identity and the trivial line bundle is the multiplicative identity. Choose a basepoint x E X. There is a semiring map rk: VectF(X) ~ Z defined by rk(E)

= dimF(Ex )

where Ex is the fiber over x. The rank of a vector bundle is constant on each connected component of X so that if X is connected, this function is independent of the choice of basepoint. Recall that if R is a semiring, there exists a ring S called the ring completion and a map R ~ S which satisfies the obvious universal property. DEFINITION B.2.1. The ring K~(X) is defined to be the ring completion of VectF(X), The functor K~ is contravariant in X. Indeed, if f then we define f* : K~(Y) ~ K~(X) by

:X

~ Y is continuous,

f*([E]) = [f*(E)],

where for a bundle E, [E] denotes the isomorphism class of E and f*{E) denotes the pullback of E. The map rk induces a ring homomorphism rk : K~(X) ~

Z.

DEFINITION B.2.2. The reduced K-theory of X is K~(X) = ker(rk : K~(X) ~ Z).

The clements of K~(X) may be characterized as follows. Call two vector bundles E, E' over X stably equivalent if there are trivial bundles 17,17' with E E£) 11 9!! E' E£) ",'. This is clearly an equivalence relation on VectF(X), THEOREM B.2.3 ([59], p. 105). Let X be a space such that for each vector bundle E over X there exists a bundle E' with EffiE' trivial. Then the elements of K~(X) are in one-to-one correspondence with stable equivalence classes in ~~~.

0

The functor K~ has a classifying space defined as follows. For each n, consider the Grassmannian G(n, F2n) of n-planes in F2n and denote by Bp the union Un>! G(n, F2n). If F ::= JR, this is usually denoted BO and if F = C it is denoted BU.

B.2. K-theory

169

THEOREM B.2.4 ([59], p. 107). Let X be a finite connected CW-complex. Then there is a natural isomorphism

K~(X) ~ [X, BFJ.

Thus, giving a stable equivalence class of an F-vector bundle on X is the same as giving a homotopy class of maps X ---+ B F. REMARK B.2.5. The notation K~ indicates that there are functors K}. This is indeed the case, but we shall not describe them here.

B.2.2. Algebraic K-theory. The definition of the group Ko(R) for a ring R is classical and dates back to the work of Grothendieck on the Riemann-Roch problem. The definition is quite easy to state, yet the group Ko(R) is often very difficult to compute. Let R be a ring. The set of isomorphism classes of finitely generated projective R-modules forms a monoid with addition given by direct sum and identity element the trivial module. If R is commutative, then we have a multiplica.tion given by tensor product with identity the rank one free module R. We define Ko(R) to be the completion of this monoid (semiring if R is commuta.tive). EXAMPLE B.2.6. If R is a field, then finitely generated projective modules are free and are determined up to isomorphism by rank. Thus, Ko(R) = Z. If R is the ring of integers in a number field, then Ko (R) = Z EB Cl( R), where CI(R) is the ideal class group of R.

Loosely speaking, projective modules over R correspond to locally free sheaves (= vector bundles) on Spec R. This is the intuitive idea behind the following result, due to Swan [121].

B.2.7. Let F

= IR

or C, let X be a finite connected CWcomplex, and let R = CF(X) be the ring of continuous F-valued functions on X. If E is a vector bundle on X, let THEOREM

r(X, E) = {s : X

-4

E: po s

= idx }

be the set of continuous sections of E. Then r(X, E) is a finitely generated projective R-module. Moreover, the map E 1--+ r(X, E) induces an isomorphism of categories from the category VectF(X) to the category of finitely generated projective R-modules and hence induces an isomorphism K~(X) --I Ko(R}.

o The groups K 1 (R) and K 2 (R) are also classical. Denote by GL(R} the infinite general linear group and by E(R} the subgroup generated by elementary matrices. By the Whitehead lemma, the group E(R) iH a perfoct normal subgroup equal to tho commutator subgroup of GL(R). The group K 1 (R) is defined H.."l GL(R)/ E(R), t.he ~lhelinnbllltio!l of GL(R). Milnor cldinf'd tlw

