Introduction to Algebraic K-Theory. (AM-72), Volume 72 9781400881796

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Annals of Mathematics Studies Number 72

INTRODUCTION TO ALGEBRAIC ^-THEORY BY

JOHN MILNOR

PRINCETON UNIVERSITY PRESS AND UNIVERSITY OF TOKYO PRESS PRINCETON, NEW JERSEY 1971

Copyright ©

1971, by Princeton University Press ALL RIG HTS RE SER VE D

L .C . C ard : 7 4 -1 6 1 1 9 7

I.S .B .N . : 0 -6 9 1 -0 8 1 0 1 -8 A .M .S. 1 9 7 0 : P rim ary, 16A 54; Secondary, 10A 15, 13D 15, 1 8 F 2 5 , 2 0 G 1 0

Published in Japan exclusively by University of Tokyo Press; in other parts of the world by Princeton University Press

Printed in the United States of A m erica

To Norman Steenrod

P R E F A C E AND GUIDE TO THE L IT E R A T U R E The name “ algebraic K-theory” d escrib es a branch of algebra which cen ters about two functors K Q and K 1? which a ssig n to each a ss o cia tiv e ring A an abelian group K QA or K jA resp ectiv ely.

The theory has been

developed by many authors, but the work of Hyman B a s s has been particu­ larly noteworthy, and B a s s ’s book A lg e b r a ic K-th eo ry (Benjam in, 1968), is the most important source of information. Here is a sele c te d lis t of further references: D. S. Rim, M odules o v e r fin it e grou ps, Annals of Math. 69 (1959), 700-712. R. Swan, P r o je c t iv e m o d u les ov er fin it e grou p s, Bull. Amer. Math. Soc. 65 (1959), 365-367. H. B a s s , K-theory and stab le algebra, Publ. Math. I.H .E .S . 22 (1964), 5-60. H, B a s s , A. H eller, and R. Swan, T h e W h iteh ead group o f a p oly n o m ia l e x ten s io n , Publ. Math. I.H .E .S . 22 (1964), 61-79. H. B a s s , T h e D ir ic h let unit theorem , in d u c ed c h a r a c te r s , an d W h iteh ead grou ps o f fin it e grou ps, Topology 4 (1966), 391-410. H. B a s s (with A. Roy), L e c t u r e s on to p ic s in a lg e b r a ic K-th eory , T ata Institu te, Bombay 1967. H. B a s s and M. P . Murthy, G ro th en d ieck g rou p s an d P ic a r d grou ps o f a b e lia n group rin gs, Annals of Math. 86 (1967), 16-73 R. Swan, A lg e b r a ic K -theory, L ectu re Notes in Math. 76, Springer 1968. R. Swan (with E . G. E v an s), K -theory o f fin it e g rou p s an d ord ers, L ectu re Notes in Math. 149, Springer 1970.

vii

viii

PREFACE

L . N. V aserstein , On the s t a b iliz a tio n o f th e g e n e r a l lin ea r group o v er a ring, Mat. Sbornik 79 (121), 405-424 (1969). v.

8

(T ran slation

, 383-400 (A .M .S.).)

The main purpose of the present notes is to define and study an analo­ gous functor K2 , also from a sso cia tiv e rings to abelian groups. definition is suggested by work of Robert Steinberg.

The

T h is functor K 2 is

related to K 1 and KQ for example bymeans of an ex a ct sequence K2q

->K2A -> K 2 (A /a)

-» K t a

->K xA ->K x(A /a)

-> K 0a

- K0A

KQ(A / a ),

a sso ciate d with any two-sided ideal a in the ring A; where K 2 and

K QCt

0

, K j Cl

are suitably defined relative groups. Here is a lis t of referen ces

for K2 : R. Steinberg, G en erateu rs, r e la tio n s e t rev e te m e n ts d e g ro u p es a lg e b r iq u e s , Colloq. Theorie des groupes algebriques, B ru xelles 1962, 113-127. R. Steinberg (with J . Faulkner and R. Wilson), L e c t u r e s on C h e v a lle y grou ps (mimeographed), Y ale 1967. C.

Moore, Group e x t e n s io n s o f p -a d ic an d a d e lic lin ea r g rou ps,

Publ. Math. I.H .E .S . 35 (1969), 5-74. H. Matsumoto, Sur l e s s o u s-g ro u p es a r ith m etiq u es d e s g ro u p es s e m i-s im p le s d e p lo y e s , Ann. Sci. E c. Norm. Sup. 4 e serie , 2 (1969), 1-62. H. B a s s , K 2 an d sy m b o ls, pp. 1-11 of L ectu re Notes in Math. 108, Springer 1969. M. Kervaire, M u ltip licateu rs d e Schur e t K ~ theorie, pp. 212-225 of E s s a y s on T o p o lo g y an d R e la t e d T o p ic s , dedicated to G. de Rham (ed. A. H aefliger and R. Narasimhan), Springer 1970. J . Wagoner, On K 2 o f the L au ren t p oly n o m ia l ring, to appear. B. J . B irch,

o f g lo b a l fie ld s , Proc. Symp. Pure Math. 20,

Amer. Math. Soc. 1970.

ix

PREFACE

J . T a te , S y m bols in arith m etic, Proc. Int. Congr. Math. N ice, to appear. M. Stein, C h e v a lle y g rou p s o v er com m u tativ e rin gs.

B u ll.

Amer. Math. Soc. 77 (1971), 247-252. It should be pointed out that definitions of Kn for a ll integers n > 0 have been proposed by sev eral authors.

See the following:

A. Nobile and 0 . Villamayor, Sur la K -th e o r ie a lg e b r iq u e , Ann. Sci. E c. Norm. Sup. 4e serie 1(1968), 581-616. R. Swan, N o n -a b elia n h o m o lo g ic a l a lg e b r a an d K -th eory , Proc. Symp. in Pure Math. 17, 88-123, A.M.S. 1970. M. Karoubi and O. Villamayor, F o n c te u r s Kn en a lg e b r e e t en to p o lo g ie , C. R. Acad. Sc. P a ris 269 (1969), 416-419. S. G ersten, S ta b le K~theory o f d is c r e t e rin g s: I and II, to appear. J . Milnor, A lg e b r a ic K -th eory an d q u a d ratic form s, Inventiones math. 9 (1970), 318-344. D. Q uillen, T h e K -th eory a s s o c i a t e d to a fin ite fie ld : I (mimeographed), 1970. R. Swan, Som e r e la t io n s b etw ee n h ig h er K -fu n ctors, to appear. T h ese definitions are not mutually com patible, in general.

Much work re­

mains to be done in clarifying the relationships between various defini­ tions.

Note also that functors Kn for n < 0 have been defined by B a s s

(A lg e b r a ic K-th eo ry , pp. 657-677). The functors K Q and

are both important to geometric topologists.

In the topological applications the ring A is alw ays an integral group ring Z ll, where II is the fundamental group of the o b je ct being studied.

T h is

theory had its beginnings in J.H .C . Whitehead’s definition of the torsion asso ciate d with a homotopy equivalence between finite com plexes. Whitehead torsion lie s in a certain factor group of K^ZIL further step was taken by C. T. C. Wall.

The

An important

Consider a topological sp ace A

X

PREFACE

which is dominated by a fin ite complex.

Then one can define a generalized

( 3) an d Sp2n (n > 2), Publ. Math. I.H .E .S . 33 (1967). L . N. V aserstein , problem , Mat. Zametki

theory an d the co n g ru en ce su bgrou p 5

(1969), 233-244 (R u ssian ).

J . -P . Serre, L e probleme des groupes de congruence pour S L 2 , Annals of Math. 92 (1970), 487-527. I want to thank Hyman B a s s , Robert Steinberg, and John T a te for many valuable conversations, and particularly for a c c e s s to their unpublished work.

A lso I want to thank Je ffre y J o e l for a number of su gg estio n s, and

for h is lecture notes (based on lectu res at P rinceton U niversity in 1967), which provided the starting point for this manuscript.

F in a lly I want to

thank Princeton U niversity, U .C .L .A ., M .I.T ., and the In stitute for Ad­ vanced Study, as w ell as the National Scien ce Foundation (grants G .P . -7917, -13630, and -23305) for their support during the preparation of th is manuscript.

CONTENTS P re fa ce and Guide to the Literature .............................

vii

§1.

P ro je ctiv e Modules and KQA ......................................................................

3

§2.

Constructing P ro je ctiv e Modules ................. ..............................................

19

§3.

The Whitehead Group K^A ......................

.............

25

§4.

The E x a ct Sequence A sso ciated with anIdeal .....................................

33

§5.

Steinberg Groups and the Functor K 2 ... ............................

39

§6 .

Extending the E x a ct Sequences .......

53

§7.

The C ase of a Commutative Banach Algebra .................................

57

§8 .

The Product I^ A ® K^A

63

§9.

Computations in the Steinberg Group

K 2A .................................................................. .....................................................

71

...................................................................................

81

§10.

Computation of K 2Z

§11.

Matsumoto’s Computation of K 2 of a F ie ld

§12.

Proof of Matsumoto’s Theorem .................................................................. 109

§13.

More about Dedekind Domains .................................................................. 123

§14.

The Transfer Homomorphism

§15.

Power Norm R esidue Symbols .........

§16.

Number F ie ld s

.................................................................

93

137 143

.................................................................................................. 155

Appendix — Continuous Steinberg Symbols Index

.......................................

.................................................... 165

.............................................................................................................................. 183

x iii

INTRODUCTION TO ALGEBRAIC K-THEORY

§1.

P rojective Modules and KQA

The word ring will alw ays mean a sso cia tiv e ring with an identity elem ent

1

.

Consider left modules over a ring A. R e c a ll that a module M is free if there e x is ts a b a sis Jm^S so that each module element can be expressed uniquely as a finite sum £ A^m^, and p r o je c t iv e if there e x is ts a module N so that the d irect sum M © N is free.

T h is is equivalent to the require­

ment that every short ex a ct sequence 0 -> X

Y -> M

0 must be sp lit

ex a c t, so that Y = X © M. The p r o je c t iv e m od u le group KQA is an additive group defined by gen­ erators and relations as follow s.

There is to be one generator [P] for each

isomorphism c la s s of finitely generated projective modules P over A, and one relation [P] + [Q] = [p ® Q] for each pair of finitely generated p ro jectiv es.

(Compare the proof of 1.1

below .) C learly every elem ent of K QA can be expressed as the difference [ P 1] — [P ] of two generators.