B. Classifying Spaces and K-theory

170

group K 2(R) to be H2(E(R), Z). When R is a field, we have the following presentation of K2(R), due to Matsumoto (see, e.g. [80]). THEORE:vI B.2.8. Let F be a field. Then the group K2(F) has a presentation with generators {x, y}, where x, y E F x, and relations 1. {x,1-x}=1forx#0,1, 2. {XIX2,y} = {Xl,y}{X2,y},

0

3. {x, Y1Y2} = {x, Yd{x, Y2}.

Furthermore, there are relative groups Ki(R,I), for i = 0,1, defined for an ideal I C R such that there is an exact sequence K 2(R)

---->

K2(RjI)

---->

K 1(R,I)

---->

K 1(R)

---->

Kl(RjI)

0. Until the work of Quillen, no one knew how to define higher K-groups Ki which extend this sequence to the left. Quillen gave two definitions of algebraic K-theory: the +-construction and the Q-construction. A hard theorem asserts that these two definitions give the same result. While K-groups can be defined for schemes, we shall concentrate on the definition for rings using the +- construction. Let R be a ring and consider the classifying space BGL(R) of the discrete group GL(R). Quillen defines a space BGL(R)+ and then defines ---->

Ko(R,I)

---->

Ko(R)

---->

Ko(RjI)

---->

Ki(R) = 11"i(K(R)),

where K(R) = BGL(R)+ x Ko(R). We describe this construction in the proof of the following result. THEOREM B.2.9. Let X be a connected CW-complex with basepoint Xo and let 11" be a perfect normal subgroup of 11"1 = 11"1(X,XO). Then there exists a CW-complex X+, which is obtained from X by attaching only 2-cells and 3-cells such that 1. The map

11"]

(X, xo)

----> 11"1 (X+

, xo) is the quotient map

11"1 ----> 11"1 j 11";

2. For any 11"I/11"-module M, we have H.(X+, X; M), where M is viewed as a local coefficient system on X+. Moreover, X+ is unique up to homotopy equivalence.

PROOF. Let hdiEI be a set of generators for 11" and let gi : (8 1 , *) ----> (X, xo) be a representing map for "Ii. Since the kernel of the Hurewicz map 11"1 ----> HI (X, Z) is the commutator subgroup [11"1,11"1], each map gi is trivial on homology. Let Y be the complex obtained by attaching a 2-cell eT for each i E I via the maps gi : aeT = 8 1 ----> X. The inclusion X ----> Y clearly satisfies 1 above.

171

B.2. K-theory

Let X ---+ Y be the covering spaces with covering group trd 7r so that Y is the universal cover ofY and 7rl(X) = 7r. Since 7r is perfect, H1(X,'1.,) = 7ra.b = O. The relative homology H.(Y,X;Z) is concentrated in degree 2, where it is the free abelian group on the [en Similarly, H.(Y, X; Z) is concentrated in degree 2 where it is the free Z[7rd7rl-module on the [ell. Since the connecting map () : H 2 (Y,X;Z) ---+ H1(X,Z) = 0 is trivial, H.(Y,Z) differs from H.(X,Z) by adding EBiEI Z[7rd7r][ell in degree 2. Since Y is simply connected, [e~l is in the image of the Hurewicz map 7r2(Y) ---+ H 2(Y, '1.,) and similarly for Y (by pushing down). Let hi : (8 2 , *) ---+ (Y,xo) be an element mapping to [en via the Hurewicz map. Attach a 3-cell ef to Y via hi for each i E 1 to obtain a space X+. Property 1 is clear. Property 2 is easy to check. The uniqueness of X+ is left to the reader. 0 Now, since BGL(R)+ is path connected, we have 7ro(K(R)) = Ko(R) and 7rl(K(R)) = 7rl(BGL(R)+) = GL(R)/ E(R) = Kt(R). Moreover, it is not difficult to see that K 2 (R) agrees with Milnor's definition. Let 1 c R be an ideal. If we define K(R,1) to be the fiber of the map K(R) ---+ K(R/1), then we obtain a long exact sequence of K-groups ... ---+

K i +1(R/I)

---+

Ki(R,I)

---+

Ki(R)

---+

Ki(R/I)

---+ ••••

Moreover, the definition of Ki (R, 1) agrees with the classical definition for i = 0,1.

We also have the following result, due to Quillen, which is known as the fundamental theorem of algebraic K-theory. THEOREM

B.2.10. Let R be a regular ring. Then there are natural isomor-

phisms Ki(R[t]) ~ Ki(R)

and Ki(R[t,C 1]) ~ Ki(R)

Ef)

Ki-1(R).

By construction, for any ring R there is a Hurewicz map hi: Ki(R)

---+

Hi(GL(R),'1.,).

This provides the motivation for studying the homology of linear groups over R. If we tensor the above groups with Q, the resulting map is injective with image equal to the primitive elements. Thus, computations of H.(GL(R), '1.,) can give information about K.(R).

Appendix C

,

Etale Cohomology This appendix provides a quick summary, mostly without proofs, of the basics of etale cohomology. A good reference for this material is J. Milne's book [79].

C.l. Etale Morphisms and Henselian rungs C.l.l. Etale morphisms. DEFINITION C.1.1. A morphism of schemes f : X ~ Yis affine if J- 1 (U) is an open affine subset of X for every open affine U ~ Y. If, in addition, rU- 1 (U), Ox) is a finite r(U, Oy)-algebra for each U, then we say that J Is finite. PROPOSITION C.1.2. (a) A closed immersion is finite. (b) The composite of two finite morphisms is finite. (c) Any base change of a finite morphism is finite. PROOF. It suffices to consider opens in some affine covering of the target. These statements then translate into statements about rings which are obvious. For example, (b) boils down to showing that a finite extension of a finite extension is finite. Statement (c) asserts that if f : X ~ Y is finite and if Z ~ Y is any morphism, then the induced morphism X x y Z ---+ Z is also finite: XXyZ-X

1

11

Z J Y. This reduces to an obvious statement about tensor products. PROPOSITION C.1.3. Any finite morphism f : X it is separated, of finite-type, and universally closed.

~

o

Y is proper; that is,

PROOF. This may be found in [79], pA. Recall that f is separated if the diagonal morphism ~:X----+XXyX

is a closed immersion. It is of finittl-type if there exists an open affine covering of Y by subsets ~ = speC!(IJi ) such that for (ladl i, f-l(~) mIl b(~ covered hy

174

C. Etale Cohomology

a finite number of affine subsets U ij = spec (Aij ), where each Aij is a finitely generated Bi-algebra. The map f is universally closed if it is closed and if any D base change of f is closed. Finite morphisms over spec(k), where k is a field, have a particularly nice description. PROPOSITION C.1.4. Let f : X ~ spec(k) be a morphism of finite-type. The following are equivalent. 1. X is affine and f(X, Ox) is an Arlin ring; 2. X is finite and discrete as a topological space; 3. X is discrete; 4. X is finite. D DEFINITION C.1.5. A morphism f : X ~ Y is quasi-finite if it is of finite-type and has finite fibers. Similarly, an A-algebra B is quasi-finite if- it is of finite-type and if B 0A k(p) is a finite k(p)-algebra for all prime ideals peA (here, k(p) is the fraction field of the domain Alp). Clearly, any immersion is quasi-finite, as the composition of two quasifinite morphisms. Also, any base change of a quasi-finite morphism is quasifinite. We know that finite morphisms are proper. Conversely, proper morphisms which are quasi-finite are finite. This is a consequence of Zariski's main theorem [79], p. 6. Recall that a homomorphism of rings f : A ~ B is flat if B is a flat A-module (via f); that is, the functor - 0A B is exact. Flatness is preserved by localization. DEFINITION C.1.6. A morphism f : X ~ Y of schemes is flat if for all x E X, the induced map OY,f(x) ----+ Ox,x is flat. Equivalently, f is flat if for any pair of open affines U ~ X, V ~ Y, with f(U) ~ V, the map f(V, Oy) ----+ qu, Ox) is fiat. As one might expect, open immersions are fiat, as is the composition of two fiat morphisms. A base change of a flat morphism is fiat. EXAMPLE C.1.7. If A is any ring, then A[Xl, ... ,Xn ] is a free A-module; thus, AA is fiat over spec(A). More generally, let Z c AA be a hypersurface; that is, Z is the zero set of a single nonzero polynomial P. Then Z is fiat over spec(A) if and only if the ideal generated by the coefficients of P is A. This may be restated by saying that Z is flat if and only if its closed fibers over spec(A) have the same dimension. Thus, flatness is the algebraic analogue of a continuously varying family.