(In fact, adding the same projective module

to P x and P 2 if n ecessary , we may even assume that P 2 is free.) We w ill give a criterion for the equality of two such d ifferen ces. F irs t another definition.

L et Ar denote the free module co n sistin g of

a ll r-tuples of elem ents of A. Two modules M and N are called s ta b ly iso m o rp h ic if there e x is ts an integer r so that M © Ar ^ N © Ar. 3

4

A L G E B R A IC K -T H E O R Y

LEMMA 1.1.

T h e g en era to r [P] o f K QA is eq u a l to th e g en era to r

[Q] i f an d on ly i f P is s t a b ly iso m o rp h ic to Q. H en ce th e d if­ fe r e n c e [ P 1] — [P 2] i s e q u a l to [Q^] — [Q^] i f an d on ly i f 0

Q 2 IS s t a My iso m o rp h ic to P^ ® Q j-

P r o o f. The group KQA can be defined more formally as a quotient group F / R , where F is free abelian with one generator

for each isomorphism c la s s of fin itely generated p ro jectiv es P , and where R is the subgroup spanned by all

+ —

.

(Thus we are reserving the symbol

[P] for the residue c la s s of

modulo R .) Note that a sum < P 1> + ... + < P k> in

F is equal to + ... + ^ anc^ on^ ^ pi -

for some permutation

77

pk -

of { l , .. ., k } .

Q„(k)

If this is the c a s e , note the resulting

isomorphism P 1 © ... © P k ^ Q 1 © ... © Qk . Now suppose that = mod R.

T h is means that

_ - S( + ~ ) - £( © -

) for appropriate modules P^, Q^, P j, Qj. Transposing all negative terms to the opposite side of the equation and then applying the remark above, we get

M © (2 (Pi © Qi) © 2 P j © 2 Qp s N © (S Pi © S Qi © S (P- © Qj)), or briefly M © X ^ N © X , sin ce the expression s inside the long paren­ th eses are clearly isomorphic. X

Now choose Y so that X © Y is free, say

© Y ~ Ar. Then adding Y to both sid es we obtain M © Ar ^ N © Ar.

Thus M is stably isomorphic to N. The rest of the proof of 1.1 is straightforward. ■ If the ring A is commutative, note that the tensor product over A of (finitely generated projective) le ft A-modules is again a (fin itely generated projective) left A module. Defining

§1 .

P R O JE C T IV E MODULES AND KQA

5

[p] * [Q] = [P ® Q] we make the additive group K QA into a commutative ring.

The identity

element of th is ring is the c la s s [A1] of the free module on one generator. In order to compute the group K QA it is n ecessary to ask two questions. Q u estion

1

. Is every finitely generated p rojective over A actually free (or at le a s t stably free)?

Q u estion 2.

Is the number of elem ents in a b a s is for a free module actually an invariant of the module? In other words if Ar “

As does it follow that r = s?

I f both q u e s tio n s h a v e an a ffirm a tiv e an sw er then c le a r ly K QA i s th e fr e e a b e lia n group g e n e r a t e d by [A1].

T h is w ill b e true, fo r ex a m p le, i f A

i s a fie ld , or a s k e w fie ld , or a p rin cip a l id e a l dom ain. Of course Q uestions 1 and 2 may have negative answ ers.

For example

if A is the ring of endomorphisms of a fin ite dimensional vector sp ace of dimension greater than

1

, then Question

1

has a negative answer; and if A

is the ring of endomorphisms of an infinite dimensional vector sp ace then Question 2 has a negative answer. (The group K QA is in finite c y c lic but not generated by [A1] in the first c a s e , and is zero in the secon d .) Here is an important example in which K QA is free c y c lic . LEM M A 1.2.

I f A is a l o c a l ring, then ev ery fin ite ly g e n e r a t e d *

p r o je c t iv e i s fr e e , an d K QA is th e fr e e c y c l ic group g e n e r a t e d by [A1]. F irs t re ca ll the relevant definitions.

A ring element u is called a

unit if there e x is ts a ring element v with uv = vu =

1.

The s e t A* con­

s istin g of all units in A evidently forms a m ultiplicative group. A is called a lo c a l ring if the s e t m = A — A* co n sistin g of a ll non­ units is a left ideal.

It follow s that m is a right ideal also .

For if some

Compare K aplansky, P r o je c t iv e m o d u le s, A nnals of M athem atics 68 (1958), 372-377.

6

A L G E B R A IC K -T H E O R Y

product mA with m c m and A e A were a unit, then clearly m would have a right inverse, say mv = 1.

T h is elem ent v certainly cannot belong to

the left ideal m . But v cannot be a unit either.

For if v were a unit,

then the computation m = m(vv~1) = (mv)v~

1

= v"

1

would show that m must be a unit. T his contradiction shows that m is indeed a two-sided ideal.

The

quotient ring A/nt is evidently a field or skew -field. Note that a square matrix with entries in A is non-singular if and only if the corresponding matrix with entries in the quotient A/m is non-singular. To prove this fact, multiply the given matrix on the left by a matrix which represents an inverse modulo m , and then apply elementary row operations to diagonalize.

T h is shows that the matrix has a left inverse, and a sim i­

lar argument constructs a right inverse. We are now ready to prove Lemma 1.2.

If the module P is finitely

generated and projective over A then we can choose Q so that P © Q ^ Ar. Thinking of the quotients P/m P and Q/ntQ as vector sp a ces over the skew -field A/m, we can choose b a se s. in P or in Q for each b a sis element.

Choose a representative

The above remark on m atrices then

im plies that the elem ents so obtained con stitute a b a sis for P © Q. C learly it follows that P and Q are free.

Since the dimension of the vector space

P/m P is an invariant of P , this com pletes the proof. ■ N ext c o n s id e r a hom om orphism

f : A between two rings.

A/

(It is always assumed that f ( l) = 1 .) Then every

module M over A gives rise to a module f#M = A'® M over A'.

Clearly if M is finitely generated, or free, or p rojective, or sp lits

as a direct sum over A, then f^M is finitely generated, or free, or p ro jec­ tive, or sp lits as as a corresponding direct sum over A'. Hence the corre­ spondence

§1.

P R O JE C T IV E MODULES AND K QA

7

[P] - [f#p] gives rise to a homomorphism f* : KqA

K0 A'

of abelian groups. Note the functorial properties (id en tity)* = identity, (f ° g )* = f* o g*. E x am p le 1. L e t Z be the ring of integers.

Then for any ring A there

is a unique homomorphism i : Z

A.

The image i* K 0Z C K qA is clearly the subgroup generated by the free module [A1]. The co-kernel K QA/(subgroup generated by [A1]) = K QA /i* K QZ is called the p r o je c t iv e c l a s s group of A. E x a m p le 2.

Suppose that A can be mapped homomorphically into a

field or skew -field F . mutative.

T h is is always p o ssib le, for example, if A is com­

Then we obtain a homomorphism j * : K0A

K qF s Z.

In the commutative c a s e , this homomorphism is clearly determined by the kernel of j, which is a prime ideal in A. Hence one can speak of the ran k of a p rojective module at a prime ideal p . If p D p 't note that the rank at p is equal to the rank at p '

For if we lo c a liz e the integral do­

main K / p ' at the ideal corresponding to p (that is adjoin the inverses of a ll elem ents not belonging to p ) we obtain a lo cal ring which embeds in the quotient field of A/ p ' and maps homomorphically into the quotient field of K / p . Using Lemma 1.2, it follow s that the ranks are equal.

In

p articu lar, i f A is an in teg ra l dom ain, then th e ran k o f a p r o je c t iv e m odu le is the sa m e a t a ll prim e id e a ls . In any c a s e , choosing some fixed homomorphism j : A -> F , sin ce j * i * is an isomorphism, we obtain a direct sum decomposition K qA - (image i* ) © (kernel j* ) .

8

A L G E B R A IC K -T H B O R Y

The first summand is free c y c lic , and the second maps b ije ctiv e ly to the p rojective c la s s group of A. In the commutative c a s e , note that (kernel j * ) is an ideal in the ring K QA.

We will denote this ideal by K QA, and write K qA s* Z © K 0 A.

E x am p le 3.

Suppose that A sp lits as a cartesian product

Ai x A2 x ••• x of rings.

Then the projection homomorphisms K qA , K qA j

give rise to a corresponding cartesian product structure KqA ^ KqAj x K 0 A 2 x ... x K 0 A^. The proof is not difficult. Such a sp litting of A occurs for example whenever A is commutative and artin ian ,* but is not lo cal.

For sin ce A is commutative, the s e t of

a ll nilpotent elem ents forms an ideal.

If A is not lo c a l, there must e x ist

an element A which is neither a unit nor a nilpotent element.

Since A is

artinian, the sequence of principal id eals (A ):)(A 2 ) d ( A 3 ) d . . . must terminate, say (An) = (An+1) = ... so that An = pA2n for some p.

But

this im plies that the elem ent e = pAn is idempotent (ee = e), and hence that A sp lits as a cartesian product A ss A/(e) x A/(l - e). T h is splitting is not trivial sin c e the hypothesis that A is neither a unit nor nilpotent im plies that e ^ 1, 0.

T h is procedure can be continued in­

ductively until A has been expressed as a ca rtesia n product of lo ca l rings. It then follow s that K qA

=

Z

x

Z

x

...

x

Z.

A ring is a rtin ian if every d escending sequ ence of id e a ls must terminate.

§1 .

9

P R O JE C T IV E MODULES AND KqA

Dedekind Domains Important exam ples in which the ring KQA has a more in terestin g stru c­ ture are provided by Dedekind domains. We w ill d iscu ss th ese in some de­ ta il, starting for variety with a non-standard version of the d efin itio n .* D E F IN IT IO N .

A D e d e k in d dom ain is a co m m u tativ e rin g w ithou t z e ro

d iv is o rs s u c h th a t, for any p air of id e a ls a C 5, th e re e x i s t s an id e a l c w ith a = be . R EM A R K 1 .3 .

N ote th a t th e id e a l c is u n iq u ely d eterm in ed , e x c e p t in

th e tr iv ia l c a s e ct = b = 0 .

In f a c t if h C = 6 c

th en c h o o s in g som e non­

z e ro p rin cip a l id e a l b QA C b w e c a n e x p r e s s b QA a s a p ro d u ct £ b and

conclude that £ b C = £ b c ', hence b QC = b QC

from which the equality

c = c ' follow s. D E F IN IT IO N .