C.l. Etale Morphisms and Henselian Rings

175

DEFINITION C.1.8. A flat morphism f : A -----+ B is faithfully flat if B ®A M is nonzero for any nonzero A-module M. Such a map is necessarily injective (take M = (a), where a E A). Note that the associated map f* : spec(B) spec(A) is surjective. A morphism f : X -----+ Y of schemes is faithfully fiat if it is flat and surjective. . DEFINITION C.1.9. Let k be a field with algebraic closure k. A k-algebra A is separable if A = A®k k has zero Jacobson radical; that is, the intersection of the maximal ideals is zero. DEFINITION C.1.10. A morphism f : Y -----+ X that is locally of finite-type is unramified at y E Y if OY,y/mxOy,y is a finite separable field extension of k(x) = OX,x/mxOx,x. where x = f(y). In terms of rings, a homomorphism f : A -----+ B of finite type is unramified at q E spec(B) if and only if p = f- 1 (q) generates the maximal ideal in Bq and k( q) is a finite separable field extension of k(p). A morphism f : Y -----+ X is unramified if it is unramified at each yEY. REMARK

C.1.11. Any closed immersion is unramified.

DEFINITION C.1.12. A morphism f : Y unramified (hence also locally of finite-type).

-----+

X is etale if it is flat and

Clearly, any open immersion is etale, as is the composition of two ~tale morphisms. Etale morphisms are also preserved by base change. Etale morphisms are the algebraic analogue of local isomorphism!!. Let A be a ring and consider the ring A[tl,"" tn ]. Let PI"'" P", be polynomials and let B = A[tl, ... , tn]/(Pt, ... , Pn). Then B is etale over A if and only if the image of det(8~/Otj) in B is a unit. If Y = spec(B) and X = spec(A) are analytic manifolds, then this criterion means that the induced map!! on tangent spaces are isomorphisms. This is made precise by the following re!!ult. PROPOSITION C.1.13. A morphism f : Y ---+ X is etale if and only if for every y E Y, there exist open affine neighborhoods V = spec( C) of y and U = spec(A) of x = f(y) such that for some polynomials PI, .. . ,Pn ,

C = A[tt, ... , tnl/(P1 , ... , Pn ) and det(8Pi/Otj) is a unit in C.

PROOF. See [79], Corollary 3.16, p. 27.

o

C.1.2. Henselian rings. Let A be a local ring with maximal ideal m and fe!!iduc field k. Denote the maps A -+ k and A[t] --+ k[t] hy a ~ a and f 1--+ 7. Two polynomial"! f,g E 8[t] (wlwrc n is any ring) mo .~t'T"ictly copr'irru~ if tho idElal genera.t()d hy .f, lJ is BU]·

C. Etale Cohomology

176

DEFINITION C.1.I4. A local ring A is Henselian if the following condition holds: If f is a monic polynomial in A[t] such that 1 factors as 1 = goho with go and ho monic and coprime, then f factors as f = gh where g and h are monic and 9 = go, Ii = h o· PROPOSITION PROOF.

C.1.I5. Any complete local ring is Henselian. D

See [79], p.35.

Let A be a local ring and denote by A its m-adic completion. Since A is a subring of A, we see that A is a subring of a Henselian ring. The smallest such ring is called the Henselization of A; denote this ring by A h. The ring A h satisfies the obvious universal mapping property; namely, if j : A --t R is a homomorphism where R is Henselian, then there is a unique map Ah --t R such that the diagram

A-Ah

~!R commutes. This characterizes A h uniquely, provided it exists. DEFINITIO~ C.1.16. Let A be a local ring. An etale neighborhood of A is a pair (B, q) where B is an etale A-algebra and q is a prime ideal lying over m such that the induced map k --t k( q) is an isomorphism.

One checks easily that the etale neighborhoods of A with connected spectra form a filtered direct system. Let the ring (A h, mh) be the limit of this system:

(A\ mh) = !!m(B, q). It is not hard to check that Ah is local with Ah /mh Henselization of A. EXAMPLE

= k, and that Ah is the

C.l.I7. Let k be a field, and let A be the localization of

k[tl, ... , tnJ at (tl,"" t n ). Then the Henselization of A is the ring of power series P E k[[tl' ... ,tnll that are algebraic over A. DEFINITION C.1.I8. Let X be a scheme and let x E X. An etale neighborhood of x is a pair (Y, y), where Y is an etale X-scheme and y is a point of Y mapping to x such that k(x) = key). The connected etale neighborhoods of x form a filtered system and !!m r(Y, Oy) = O~,x'

Note that by definition a Henselian ring A has no finite etale extensions with trivial residue field extensions except those of the form A --t Ar for some r. Thus, if the residue field of A is separably algebraically closed, then A has no finite etale extensions. Such a ring is called strictly Henselian. Every local

C.2. Etale Cohomology

177

ring A has a strict Henselization Ash satisfying the obvious universal mapping property. It can be constructed as follows. Fix a separable closure ks of k. Then Ash = ~B, where the limit runs over all diagrams

B--ks

1/

A

with A --t B etale. If A = k is a field, then Ash is any separable closure of k. Let x --t X be a geometric point of a scheme X. An etale neighborhood of x is a commutative diagram

x--u

~!

X

with U --t X etale. Then OJl';x = ~r(U,Ou), where the limit is taken over all €talc neighborhoods of x. This is the analogue for the etale topology of the local ring for the Zariski topology. Indeed, the two definitions are the same: take the direct limit over "open" sets containing x.

C.2. Etale Cohomology C.2.t. Sheaves. We assume that the reader is familiar with the concept of a sheaf on a topological space. It is possihle to define "topologies" on the category of schemes which are more general than the standard Zariski topology. With this generalized notion of "open covering" , we can extend the formal properties of sheaves to obtain sheaves for these topologies. Let E be a class of morphisms of schemes satisfying 1. All isomorphisms are in E; 2. The composite of two morphisms in E is in E; 3. Any base change of a morphism in E is in E. The full subcategory of Schj X (the category of schemes over a fixed base scheme X) whose structure morphism is in E will be denoted by E j X. Ther(l three obvious examples: 1. E = (Zar) consists of all open immersions;

2. E = (et) consists of all etalc lJIorphisms of finite-type; 3. E = (fl) conHiHtH of all flat morphismH locally of finite-type.