T w o n o n -zero id e a ls a and b in th e D edekind dom ain

A b elo n g to th e s am e id e a l c l a s s if th e re e x i s t n o n -zero rin g e le m e n ts x and y s o th a t x a = y b .

Clearly the ideal c l a s s e s of A form an abelian group under m ultiplica­ tion, with the c la s s of principal id eals as identity elem ent.

We w ill use

the notation C(A) for the ideal c la s s group of A, and the notation \ct \ e C(A) for the ideal c la s s of a . Note that \ a \ = l b ! if and only if a is isomorphic, as A-module, to b . For if cj) : a -» b is an isomorphism, then choosing aQ e a , the computation a 0 (a) = ^ (aoa) = Ak and Ak -» by the formulas b h> ( b c 1/ a Q, . . . , b ck/aQ) and

6

xk) h b jX j + ... + bkxk .

S in c e th e co m p o s itio n is th e id e n tity map of ft, th is p ro v e s th a t 6 is

finitely generated and p rojective. Any finitely generated p rojective P can be embedded in the free module A^ for some k. P ro jectin g to the k-th factor we obtain a homomorphism cf) : P -» A with

(kernel ) C A^” 1.

D ED EKIN D DOMAINS

Sin ce the image 0 ( P ) =

11

is an ideal, hence p ro jectiv e, we have

P ~ (kernel 0 ) © a^.. An easy induction now com pletes the proof, m Rem

ark

. More g e n e ra lly , any m odule w h ich is fin ite ly g e n e ra te d and

to rsio n fre e o v e r A c a n e a s i ly b e em bedded in so m e A^ and h e n c e , by th is argu m en t, i s p r o je c tiv e . T H E O R E M 1.6. (Steinitz).

Tw o d ir e c t su m s a l © . .. © Ctr an d

© ... © bg o f n on -z ero id e a ls a r e iso m o rp h ic a s A -m od u les if an d on ly if r = s an d the id e a l c l a s s l a

1

Ct2 ... Ctf S is e q u a l

to \h1 b 2 ... Br s. (Compare Kaplansky, M odu les o v e r D ed e k in d rin g s an d v alu a tion rin gs, T rans. Amer. Math. Soc. 72 (1952), 327-340.) For the first half of the proof, the ring A can be any integral domain. F ir s t note that, if a C A is a non-zero id eal, then any A -linear mapping : a -> b C A determines a unique elem ent q of the quotient field of A

0

such that 0

(a) = qa for all a £ a.

To prove this it is only n ecessary to divide the equation aQ0 ( a ) = 0 ( a Qa) = 0

(a Q)a by a Q, settin g q = 0

0

(ao)/aQ. Sim ilarly, if the mapping

: a 1 ® ... © a r -> h 1 © ... © bs

is A -linear, then there is a unique s x r matrix Q = (q^-) with en tries in the quotient field so that the i-th component of bi = for a ll (aj^j-.^aj.) e

2

0

( a 1 , . . . , a f) = (b^^^.^bg) is

qija j

© ... © Ctf . If 0 is an isomorphism, then th is matrix

Q has an inverse, hence r = s.

We then a sse rt that the product ideal

h 1 ... b f is equal to (det Q) a 1 ... a f . In fact for each generator a 1 ... af of

... a r, the product (det Q) a 1 ... af can be expressed as the determi­

nant of the product matrix

A L G E B R A IC K -T H E O R Y

12

(

ax0 ... 0

0

Q

a 2“ * 0

0 0 ... ar

whose i-th row c o n sists com pletely of elem ents q - a j of h^. T h is proves that (det 0)0, ... a. C b, ... b„. A sim ilar argument shows that (det Q - 1) ^

... 5 f c Oj ... a r.

Multiplying this la s t inclu sion by det Q and comparing, it follow s that b j ... br is equal to (det Q)Q j ... Ctf ; and hence belongs to the ideal c la s s

\a 1 ... a r l. T h is proves the first half of 1.6. To prove that the rank r and the id eal c la s s \Q1 ... Ctr l form a com­ plete invariant for LEMMA 1.7.

C^©... © af , it clearly su ffic e s to prove the following.

If

Cl

an d b a r e n on -zero id e a ls in a D e d e k in d dom ain

A, then the m odu le a © b is iso m o rp h ic to A1 © (Ctb). If a and b happen to be relatively prime (a + b = A), the proof pro­ ceed s as follow s. Map a ©b onto A1 by the correspondence a ©b

a + b.

The kernel is clearly isomorphic to the module a n b. Sin ce A1 is pro­ je c tiv e , the sequence 0 - ^ a n b - ^ a e b ^ A 1 ->0 is sp lit ex a ct, and. there­ fore 0 © b = A1 © (a n b).

But the intersection ft n b is equal to the

product ideal Ctb. For the inclusion Ctb C a n b is clear; and if 1 = a Q + bQ then every x e ft n b can be expressed as x = a Qx + xbQ, and hence belongs to ab.

Thus a © b s A1 © ftb as required.

For the general c a s e , the hypothesis that A is a Dedekind domain will be needed in order to replace a by an ideal which is relatively prime to b. It clearly su ffic e s to prove the following. LEMMA 1.8.

G iven n on -z ero id e a ls a an d b in a D ed e k in d dom ain

A th ere e x is t s an id e a l a' in the id e a l c l a s s o f a w h ich i s prim e to b.

D ED EKIN D DOMAINS

13

To prove this we must first e sta b lish two of the standard properties of Dedekind domains. LEMMA 1.9.

E v ery n on -z ero id e a l in a D e d e k in d dom ain A can b e

e x p r e s s e d u n iqu ely a s a produ ct o f m axim al id e a ls . In fact choosing any maximal ideal some ideal

D Cl we have a =

then sim ilarly

using the C hinese Remainder Theorem.

1

,

(See for example Lang, A lg eb ra,

Addison-Wesley 1965.) Then the ideal generated by y^ and a is contained in TIL, but is not contained in m 1 2 or in any other maximal ideal. using

1 .9

., this ideal can only be

So,

its e lf.

T h is proves that m 1 is a principal ideal modulo a.

But every ideal of

A/a is a product of maximal id eals, so this com pletes the proof of We are now ready to prove Lemma 1.8.

1 .1 0 .

H

Given non-zero id eals a and

b, choose 0 / a 0 £ a and define £ by the equation £ a = aQA.

Applying

14

A L G E B R A IC K -T H E O R Y

1.10

to the ideal £ modulo f>£, we se e that £ is generated by f)£ together

with some element x . Now multiplying the equation S = B e + x0A by a, and then dividing by aQ, we obtain A =

6

+ a x 0 /aQ.

Since a x 0 /aQ is clearly an ideal in the ideal c la s s \a S, this proves 1.8, and com pletes the proof of Theorem 1.6. M C O R O L L A R Y 1.11.

I f A is a D ed e k in d dom ain, then KQA = Z © KQA,

w h ere the a d d itiv e group o f K QA is c a n o n ic a lly iso m o rp h ic to the id e a l c l a s s group C(A), an d w h ere th e prod u ct o f any two e le m e n ts in the id e a l K QA is zero. In fact the correspondence [ a x © ... © a r] M. (r, maps K QA isom orphically onto Z © C(A).

... a r i) R e ca ll that KQA can be identi­

fied with the se t of differences [P] — [Q] with

rank P = rank Q. Then

each element of KQA can be written as a difference [ a ] — [A1], and we must prove that ( [ a ] - [A1]) ( [ 6 ] - [A1]) = 0. But the product [ a ] [ h ] ^ [ a ® modules a

6

] is equal to [ a h ] .

and ah both have rank

1

, so the natural su rjectio n from the

tensor product to the product ideal is an isomorphism. follows from R em ark s.

1 .7

In fa ct the projective

The conclusion now

. ■

The ideal c la s s group C(A) can be naturally identified with

a m ultiplicative group,

1

+ K QA, of units in the ring KQA. Something sim i­

lar happens for an arbitrary commutative ring.

C all a module M, over a com­

mutative ring A, in v e r tib le if there e x is ts a module N so that M ® N is free on one generator.

The s e t of isomorphism c la s s e s of invertible modules

clearly forms a group under the tensor product. P ic a r d group, denoted by P ic(A ).

T h is group is called the

D ED EKIN D DOMAINS

15

It can be shown that a module is invertible if and only if it is projectiv e, finitely generated, and has rank 1 at every prime ideal.

(Compare Bourbaki,

XXV II A lg eb re com m u tative, Ch. 2 , p. 1 4 3 .) Furthermore the second exterior power E^M of an invertible module is zero.

For this exterior power is a

p rojective module which has rank zero at every prime. (A lg eb r e com m u tative, Ch. 2, p. 1 1 2 .) It follow s that the Picard group embeds as a subgroup of the group of units in K QA.

For if two invertible modules M and M' are

stably isomorphic, M © Ar ^ M' © Ar, then taking the (r + l) - s t exterior power of each sid e, we se e that M — IVT. (B a s s proves the sharper s ta te ­ ment that there e x is ts a can onical retracting homomorphism from the addi­ tive group of K qA to the m ultiplicative group P ic(A ) C KqA.) In the c a s e of a Dedekind domain, it is clea r that P ic(A ) is canonically isomorphic to the ideal c la s s group C(A). To conclude § 1 , le t us prove Theorem 1 .4 .

If F is a finite extension

of the field of rational numbers, we must show that the se t A, co n sistin g of a ll algebraic integers in F , is a Dedekind domain. L e t n be the degree of F over Q. It w ill be convenient to use the word la t t ic e to mean an additive subgroup of F which has a finite b a sis. Thus every la ttic e L in F is a free abelian additive group of rank < n. The produ ct L L / of two la ttic e s in F is the la ttic e generated by a ll prod­ u cts 11' with £ e L and £' e L'. LEMMA 1 .1 2 .

An e le m en t f o f F i s an a lg e b r a ic in te g er i f an d

on ly i f th ere e x is t s a n on -zero la t t ic e L C F w ith fL C L. For if f is a root of the polynomial x^ + a 1 x ^ “

1

cien ts in Z, then the field elem ents 1 , f, f 2 , ... fk with fL C L.

+ ... + a^ with co e ffi­ 1

span a la ttice L = Z[f]

C onversely, if fL C L where L is spanned by b 1, . .., b^,

then we can s e t

. %

■

“ « bi j for some matrix ( a - ) of rational integers. Writing this as 2

( ® ij- .„ ) !■ ) -

2

j

0

.