C. Etale Cohomology

178

Fix a base scheme X, a class E of morphisms, and a full subcategory C / X of Sch/ X that is closed under fiber products and is such that for any Y ---+ X in C/X and any E-morphism U ---+ Y, the composite U ---+ X is in C/X. An E-covering of Y E C / X is a family (Ui ~ Y)iEI of E-morphisms such that Y = Ugi(Ui ). The class of all such coverings is the E-topology on C/X. The category C / X with the E-topology is called the E-site and is denoted by (C/X)E or XE. A presheaf P on a site (C / X) E is a contravariant functor C / X ---+ Ab, where Ab is the category of abelian groups. A prcsheaf P is a sheaf if it satisfies 1. If s E P(U), and (Ui ---+ U)iEI is a covering of U such that resu.,u(s) = 0 for all i, then s = OJ 2. If (Ui ---+ U)iEI is a covering and the family (SdiEI' Si E P(Ui ) is such that

resuiXUUj,U.(Si) = resu.XUuj,Uj(Sj) for all i,j E I, then there exists s E P(U) such that resu.,u(s) all i E I.

= Si for

Note that these axioms are the usual sheaf axioms in the case E = (Zar)j indeed, the fiber product Ui Xu Uj is just the intersection Ui n Uj. The usual properties of sheaves on a topological space generalize to this setting. For further details, see Chapter II of [79]. C.2.2. Cohomology. Let A be an abelian category. An object I of A is injective if the functor

M

1--+

HomA(M, I)

is exact. The category A has enough injectives if for every M in A there is a monomorphism M ---+ I, with I injective. If A has enough injectives and if f : A ---+ B is a left exact functor into another abelian category, then there are functors Ri f : A ---+ B satisfying 1. R O f = fj 2. f(I) = 0 if I is injective and i > OJ 3. For any exact sequence 0 ---+ M' ---+ M ---+ M" ---+ 0 in A, there are morphisms 8i : f(M") ---+ Ri+l f(M'), i ~ 0 such that the sequence

m

m

••• ---+

Ri f(M)

---+

Ri f(M")

.! Ri+l f(M') ---+ Ri+l f(M) ---+ •••

is exact. Moreover, this construction is functorial.

The derived functors are defined as follows. If MEA, choose an injective resolution O--+M--+Io--+h--+'"

C.2. Etale Cohomology

179

(that is, each I j is an injective object of A). Such resolutions are essentially unique. Consider the complex

C :0

-----+

f(Io) -----+ f(h)

-----+ ••••

The objccts Rf f(M) are the cohomology objects of this complex:

Ri f(M) = Hi(C). PROPOSITION C.2.I. Let X be a scheme and let XE be the E-site on X. Denote by S(XE) the category of sheaves of abelian groups on X E . Then S(XE) has enough injectives.

0

PROOF. See [79], p. 83. DEFINITION C.2.2. (a) The global sections functor

r(X,-): S(XE) __ Ab with r(X, F) = F(X) is left exact and its derived functors are written

Rir(X, -) = Hi(X, -) = Hi(XE, -). The group Hi(XE, F) is called the ith cohomology group of XE with values in F. (b) For any U ---t X in C/X, the derived functors of F 1--+ F(U) are written

Hi(U,F). (c) For any fixed sheaf F o on XE, the functor F

1--+

Horns (Fo, F)

is left exact. Its derived functors are written Ext~(Fo, -). REMARK C.2.3. 1. Hi(XE, F) is a contravariant functor on XE; that is, if 71"* : S(XE) ---t S(XE,) is exact, then the maps Hi (XE, F) ---t Hi(XE" 71"" F) are induced by the obvious map HO(X,F) ---t HO(X', 7r" F), where 7r : X' - X. 2. There is an isomorphism of functors r(X, -) ~ Hom(Z, -), where Z is the constant sheaf on XE. It follows that Hi(X, -) ~ Exti(Z, -). EXAMPLE C.2.4. Let X = spec(k), where k is a field, and consider the etale site on X. Let ks be a separable closure of k and let G be the Galois group of ks over k. Then there is an isomorphism

S(XE )

~

G - mod

of categories, where G-mod is the category of continuous G-modules. If the sheaf F corresponds to the module M, then reX, F) = MG and

Hi(X, F)

= Hi(G, M) = Hi(k, M),

where the groups OIl the right are the Galois cohomology groupH of k with coefficients ill M. These are defined to be Hi (G( k' / k), MAl' ), where k' / k iH a finite galois ext.em!iou of k (note t.hat k" = Uk').

lim

C. Etale Cohomology

180 EXAYIPLE

C.2.5. (Hilbert's Theorem 90) The canonical map HI (XZafl O~)

------>

HI (Xet' rG m

)

is an isomorphism, where rG m is the sheaf of multiplicative groups. It follows that HI (Xct, rG m ) = Pic(X). The functors H~t(X, -) = Hi(Xct, -) satisfy many of the properties of a cohomology theory, such as excision. The cohomology groups Hi(XE' -) agree, in most cases of interest, with the Cech cohomology groups defined in the usual manner via coverings of X. This is the case, for example, if F is a quasi-coherent sheaf of Ox-modules in the Zariski topology, assuming X is separated. The same is true for etale cohomology if the following condition holds: X is quasi-compact and every finite subset of X is contained in an affine open subset (e.g., X quasi-projective over an affine scheme). As usual, there is a spectral sequence relating Cech cohomology to derived functor cohomology. For full details, see [79], Chapter III, Section 2. Given two topologies on C j X, one would like to know the relationship between the two cohomology theories. We present one such result, which is used in Chapter 5. THEOREM C.2.6. Let X be a smooth scheme over
Hi(X(C), M) ~ Hi(Xct , M). REMARK C.2.7. This is absolutely not true with integral coefficients. For example, if X is a smooth complete curve over
HI(X(C),Z)

=

Z2g

=

Hom(JrI(X),Z),

For a scheme X, there is an algebraic fundamental group Jrflg(X) which classifies etale coverings of X ([79], Chapter I). We have HI(Xet' Z) = Homcont(Jrflg(X), Z) = 0 in this case since for a complex variety, the group Jrf1g(X) is the profinite completion of the topological fundamental group, and hence has only finite discrete quotients. However, we do have

(Zjn)2g Homcont(Jrflg(X), Zjn)

HJt(X, Zjn). Let l be a prime number. A sheaf F is torsion (resp. l-torsion) if, for all quasi-compact U, F(U) is torsion (resp. l-torsion).

181

C.3. Simplicial Schemes

PROPER BASE CHANGE THEOREM C.2.8 ([79], p.224). Let kcK beseparably closed fields and let X be a scheme which is proper over k. Let XK be the scheme X x It; K and let F be a torsion sheaf on X et . Then the natural map

Hi(X, F) ~ Hi(XK,FlxK) is an isomorphism for i

~

o

O.

C.3. Simplicial Schemes Most of the above theory can be extended to simplicial schemes. A thorough discussion of this may be found in [40]. The etale site of a simplicial scheme X., denoted Et(X.), is the category whose objects are etale maps U - Xn for some n ~ 0, and whose maps are c0IIl:mutative diagrams U

1

>V

1

Xn-Xm

where the bottom arrOlN is a structure map of X •. A covering of an object U - Xn is a family of etale maps (Ui ~ U)iEI over Xn with U = Ug~(U~). Note that this definition does not imply that there is a map of simplicial schemes U. - - X. given in dimension n by U -'> X n . A presheaf on Et(X.) is a contravariant functor on Et(X.); that is, to each object U -'> Xn we associate a set (or group, ring, etc.) P(U). The presheaf F is a sheaf if the usual axioms hold. The category of sheaves of abelian groups on Et(X.) has enough injectives. DEFINITION C.3.1. Let X. be a simplicial scheme. For i mology group functor

~

0, the coho-

Hi(X., -) is the ith derived functor of the functor sending a sheaf F to the abelian group given as the kernel of the map

do - dr : F(Xo) ~ F(Xl)' Equivalently, Hi(X., -) = Ext~(x.)(Z, -), where Z is the constant sheaf.