16

A L G E B R A IC K -T H E O R Y

where ( S - ) denotes the k x k identity matrix, it follow s that the columns of the matrix (f T h ese determine corresponding b a se s | jj* x^| for j ^ P 1 and {j 24;y ! for j 2^ P 2 over A'. The isomorphism k : h it P 1

h ff P 2

12-

CO N STRU CTIN G P R O JE C T IV E MODULES

is now completely described by the matrix

21

5fs

A = (* « £ )

over A', where h0 ' i * v

= £ a a j 8 i 2*

A given matrix ( a ^ ) can occur in this construction if and only if it is in­ vertible ( i.e ., has a two sided inverse matrix ( b ^ ) ) . LEMMA 2.4.

I f th is m atrix A = (aa ^g) is the im a g e under j 2 o f an

in v e r tib le m atrix o v er A^, then the m odu le M = M (Pj ,P 2 ,h) is fr e e . P ro o f.

Let a ^ = j 2 c ^

where (c a ^g) is invertible.

ya = S

Caf3 ^

Set

e P 2’

Clearly these elem ents ly^S form a b a sis for P . The identity hh * xa = j 2* y a now shows that the pair z

a

= (x ,y ' j e P i x P 0 K a J aJ 1 2

belongs to the submodule M (P 1 ,P 2 ,h) C P^ x P 2> It is now easy to verify that M(P 1 ,P 2 ,h) is free over A with b a sis {z^l.m LEMMA 2 .5 .

I f P j an d P 2

a r e fr e e , an d j 2 is s u r je c tiv e , then

M (P 1 ,P 2 ,h) is p r o je c t iv e . P ro o f.

L et

be free over A 1 with one b a s is element u^ for each

elem ent y^ of the b a sis of P 2 » Similarly le t Q 2 be free over A 2 with b a sis {v^|, corresponding to the b a sis {x^l for P ^ h#Qi

L et

- i2 # Q 2

We w ill be mainly in te re ste d in the c a s e where the index s e ts {a.! and \(3\ are fin ite. However the argument works ju s t as w ell for in fin ite index s e ts , providing that we make the convention that a ll in fin ite “ m a tric e s” are to have only fin itely many non-zero e n tries in each row.

22

A L G E B R A IC K -T H E O R Y

be the isomorphism with matrix A” 1 where A =

*s *he m a^ x ° f h.

Then M(Pr P 2 ,h) © M(Q1,Q2 ,g) -

M(P1 © QX, P 2 © Q2 ,h © g),

where h © g corresponds to a matrix of the form

C over A'

:-)

We will prove that this compound matrix s a t is f ie s the hypothesis

of Lemma 2.4.

Hence M(P;f © Q i >P2 0 Q2 ,h © g) is free, and therefore

M ( ? i , ? 2 'h) i s projective. Start with the identity

c :-)■(: x.- x :x:x Since j 2 is surjective, the first factor on the right can clearly be lifted to some matrix of the form ( *

\0 invertible.

* ) over A0 . But any matrix of this form is

l)

2

Since the other factors lift similarly, this proves 2.5. ■

Now consider the general c a s e of Theorem 2.1, where the modules P 1 and P 2 are only assumed to be projective, with h : J‘ i # p i

2.6.

LEMMA

A2

so

~

h # P 2'

T h ere e x is t p r o je c t iv e s Q j ov er A 1 a n d Q2 o v er

that P j © Q j an d P 2 © Q2 a r e fr e e , an d s o that

h#

~

h#

Proof.

Since P^ is projective we can certainly choose some module

^

2‘

N1 over A 1 so that P 1 © N1 is free, say of rank r, over A 1 . Here r can be any cardinal number.* We will write

P x © N t s (A1)r. &

Note however that if r is finite.

is finitely generated then Nj can be chosen so that

§2.

CO N STRU CTIN G P R O JE C T IV E MODULES

23

Similarly we can choose N2 so that p 2 ® n2 Now applying j ^

(A2 )s .

and j 2^ resp ectiv ely , and settin g p = h# p i ~ h #

P 2

we have P ' © j 1# Nt a p ' * i2#

(A ')r

N2 =

and therefore

h t t N 1 ® (A ' )S = hit N 1 ® P ' ® j 2# N2 = hit N2 ® ^ Now defining Qx = N1 © ( A ^ 8, Q 2 = N 2 © (A 2)r it follows that i i f f Q1 = j 2 # P r o o f o f T h eorem 2.1.

^2

as reclui re(^-■

Choose Q j and Q2 as above and choose some

isomorphism k : h#Qi Then, sin ce j 2 is su rjectiv e, the module M(Pl f P 2 ,h) ©M(Q 1 ,Q 2 ,k) s M(P1 © Q 1 ?P 2 ® Q 2 ,h © k) is projective by Lemma 2.5.

Hence M(P 1 ,P 2 ,h) must its e lf be projective.

If P 2 and P 2 are fin itely generated, we must prove that M (P 1 ,P 2 ,h) is also finitely generated. fin itely generated a lso . 2 .5

,

But we can certainly choose Q 1 and Q 2 to be

Looking through the proofs of Lemmas 2 .4 and

we se e that all of the constructions used preserve fin ite generation.

Hence M (P 1 ,P 2 ,h) is a direct summand of a finitely generated free module, and therefore is finitely generated. T h is com pletes the proof of Theorem

2 .1. ■

24

A L G E B R A IC K -T H E O R Y

P r o o f o f T heorem 2.2*

If P is projective over A s e t

P 1

= *1 # P ’

= *2 # P '

P 2

Since j ^i^ = j 2 i 2 there is a canonical isomorphism h : h it p i

h i f P 2'

The diagram

hL

2

* *

ID

P2“

^2 ^ ;

^ J2 #

T3 2

now clearly s a tis fie s the analogues of Hypotheses 1 and 2. P

is the product of P j and

P 2 over P -

P 2< Hence

M(Pr P 2 ,h). I

P r o o f o f T heorem 2.3* For any M = M(P 1 ,P 2 ,h) map M -» P j

gives rise

In particular,

the natural A -linear

to aA 1 -linear map ~>Pr

We must prove that f is an isomorphism. S p e c ia l c a s e .

Under the hypotheses of Lemma 2 .4 , the modules i^ M

and P 1 are free over A 1 with b ases which correspond under f. So cer­ tainly f is an isomorphism. G en eral c a s e .

The proof of Theorem 2.1 co n sisted in showing that

every M (P 1 ,P 2 ,h) is a direct summand of some M (P 1 © P l ,P 2 © P 2,h © h) which s a tis fie s the hypotheses of Lemma 2.4.

It follows that the map

f © f asso ciated with this d irect sum is an isomorphism; hence f its e lf must be an isomorphism.

T his com pletes the proof, a

§3.

The Whitehead Group K^A

L e t G L(n,A) denote the general linear group co n sistin g of a ll n x n invertible m atrices over A, and let GL(A) denote the direct limit (or union) of the sequence G L(1,A ) C G L(2,A ) C G L(3,A ) C ..., where each GL(n,A) is in jected into GL(n + 1,A) by the correspondence

A matrix in GL(A) is called elem en ta ry if it coin cid es with the identity matrix excep t for a sing le off-diagonal entry. Wh

it e h e a d

L

em m a

3.1 .

T h e su bgrou p E(A) c GL(A) g e n e r­

a t e d by a l l elem en ta ry m a tr ic e s is p r e c is e ly eq u a l to th e com ­ m utator su bgrou p o f GL(A). P ro o f,

It is ea sily verified that each elementary matrix can be expressed

as a commutator of two other elementary m atrices. ly every commutator ABA ”

1!}- 1

(Compare § 5.) C onverse­

in G L(n,A) can be expressed as a product

in G L(2n,A ), and it follows from the proof of 2 .5 that each of th ese factors can be expressed as a product of elementary m atrices. (Compare § 4 .3 .) m Hence E(A) is a normal subgroup, and the quotient GL(A)/E(A) is a w ell defined abelian group. De f i n

it io n

. T h is abelian quotient group GL(A)/E(A) is called the

W h iteh ead group K^A. 25

26

A L G E B R A IC K -T H E O R Y

We will sometimes think of K j A as an additive group, and sometimes as a m ultiplicative group. C learly any ring homomorphism f : A -> A ' gives rise to a group homo­ morphism f* ; K jA -» K jA ' REM ARK.

Thus K 1 is a covariant functor.

L et P be any finitely generated p rojective over A, and let

Aut(P) denote the group of A -linear automorphisms of P .

It is interesting

to note that there is a canonical homomorphism Aut(P) -> K XA. In the c la s s ic a l c a s e , linear transformations of a vector sp ace, this turns out to be ju st the determinant homomorphism. The construction is as follow s.

Choose a fin itely generated projective

Q so that P © Q is free, and choose a b a sis b ^ . . . , ^ for P © Q over A. E ach automorphism a of P gives rise to an automorphism a © 1q of P © Q. Using the chosen b a sis, this automorphism is represented by a matrix in the group GL(r,A). T h e resu ltin g em bed d in g

LEMMA 3 .2 .

Aut(P) C Aut(P © Q) ^ GL(r,A) C GL(A) is w e ll d e fin e d up to inner au tom orphism o f GL(A); a n d h e n c e g iv e s r is e to a w e ll d e fin e d hom om orphism Aut(P) -» K-^A. L e t b ^ ,...,b g be a different b a sis for P © Q. (We must allow

P ro o f.

the p o ssib ility that s ^ r.) Then b- = 2 c -jb j where the s x r matrix C = ( c - •) is invertible. J

L et A be the matrix of a © l n with resp ect to the V

original b a sis I b j}. Conjugating A within GL(r + s,A ) by the square matrix C

0

\

O

c~V

A

we obtain the matrix CAC - 1 e G L (s,A ), which d escrib es the

automorphism a

©

1q with resp ect to the new b a s is . Thus a change of b a sis

alters our embedding only by an inner automorphism of GL(A).

Now if we

choose some other module Q ' in place of Q, with say P © Q ' ~ A^, then Q © A^ ^ Q'© Ar. Hence a different ch o ice for Q can also alter our embed­ ding only by an inner automorphism.

T his com pletes the proof, m

13.

27

TH E WHITEHEAD GRO U P K j A

Now suppose that the ring A is commutative.

T h en a n atural produ ct

o p era tio n K qA ® K xA -a K XA can b e d e fin e d a s fo llo w s .

L et [P] be any generator of K QA. The corre­

spondence cl

lp ®

a

defines a homomorphism from the automorphism group Aut(An) = GL(n,A) to Aut(P < 8>An).

(Here the symbol © stands for the tensor product over A.)

Combining this construction with 3 .2 , we obtain a com posite homomorphism G L(n,A) which w ill be called h(P).

Aut(P ® An) -> K^A,

The identity

h(P © P ') = h(P) + h (P /) shows that the homomorphism h(P) depends only on the sta b le isomorphism c la s s of P , and hence depends only on the element [P] e K QA. Now pass to the direct limit as n -» oo, and abelian ize.

By definition, the resulting

homomorphism from K jA to K-^A carries each element k to the product [P] •k. Thus we obtain a product operation K QA © K 1A -» K j A, making K jA into a module over the ring K QA. Another d istin ctiv e feature of the commutative ca s e is that the determi­ nant operation is defined, and can be used to sp lit K jA into a direct sum. L e t A* denote the m ultiplicative group co n sistin g of a ll units of A. Then the composition A* = G L (1,A ) C GL(A) —

> A*

is evidently the identity map. Hence, if SL(A) C GL(A) denotes the kernel of the determinant homomorphism, we obtain a d irect sum decomposition K jA s A* © (SL(A )/E(A)). The notation SK-^A w ill sometimes be used for the second summand.

28

A L G E B R A IC K -T H E O R Y

In many interesting c a s e s the sp e cia l linear group SL(A) is generated by elementary m atrices, and therefore K^A s A@. T h is is the c a s e , for example, if A is a field, or a lo ca l ring, or if A p o s s e s s e s a euclidean a l­ gorithm, or if A is the ring of integers in a fin ite extension of the rational numbers.

(Compare § 1 6 .3 .)

For further information and references se e Milnor, W h iteh ead torsion , Bull. Amer. Math. S o c ., 72 (1966), 358-426, or Kervaire, L e g rou p e d e W hite­ h ea d , (notes by J . M. Arnaudies, mimeographed), E co le Norm. Sup. P a ris, 1966, or de Rham, Maumary, Kervaire, T orsion e t T y p e S im p le d ’H om otop ie, Springer L ectu re Notes 48 (1967). The

4 ‘M ayer-Vietoris’ ’

E x act Sequence

Now consider a square of ring homomorphisms sa tisfy in g Hypotheses 1 and 2 of §2. We w ill construct an ex act sequence K xA -> K 1 A 1 © K xA 2 -» K jA '-* K qA -* K 0\ 1 © K 0 A 2 -> K 0 A ' of length six. Define the homomorphisms K A -> K A , © K A 0 -> K A '

a

a

1

a

a

2

by x ** ( 4i * * © * x )>

and (y>z ) » i j * y -

resp ectively.

j2*

z

Define the homomorphism d : K j A '

K QA as follow s. Repre­

sent the element x of K 1 A/ by a matrix in G L (n,A '). an isomorphism h from the free A-module

T h is matrix determines

A^ to the free A-module

j 2# ^ 2 " Hence, in the notation of § 2 , we can form the projective module M - M(A*, A2 , h) over A.

L et

5(x) = [M] - [An] e K 0 A. It is not d ifficu lt to verify that d is a well defined homomorphism. T H E O R E M 3 .3 .

T h e resu ltin g s e q u e n c e o f len g th s ix i s e x a c t.

TH E “ M A Y E R -V IE T O R IS ” E X A C T SE Q U E N C E

T h e proof is n ot d iffic u lt.

29

D e ta ils w ill be o m itted .

There is an evident analogy between this sequence and the MayerV ietoris sequence of algebraic topology. Steenrod,

(See for example E ilenberg and

F o u n d a tio n s o f A lg e b r a ic T o p o lo g y , p. 3 9 .) Hence we w ill refer

to our sequence as a M ay er-V ietoris s e q u e n c e also . E x am p le.

L e t H be a c y c l i c group of prim e ord er p w ith g e n e ra to r t ,

and le t £ = e 277i/p .

Then it is known that Z[£] is the integral closure of Z in the cyclotom ic field Q[ K QZ[g] is

an isom orp h ism . Hence the computation of K QZ n is reduced to a study of the ideal c la s s group of the Dedekind domain Z[ 3 is needed sin ce th ese relations are completely inadequate when n = 2. Define the can on ical homomorphism : S t(n ,A ) -> G L (n ,A )

by the formula : St(A) Note that the image

GL(A).

^ (St(A )) = E(A) is equal to the commutator subgroup

of GL(A). (Compare 3 .1 .) D EFIN ITIO N .

The kernel of the homomorphism

: St(A) -> GL(A)

will be called K2A. We w ill prove: T H E O R E M 5.1.

T he group K 2A = kernel ((/>) is p r e c is e ly the

cen ter o f the Steinberg group St(A). Thus K 2A is anabelian group which fits into the e x a ct sequence 1 -> K2A - St (A) -» GL(A) -» K jA ->1.

§5.

S T E IN B E R G GRO U PS AND THE F U N C T O R K 2

41

Intuitively speaking we may think of K2A as the s e t of a ll nontrivial relations between elementary m atrices,

the con sequen ces of relations (1),

(2), and (3) being the “ triv ia l” relations.

In fa ct any relation

A■) e- .

Ar» e-

A« e- • = I

\ h

\ h

% jr

X-i A2 between elementary m atrices gives rise to an element xx2^2 lh of K2A, K2A, and every element of K2A can be obtained in this way.

Ar *r^r

As an example the matrix 1 —1 1

in E (2 ,Z )

/ 0

1

represents a 90° rotation, and hence has period 4.

The relation

(e i2 ®21 e i2 ')4 = 1 in E (Z ) gives rise to an elem ent ( x * 2 x ” * x * 2)4 in K2Z.

We w ill s e e in

§10 that the group K2 Z is c y c lic of order 2, generated by this elem ent (x u xn x i 2)4 Note that K2 is a covariant functor from rings to abelian groups.

In

fa ct every ring homomorphism A -> A ' clearly gives rise to a commutative diagram 1 -* K0A -» St(A) - GL(A) -> K .A -> 1

i

1

!

1

1 -> K^A'-> StfA ')^ GL(A')-> K^A'-* 1 P r o o f o f T h eorem 5-1-

F ir s t re ca ll the well known fa ct that an n x n

matrix ( a - ) commutes with every n x n elementary matrix e^g if and only if ( a - ) is a diagonal matrix, with a n = a 2 2 .= ••• = ann belonging to the center of A.

For if (a -j) commutes with e£g then direct computation shows

that a^g = 0 and a j^ = agg. In particular note that no elem ent of the subgroup E(n - 1, A) C E(n, A), Compare the t h e s is of S. Gersten, Cambridge, 1965-

42

A L G E B R A IC K -T H E O R Y

other than I, belongs to the center of E(n,A ).

P a ssin g to the direct limit

as n -» oo, it follow s that the limit group E(A) has trivial center. Now if c belongs to the center of St(A) then 0 ( c ) belongs to the center of E(A ), hence 0 ( c ) = I. Conversely if 0 (y ) = I we must prove that y commutes with every generator Xy of the Steinberg group. Choose an integer n large enough so that y can be expressed as a word in the generators x-1* with i < n and J

j < n. L et P n denote the subgroup of St (A) generated by the elem ents x ln* X2n?*‘ '' anc^ xn _ i n’ w^ere

range s over A.

T h is group is commuta­

tive by R elation (3). LEMMA

5.2.

E a c h e le m en t o f

Pn

can b e w ritten u n iqu ely a s a

product VM1 in

,l2 x Mn _ i 2 n •’ ' n - i , n *

H en ce the c a n o n ic a l hom om orphism 0 m aps P n is o m o r p h ic a lly in to the group E(A). The proof is immediate. Next note that x^ P n

C P n, providing that i,j < n.

For each con­

jugate X|j xj^n x ^ X of a generator of P n is equal either to xj^n or to x *£ x^n according as j ^ k or j = k. But both of th ese exp ression s belong to P n . Now sin ce y is a product of x^j with i,j < n it follow s that y P n y” 1 C P fl. But 0 (y ) = I, so y actually commutes with each element p of P n. For

0 (y

p y_ 1 ) -

5,

we can choose an index h d istin ct from i, j, k,

£. Choosing y € GL(A). Mapping the ex a ct sequence 1* 1 -» K2D -> St(D) -> GL(D) -» K jD -> 1 by P1>K onto the corresponding sequence for A, we obtain as kernel the ex act sequence 1

K2 a -> St(a) -> G L(a) -> K xa -> 1.

Here K 2 a is defined ju st as on p. 33.

By inspection of the commutative

diagram

^ 1 1

-» K 2 a

-» S ta

G L a ->

1 --> K j A -> StA -» GLA ->

K^ci -a l k |a

-» 1

1 1 I I 1 -> K2A ^ StA'-» G L A ' h, K xA ^ 1 , I 1 where A / -- A / a , we now prove the following. THEOREM 6.2.

T h ere i s an e x a c t s e q u e n c e

K2a - K2A

K ^A '^K jC t -» K jA -> ...

ex ten d in g the e x a c t s e q u e n c e o f §4. Here th e homomorphism 3 is obtained by proceeding from K^A' across to

St A', up to

St A , acro ss to G L A ,

up to

G L a (uniquely), and across

to

K 1a. T h is y ield s a well defined homomorphism, sin c e any two elem ents

o f St A. with the same image in StA ' differ only by an elem ent from S ta

which maps into 1 in K^a.

Further d etails will be left to the reader. ■

Next consider a ring homomorphism f : A -* F . b e two-sided id eals with f ( a ) C b.

L e t a C A and b C T

Then there is a corresponding commuta­

tive diagram 1 -» K ^ a

-> S ta ~> G L a -» K ^ a

■1 1 -» K2 b -> Stb

-1 >t GLb - K xb

-> 1

1.

E X T E N D IN G TH E E X A C T SE Q U E N C ES

§6-

E X C I S I O N L e m m a 6 . 3 . I f f : A -> T

th e id e a l o o n e-to -o n e on to b,

55

i s s u r je c t iv e , an d m aps

then the in d u c ed hom om orphism

f * : K 2 a . K 2b is s u r je c t iv e an d f * : K^a -» K^B is an iso m o r­ phism . P roo f.