As usual, there is a definition of Cech cohomology via coverings. This theory agrees with the above definition if, for example, each Xn is quasiprojective over a noetherian ring. The reader is referred to Chapter 3 of [40] for further details. Alfm, the cohomology of a simplicial scheme over C agrees with that of the corresponding simplicial manifold with coefficients in a torsion sheaf. Moreover, the proper hase change theorem holds in this more general context as well.

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Index Adams operation, 2 affine group, 38, 44 algebraic K-thcory, 169 amalgam, 112 arithmetic subgroup, 12, 19, 22

etale etale etale etale etale

bar resolution, 151 Becker-Gottlieb transfer, 119 Beilinson's conjecture, 103 Bloch group, 70 is uniquely divisible, 81 Bloch invariant, 88 Brauer lifting, 4 Bruhat-Tits building, 95

finite subgroup conjecture, 129 five-term relation, 70 frame, 35 Friedlander-Milnor conjecture, 117 Frobenius map, 4, 124 functor with transfers, 132

center kills, 39, 42, 97, 154 Chern class, 3, 13 Chern-Wei! map, 121 Chern-Simons invariant, 88 classifying scheme, 117 classifying space, 149, 165 coefficient system, 98, 162 coinvariants, 152 cone, 76 conjugate homomorphisms, 14 continuous cohomology, 21, 28 cross ratio, 87 cuspidal cohomology, 22 Dedekind domain, 58, 99 divided power algebra, 40 Eilenberg-Moore spectral sequence, 3 elementary abelian Rubgroup, 13 equivariant homology, 161

approximation, 15 cohomology, 117, 173 homotopy theory, 14 K-theory, 17 morphism, 173

G-invariant elements, 127 general position, 47 Grassmannian, 1, 13, 166 HI-ring, 83 Hensel ring, 83, 175 henselization, 139 higher Bloch groups, 84 Hochschild-Serre spectral sequence, 159 homotopy invariance, 110 homotopy invariant functor, 132 ideal triangulQ.tion, 87 indecomposable part, 80 interior disjoint cells, 65 jointly unimodular, 35 K-theory of an algebraically closed field, 12, 133 of an olliptic curve, 102

192 of a finite field, 11 fundamental theorem, 171 of a local ring, 56 of number rings, 20 Lang isomorphism, 124 lattice, 95 Lie group cohomology, 117 link, 35 Malcev completion, 27 Milnor K-theory, 46 monomial matrices, 71 Nagao's theorem, 96 orthogonal group, 56 +-construction, 170 polar basis, 57 polytope, 65 pre-Bloch group, 70 presheaf, 178 principal bundle, 165 principal congruence subgroup, 23 projective resolution, 150 proper base change theorem, 181 pseudonilpotent group, 28

Index singular simplices, 141 site, 178

SL2(kl:t]), 96 SL2(k[t, rl D, 97 SL2('L),91 SL2('L[1/pJ), 92 SLa('L), cohomology of, 106 SLn(k[tJ), homology of, 107 solvable Lie group, 119 special unimodular frame, 43 specialization map, 134 spectral sequence, 156 split building, 58 stability, 33 for local rings, 47 stable rank, 34 Steinberg module, 67 Tits building, 57, 67 topological K-theory, 168 totally isotropic subspace, 57 transfer, 155 transversal frames, 36 twisted coefficients, 38, 113 ultrafilter, 131 ultraproduct, 131 unimodular vector, 34

Quillen's conjecture, 12 van der Kallen's theorem, 37 rank conjecture, 61 rank filtration, 61, 103 regulator map, 88 relative completion, 26 relative homology, 153 rigid functor, 132 rigidity, 132 ring with many units, 35, 38 S(n) ring, 38 S-integers, 13 scissors congruence, 65 scissors congruence group, 66 Shapiro's lemma, 155 sheaf, 178 simple pasting, 65 simplicial scheme, 181

wreath product, 5