Since a maps one-to-one onto b it is clear that GL(ct) maps

isom orphically onto G L(b).

Furthermore, recallin g from 6-1 that

generated by exp ression s A^(s)X|j),^ A ^ (s’“1) with s e S t(D , maps onto S t(D St(b).

St(B)

is

sin ce St(A)

and a maps onto b it is clear that St(d) maps onto

The lemma now follow s ea sily .

(Compare B a s s , A lg e b r a ic K-Theory,

p. 3 8 2 .) « S till assuming that f is su rje ctiv e , and maps a isom orphically onto h, we have the following commutative diagram: K2ct

->

K2A -> K2A ' -> K XQ ->

->

K2r ^ K2 r %

I onto k

2b

I

I

Here A ' = A/a and r ' = F/b.

K xA -> K 1A/

1“

I

k xb

k xf

I -> K 1r /.

By insp ection of th is diagram we ea sily

obtain the ex a ct sequence K2A -> K 2F © K2A ^ K2r '-> K XA - K xr © K jA '-* K j F '. T h u s we h a v e proved the fo llow ing. T H E O R E M 6-4*

I f a com m u tative sq u a re o f s u r je c t iv e ring h om o-

m orphism s A ------------------^ A'

r

------- ^

r'

s a t i s f i e s the h y p o t h e s e s o f §2, then the May er- V ie tori s e x a c t s e q u e n c e o f §3 can b e e x te n d e d by the term s k

2a

->

k

2 f © K 2A '

K 2 r ' -> K XA

56

A L G E B R A IC K -T H E O R Y

In fact the two hypotheses of §2 were that A should be equal to the product of F

and lY over Y', and that at le a s t one of the two homomor­

phisms to F / should be su rjectiv e.

But if A ' = A/a and r ' = F/b then

the first hypothesis is clearly equivalent to the requirement that a should map one-to-one onto b. m R e m a r k 6-5-

Swan has recently shown that it is not p o ssible to de­

fine any functor Kg so that the ex act seq u en ces of 6 .2 and 6 .4 are ex ­ tended by appropriate Kg-terms.

(See Swan, E x c is io n in a lg e b r a ic K -th eory ,

to appear.) T his result su ggests that our definition of K^a may not be too useful. REMARK 6-6

To conclude this sectio n , let us consider the rather dif­

ferent situation asso ciated with a pair of two-sided id eals a C b in the same ring A.

The image of the ideal b in the quotient ring A/a will be

denoted by b/a. Then it is not d ifficu lt to construct homomorphisms K2a -> K2 b -» K2 b /a

(*)

Kxa -> ... ,

the composition of any two su cce ssiv e homomorphisms being zero. sider

the following K2 o^ \

commutative

\ 2A ^ / K j6

K2A / b ^

\

\

diagram of

/ K2A/o

/

K 2V tt

\

\

/

Kt a

Con­

interlocked

^ K xb/a / Kj 6

^

sequen ces

...

\

\

/

K XA

By inspection of this diagram one can verify that at le a s t the following portion of the sequence (*) is exact:

K2b / a -* Kxa -» K xb

K jb / a -> KQa -> K0b -> KQb/a.

As a typical application, due to B a s s , if A is a Dedekind domain and if a / 0, then it is not difficu lt to show that S K jb / a = 0. So it follows that the homomorphism S K ^ -> S K jb

must be su rjectiv e.

I do not know whether or not the in itial portion of the sequence (*) is ex act.

§7.

The C ase of a Commutative Banach Algebra

L et A be a commutative Banach algebra over the real or complex y numbers. For example A could be the ring R of continuous real valued functions on a compact sp ace X.

T h is sectio n will use the topological

structure of G L(n,A ) to compute K^A and to estim ate K2A. LEM M A 7 .1.

T he group E(n,A ) is an open, path -con n ected su b­

group o f the s p e c ia l linear group SL(n,A ). Hence E(n,A ) is the component of the identity in SL(n,A ), and the quotient SL(n,A )/E(n,A ) can be identified with the group

77QSL(n,A )

of

path components. Now recall that K 1A sp lits as the direct sum of the group A* of units and the group SK 1(A) = SL(A)/E(A) = lim SL(n,A )/E(n,A ) - lim rrn SL(n,A ). If we give SL(A) the direct limit (= fine) topology, then clearly the group 77qSL(A)

of path components can be identified with lim 7rQSL(n,A ). Thus

we obtain: COROLLARY 7.2.

The group K XA s p lits a s the direct sum o f

the group A* o f units o f A and the group SK 1(A) =

57Q SL(A)

o f path com ponents o f SL(A ). In the sp e cia l ca s e A = R as follow s.

Y

, th is description can be further sim plified

Since the m ultiplicative group R # is topologically isomorphic

to the additive group F 2 © R, we have (R X )* as (R*)X s ( F 2)X ® R X 57

58

A L G E B R A IC K -T H E O R Y

Furthermore the sp e cia l linear group SL (n ,R ) contains the rotation group SO(n) as deformation retract,

and hence the sp ace

S L (n ,R X ) s contains the function space SO(n)

V

S L (n ,R )X as deformation retract.

P a ssin g to

the limit as n --> oo we obtain: COROLLARY

7.3.

o f the group (R*)

Y

T h e group K 1R

X

s p lit s a s th e d ir e c t sum Y

o f units an d the group n0(SO ) o f h om otopy

c l a s s e s o f m appin gs from X Sim ilarly, for the ring C

X

to th e in fin ite ro ta tion group SO.

of continuous complex valued functions, we

obtain K 1C X s ( C * ) X ® 770(SUX), where SU denotes the infinite sp e cia l unitary group. Now let us prove Lemma 7.1.

Since each elementary matrix e^j can

be joined to I by the path t b e jp

0 < t < 1,

it is clear that E(n,A ) is path connected.

To prove that E(n,A ) is open

(and hence clo sed ), we will prove a sharper statem ent.

L et I + A be an

n x n matrix with determinant 1. LEMMA 7.4.

I f e a c h entry a -

o f A s a tis fie s

||a-1| < l/(n _ 1)

then I + A can b e e x p r e s s e d c a n o n ic a lly a s a p rod u ct o f n2 + 5n — 6 elem en ta ry m a trices, e a c h o f w h ich d e p e n d s co n ­ tin u ou sly on A. P roo f.

The entry 1 + &11 has an inverse u satisfy in g llujl < ( i

V

L V 1 =

n - 1/

n _ 2

.

T h is i s proved, for example, using a form of the Gram-Schmidt orthonormaliza­ tion process.

§7.

Subtracting u a ^

COMMUTATIVE BA N A CH A L G E B R A S

59

times the first row from the k-th for k = 2, 3,

obtain a matrix I + (a'^j) satisfy in g

a'21 = a '^ = ... = a'nJ =

n, we and

0

lla kj'! < liakjll + Huaki a ijH
bl 63

64

A L G E B R A IC K -T H E O R Y

and [pap- 1 , p b p -1] = p f a ^ l p - 1 respectively. ■ Now suppose that the ring A is commutative.

L et u and v be any

two units of A. Then the diagonal m atrices 'u du = [ o

0

0\

u -1

o I,

0 0

/v d ;

=

1/

0

0

o

i

o

\0

0

v -1

commute, and belong to E (3,A ). D E F IN I T IO N .

T h e c om m utato r

Du *

e K ? A w ill be d eno ted briefly

by ju, v}. LEMMA 8 . 2 .

T h e sy m b o l {u,v| is a l s o sk ew -sy m m etric Su,v| = I v , u 1 ,

an d b im u ltip lic a tiv e Su1 u 2 ,v| = Sul ; v } { u

P ro o f.

in E (3,A )

v}.

L et

so th at p d up

-1

=

d;,

pd

;

p

- 1 =

d

v.

Then S u .v S -1 = (Du *

d;

) - 1 =

d;

*

du

= (p d ; ? - 1) * (p d up _ 1 ) = d v * d ;

= {v ,u }.

The rest of the proof is clear. ■ Now we can com pletely describe D * D ' for any pair of diagonal ma­ trices in SL(n,A ).

L e t diag(u1,...,u n) denote the diagonal matrix with

diagonal entries u 1,...,u n.

§8.

L e m m a 8 -3 .

TH E P R O D U C T I ^ A ® K 1A -» K2 A

65

I f UjU2 ... u fl = v 1v 2 ... v n = 1 then d ia g ( u 1 , . . . , u n ) * d ia g ( v l f . . . , v n )

is e q u a l to the p rodu ct iu2 ’v2 !

Sun-vn S •

The proof w ill be le ft as an e x e rcise. Here are some exam ples to show that the symbol lu,vl

is not identi­

ca lly equal to 1. R e c a ll from §7 that there is a natural homomorphism K2 A

67

It may be conjectured that the homomorphism Ki CX - K—i X of §7, from algebraic K-theory to topological K-theory, is com patible with the product operations in the two theories.

If true, this would lead to a

quite direct proof of 8.5. Now let us define the product operation K j A K j A -» K2A. We con­ tinue to assume that A is commutative. Given an automorphism a

of the free module Am and an automorphism

j8 of the free module An, we can form the automorphism a ® (3 of Am ®An . In order to translate this construction into matrix notation, it is only n e c e s ­ sary to choose some fixed ordering for the can on ical b a sis of Am ® An. Then to each matrix A of GL(m,A) and each matrix B of G L(n,A) there corresponds the matrix of the corresponding automorphism of Am ® An which we denote by A ® B e GL(mn,A). Using this notation, we w ill con­ struct a bim ultiplicative symbol |A,Bi h l 3 (u) is not a homomorphism. If A is commutative, then clearly th ese exp ression s belong to the central group Cn = W n k ern el(0).

Ju s t as in §8 we can then prove:

74

A L G E B R A IC K~ T H E O R Y

LEMMA 9.7.

I f A is com m u tative, then th e sy m b o l iu,vi = [h jj(u ),h ik(v)]

is sk ew -sy m m etric an d b i-m u ltip lica tiv e , w ith v a lu e s in th e com ­ m u tative group Cn . No confusion will arise in thus giving the symbol {u,v| a new and sharper meaning, sin ce th is new symbol clearly corresponds to the old one under the natural homomorphism Cn -> lim Cn C K2A. C learly this symbol |u,v| does not depend on the ch o ice of ind ices i

j ^ k ^ i. Here is a fundamental lemma which relates the symbol {u,v! to the

additive structure of the ring A. LEMMA 9.3.

I f both u an d 1—u a re units, then {u,l~u| = 1.

F urtherm ore, for any unit u the id en tity {u,-~u| = 1 is v a lid . P roo f.

If v is equal to either 1—u or —u, we must verify the equality

(1)

h12(u)h12(v) = h12(uv).

Substituting the definition h12(x) == wi2^x)

^ su ffices to

prove that the expression (2)

wl 2 ^ w12("”1 )w 12(v)

is equal to w12(uv). If v = 1—u, this can be proved by substituting the identity wi 2( - 1) = into (2).

= 4 i x r 2l x 2 i

(Compare 9 .5 .) Using the formulas 2

wi2(u)x2i =xr2u'wi2 and

§9.

COMPUTATIONS IN TH E S T E IN B E R G GRO U P

X21W12(V) = W12(V) X'12

75

.2

V'

of §9.4, the expression (2) becom es

(20

12

*

Now substituting the definitions of w12(u) and w12(v), and using the eq u alities —u2+u = uv, u—1+v = 0, v—v2 = uv, and then —u —(uv)” 1, the expression (2 ') becom es

as required. The proof that wl 2(u)W i2( - 1) w12( - u) = w12( - u 2) is ea sier.

Writing the le ft hand side as w12(u) w12(—1) w12(u)” 1, it is

equal to w21(u ~ 2) by 9 .4 , and hence to w12(—u2) by 9 .5 . p letes the proof of Lemma 9 .8 .

T h is com­

(I want to thank Steinberg for this version

of the p ro o f.)! C O R O L L A R Y 9.9.

If A

IS

a fin ite fie ld , or if A

is

the ring o f

in te g e r s m odulo a pow er o f an o d d prim e, * then lu,v| = 1 fo r a ll u an d v. P ro o f.

F irs t suppose that A is a field with q elem ents,

q being

odd. Then the group A* of units is c y c lic of order q—1. Half of its e le ­ ments are squares, and h alf are non-squares. Note that we can find a non­ square uQ

so

that

1 — uQ

is also a non-square.

For otherw ise the corre­

spondence u h> 1—u from A * - { l i to its e lf would carry each of the (q—1)/2 non-squares to a square.

Thus we would find (q—1 ) / 2

addition to 1; which is im possible.

Compare p. 92.

squares in A*, in

76

A L G E B R A IC K -T H E O R Y

L et v be a generator of the group A*. Then uQ = v1, l —u0 =

with

i and j odd, and it follow s that !v,v|^ = 1. Sin ce {v ,v }2 = 1 by skew symmetry, this proves that Sv,vS its e lf is trivial. If A = Z/pnZ with p odd, then a sim ilar argument applies sin ce A® is c y c lic , and sin ce an integer is a quadratic residue modulo pn if and only if it is a quadratic residue modulo p.

F in a lly , if A is a field of

order q = 2^, then the equations {v , v 1 = 1 and |v,vP = 1 imply that {v,v} = 1.

T h is proves the corollary. ■

Next we will show that the group Cn is generated by the symbols {u,v}. As a first step we w ill prove the following. LEM M A 9 . 1 0 .

A ll o f the sy m b o ls h j^ (u )

can b e e x p r e s s e d in

term s o f th e h^j^u) w ith j = 1. In fa c t t h e s e s y m b o ls s a t is fy th e id e n t itie s hjk (u) hk j(u> = 1 > P roo f. commutator hjk(u).

an d

hi j( u) _ l h jk (u) _ l h k i(u) _1 = L

Using 9 . 4 to simplify its first three factors, we se e that the Hi k ( u ) W j k ( l ) h i k ( u ) - 1 W j k ( - l )

is equal to

W jk ( u ) W j k ( - 1 )

=

On the other hand, using 9 . 4 to simplify its la s t three factors, this

commutator is equal to h ^ (u )h -j(u )- 1 . Thus: (*)

hj k < » = hik (u) hi j ( u) _ 1 -

Setting i = 1,

we have proved the first assertion of 9 . 1 0 (at le a s t in the

c a s e j,k > 1). Multiplying the identity (*) by the corresponding identity with j

and k interchanged, we obtain the second assertio n of 9 .1 0 , and

substituting h ^ (u )“ 1 for h ^ (u ) we obtain the third, m REMARK.

E v id e n tly the sym bols

h ^ u ) ” 1 would have been som e­

what e a s i e r to work with than the h^j(u) th e m s e lv e s .

L e t A be any commutative ring. T h e o r e m 9.11.

T h e cen tra l su bgrou p C n = W n (kernel 1 to push a ll occurrences of w ^ to the left. the permutation which interchanges i and j.)

(Here

77

denotes

Then use the relations

wig wi e s 1 and w 1 jw 1 £ to elim inate the w ^ ,

EE

wx£W£j for j ^ I

one or two at a time. At the end there w ill be at

most a sing le w ^ at the left.

But this sing le w ^ cannot occur sin ce

otherw ise c£(c) would not correspond to the identity permutation.

Sim ilar­

ly we can elim inate a ll occurrences of w2g, and so on, continuing induc­ tively until we have proved that c = 1 modulo H. Thus c can be written as a product of symbols h^j(u). Using 9 .1 0 it follow s that c can be written as a product of the symbols h ^ u )

and

their in verses. If C * denotes the subgroup of Cn generated by the {u,v|, note that hj£(uv) = h 1£(u)h1£(v) mod C * and h l j ( u) hi f ( v) = V W S j W

m o d C *.

78

A L G E B R A IC K -T H E O R Y

It follows ea sily that the elem ent c can be expressed as a product of the form c = h i 2 (u2 ) hl3 ( u3) ••• hln (un) mod C * ' T his im plies that c/>(c) is the diagonal matrix diag(u2>.. un, u j 1, ...,u “ 1). But 0 ( c ) = I by hypothesis, hence u2 = u3 = ... = un = 1, c == 1 mod C *.

and therefore

T h is com pletes the proof of 9.11. ■

T H E O R E M 9 . 12 .

I f A is a f i e l d or s k e w - fie ld , then th e en tire

kern el o f 0 : St(n,A) - GL(n,A) i s c o n ta in e d in W,

an d h e n c e is e q u a l to C n .

T h e r e fo r e the

S tein b erg group St(n,A) i s a cen tra l e x te n s io n o f E(n,A ) for ev ery n > 3. Combining 9 . 1 1 , 9 . 1 2 , and 9 . 9 and passing to the d irect lim it as n -> oo, this im plies the following. CO R O LLAR Y 9 .1 3 .

by the sy m b o ls {u ,v } . k

I f A i s a f i e l d then K 2 A i s g e n e r a t e d In p articu la r i f A i s a fin it e f i e l d then

2a = 1.

R e m a r k . M. Stein has recently generalized this result by showing that K2A is generated by the symbols {u,vi for any sem i-lo cal ring A which is additively generated by the s e t A* of units. The proof of 9.12 w ill be based on two lemmas. L e t T denote the subgroup of St(n,A) generated by a ll

for which

i < j. LEMMA 9.14.

a produ ct

E v ery e le m en t o f the group T

can b e w ritten a s

__

ni j - i < j

x * .« ) ij

*

the fa c t o r s b ein g arran g ed in le x ic o g r a p h ic a l order. m aps T

H en ce 0

iso m o r p h ic a lly on to the n ilp o ten t group c o n s is tin g o f

a l l upper triangular m a tr ic e s w ith l ’s alon g th e d ia g o n a l.

§9.

79

COM PUTATIONS IN TH E S T E IN B E R G GRO U P

T h e p ro o f i s no t d if f ic u lt.

Lem m a 9.15. o f S t(n ,A ) P roo f.

//A

b e lo n g s to the produ ct T W T .

C le a r ly

j = i ± 1.

is a f i e l d or s k e w - fi e ld then ev e ry ele m e n t

S t(n ,A )

i s g e n e r a te d by th o s e e le m e n ts

x^j

fo r w h ich

S in c e wi j ( 1 ) x i j wi j ( - 1 ) = x j i A’

i t fo llo w s th a t

S t(n ,A )

i s g e n e r a te d by

p ro v e 9.15 i t s u f f i c e s to c h e c k th a t tio n by

w rite

t2

TW T

and th e

w^ j + 1 (l)«

th e n o ta tio n , s e t

x ^ t'

a s a p ro d u ct

k < I

and

i s c lo s e d un der rig h t m u ltip lic a -

j = i + 1. G iv e n any t j w t2

w h ere

= t x (W X jj

W jjd ))^ ,

w h ere th e e le m e n t

t" = w i j ( - l ) t ' w i j ( l ) c le a r l y b e lo n g s to

T.

T h u s i t s u f f i c e s to c h e c k th a t w x jj

W jj(l)

f TW T.

b e th e p e rm u ta tio n c o rre s p o n d in g to

77

C a s e 1.

w.

If 77-(i) < 770) = 7r(i + 1 ),

th e n w xi j wi j ( ! ) = x i ( i j ) wwi j ( ! ) f TW '

C a se 2.

If

7r(i)

> n( j )

and

A i s a u n it, th e n

* i j = WjjCA)

xj- 1

hence WXj^j W j j ( l )

=

in T W T ,

t ' c a n b e w ritte n a s a p ro d u ct o f

( k ,£) ^ ( i , j ) . T h e n t l w t2 wy ( l )

L et

T h u s to

wi> i+ 1( l ) .

T o s im p lify

w ith

T

(w W jjU ) x j j * ) ( x ^

1 W jjd ))

= ( x j (ji) w w jjU )) (W ijd )

x ~ -x

) £ TW T.

x^g

80

A L G E B R A IC K -T H E O R Y

C a s e 3.

If A is not a unit, then A = 0 and the statem ent is clear.

T his proves Lemma 9.15. ■ P r o o f o f T h eorem 9.12.

If 0 ( t xwt2) = I then the monomial matrix 1

w h ere Cn is the c y c l ic group o f ord er 2 g e n e r a te d by th e sym ­ bol S - l , - I S = ( x l 2 * 2 1 x 12 ) 4 -

(Compare §§ 9 .3 , 9 .7 , and 9 .1 1 .) An immediate consequence is the fo l­ lowing. C O R O L L A R Y 10.2.

T h e group K2Z is c y c l ic o f ord er 2, g e n e r ­

a t e d by {—1, —1}. For K2Z is the d irect limit of th ese groups Cn. The following statem ent is equivalent to 1 0.1.

L e t e - = c£(x?j) be the

elementary matrix with entry +1 in the (i, j)-th p lace, C o ro lla ry

10.3.

F or n > 3

i ^ j.

the group SL(n, Z) = E(n, Z)

h a s a p rese n ta tio n w ith n(n—1) g en e r a to r s e^- s u b je c t on ly to the S tein b erg r e la t io n s [e ij> e k£] = 1 i f j 4= k, i £ i , ^e i j ’ ejk l = e ik

k a r e d is tin c t>

an d to th e re la tio n (e i 2 e 2 1 e l2^4 = P r o o f o f 10.3. D efining e^= to be the A-th power of e^.-, a ll of the A Steinberg relations between th ese elem ents eA follow immediately from the relations listed .

Together with 10.1, this com pletes the proof. ■ 81

82

A L G E B R A IC K -T H E O R Y

The proof of 10.1 is based on c la s s ic a l work by N ielsen and Magnus. In fact 10.1 can be derived from Magnus' paper U ber n -d im en sio n a le G ittertran sform ation en , Acta Math. 64 (1935), 353-367. proof, based on recent work by J . R. Silv ester. tion o f G Ln(Z) an d G Ln(k[X]),

We w ill give a sim plified (See Silv ester, A P r e s e n t a ­

to appear.) It should be noted that S il­

vester, and also K. D ennis, have used this same method to prove the is o ­ morphism K 2 F [ x ] ^ K2F ,

where F [x ] denotes the ring of polynomials

in one indeterminate over the field F .

Dennis has extended this to the

ca se of a skew -field. Since S ilv e ste r's proof is inductive, we must start with the c a s e s For n = 1, we simply define S t(l,A ) to be trivial. tion according to Steinberg is as follow s. D e f i n i t i o n 10.4.

For n = 2, the defini­

(Compare p. 7 3.)

For any ring A ( a s s o c i a t i v e with 1) let

be the group with generators

n = 1,

S t(2,A )

x^2 and x 2 1 , as A v a r ie s over A,

and

defining re latio n s

VA _ VA+i± ij ij “ ij and Wij(u) xj i wi j( —u) = ^ uXu for u e A®; where w -(u ) by definition is equal to xjjx j^ u The analogue of 10.1 is then the following. tion x-j

x-A .

We introduce the abbrevia­

for the Steinberg generator x^j .

TH EO R EM 10.5.

T h e grou p

St(2,Z ) i s a c e n t r a l e x t e n s i o n

1 -> C2 -> St(2,Z ) ^ E (2 ,Z ) -> 1 w here

C2

is a fr e e c y c l i c g rou p g e n e r a t e d hy th e e le m e n t

(x 12x~11x 12)4 . H e n c e S L (2 ,Z ) = E (2 ,Z ) h a s p r e s e n t a t i o n w ith g e n e r a t o r s

e 12 a n d e 21 a n d d e f i n i n g r e l a t i o n s

e12e2l'el2 = e21e12e21 and a a+ 1 ? be the largest value of cra for which this occurs.

Sin ce there may be

sev eral d istin ct values of a for which this same maximum is attained, we set (i = MaxSa such that oa - A > a a + 1 } . The pairs (A,/z) are to be ordered lexicograp h ically; so that (A,//) > (A\ fi) if and only if either A > \\ or A = A' and (i > fi\ The proof of 10.6 will co n sist in showing that each word g1 ... gfw with (A,fi) > (1, 1) can be altered, by means of Steinberg relatio n s, so as to d ecrease the pair (A?/u). After repeating this procedure a finite number of tim es, we must achieve the required word with (A,/z) = (1, 1). Suppose then that the pair (A,pt) > (1, 1).

A= Note that fi > I

v

Then

i-

(sin ce if fx = 0 we would have 1 = orQ > a 1 which is im­

p o ssib le); and that a

^< a •

We will suppose, to fix our ideas and sim plify the notation, that the generator g^ is equal to x J 2 . (The general c a s e can e a sily be reduced to this sp e cia l c a s e as follow s.

If g^ = x - ,

we can simply renumber the

coordinates to replace i and j by 1 and 2 resp ectiv ely. then conjugating every g^ by w -,

If g^ = x^j1 ,

and replacing /3 by /3wjj1 and w

by w^jW, we obtain an equivalent problem, with g^ replaced by Xj^. We then proceed as before.) Assume then that g^ = x 12. Setting P&1&2 it follow s that

= (a ’ b’ C’ "■) f Z ° '

87

COM PUTATION O F K2 Z

§10.

J 3 g jg 2 ... g ^ _ j = (a, b - a , c , ...). Hence the inequality a fj. ,I < a P can be written as |b — a| < |b| . T h is is clearly equivalent to the following statem ent. (1)

|a I< 2 1b |, an d if a ^ 0 then ab > 0. The proof of 10.6 w ill be divided into seven c a s e s , depending on g^+ 1-

F irs t we consider the four c a s e s where g C a s e 1.

.

commutes with g = x.

0

Suppose that g^+1 = X jj with j > 3. Without lo s s of gener­

ality , we may assum e that j = 3, and n = 3. Thus g^+1 transforms (a, b, c) to (a, b, s a + c), with |c| > |s a + c| . Now alter the word g 1 ... grw by replacing the product g^g^+1 = x 12x l3 by x^3x 12. Then the transformation (a, b—a, c) s-~-* (a, b, c)

—>(a, b,

a + c)

w ill be replaced by (a, b—a, c) »-» (a, b—a, s a + c)

m-

(a, b, s a + c).

The asso cia te d numbers o a are unchanged, excep t that o^ - ||(a, b, c)|| is replaced by a number o' - ||(a, b—a, s a + c)|| which s a tis fie s p °p- 1 > V Inspection shows that the pair (A', p') asso cia te d with the new sequence is le s s than (A, p). C a s e 2.

If g^+1 = x?j with i and j

greater than 2, the proof pro­

ceed s ex actly as in C ase 1. C a s e 3.

s

ca n cel the factor g g

then we can simply

1, thus reducing (A, p). But s cannot be +1,

P P'~l

for if

s

Suppose that g^+1 = g^ = x 12- If s = - 1 ,

88

A L G E B R A IC K -T H E O R Y

(a, b, c) g

= (a, b+a, c)

then we must have |b—-a| < |b| > |b+a| which is clearly im possible. C a s e 4.

£ £ If g^+1 = x 32 (or x*2

with i > 3) then a rather more com-

plicated argument is needed. Note that the product x 12x 32 in the Stein£

p

£

P

berg group can also be written either as X3 2x i 2 > or as x i 3 x 32x l 3 ? or as £ —£ X 3 1 X 1 2 X 3 1 • (The proof of these id en tities is straightforward.) T h is means that the transformation (a, b—a, c) b> (a, b, c) h (a, b + s c , c) £ corresponding to the product %12x 32 can be replaced either by (a, b—a, c) i-> (a, b~a + £ c , c) t~> (a, b + s c , c), or by (a, b—a, c) i-> (a, b—a, c+ea)

(a, b+£c, c+£a)

h> (a,

b+sc, c),

or by (a, b—a, c) i-» (a+£c, b—a, c) i-> (a+ sc, b+£c, c)

(a, b+£c, c).

Inspection shows that the first replacement will reduce (A,fj) providing that (2)

|b—a | > |b-a+£c| .

Similarly the second will reduce (A, (i) if (3)

|c| > |c+£a| ,

and the third w ill reduce (A, (i) if |a| > |a+£c| .

(4)

We must show that at le a s t one of th ese three in equ alities is sa tisfie d . Note first that the inequality

>o

im plies that

|b ! > |b+£c| Hence b and £c must have opposite sign.

.

If a ^ 0,

then it follows from

(1) that a and £ c have opposite sign, and therefore that either (3) or (4) is sa tisfie d .

But if a = 0, then (2) is sa tisfie d .

cu ssion for C ase 4.

T h is com pletes the d is­

§10.

89

COM PUTATION O F K 2 Z

Now we must consider the various c a s e s in which g and g ,, do ^(1 &/x+l not commute. C a s e 5.

If g^+1 = x 2 l ’ ^ en

product

corresponds to the

transformation (5)

(a, b - a )

h-

(a, b)

(a+eb, b)

with Ia j + |a+eb | . If e were +1, then a and b would have opposite sign, contradicting (1); so we may assume that x2 l w2 i ( —1),

8

= —1. R eplacing the product x ^ x " ^

and noting that the element w21( -- l)

ously past g^

2

^1 1+ 3 •••

^

by

can be pushed innocu­

§9.4 or §10.4, we se e that the transforma­

tion (5) can be replaced by X

(a, b—a) t - - i (b, b—a). Evidently th is reduces (A, fi). 8

8

Suppose that g +1 equals x 23 (or x 2j with j > 3). In this 8 8 £ c a s e the word x 12x 23 can be replaced either by x i 3x 2 3 x i 2 or fry C a s e 6.

0

0

0

_H

x 23x l 3 x 12 or by x 2 i x i 3x i2

(a, b—a, c)

12* C orresPonc*ingly h> (a,

transformation

b, c) i-> (a, b, c+eb)

can be replaced by one of the following: (a, b - a , c) h> (a, b - a , c+ea) »-> (a, b - a , c+eb) h> (a, b, c+eb), (a, b - a , c)

h

(a, b - a , c + e b -e a ) ^ (a, b - a , c+eb)

(a, b, c+eb),.

or (a, b - a , c)

(b, b—a, c)

(b, b - a , c+eb)

h

(b, - a , c+eb).

Thus (A,fi) can be reduced if either (6)

!c| > lc+ea| ,

(7)

|c ! > |c+eb—ea | , or

(8)

Ia | > |b| ;

using the inequality

jb—a| < |b|. The inequality a

>

j

im plies that

90

A L G E B R A IC K -T H E O R Y

|c | > jc+eb|, so that c and sb have opposite sign, and |b! < 2 1c|. If a = 0, then the equality (7) is sa tisfie d .

But if a ^ 0,

by (1), then ea and c have opposite sign.

Now either

so that ab > 0 |a| < 2|c|, which

im plies (6), or 1a| > 2|c| which im plies (8). C a s e 7.

£

If g ^ = x 31

(or

£

with i > 3),

£

then x 12x 31 is equal to

x31x 12x 32e anc* to x3 l x T3 x 32 x 13‘ ^ ence ^ e transformation (a, b—a, c)

(a, b, c)

h>

(a+ec, b, c)

can be replaced either by (a, b—a, c) i-» (a+ec, b—a, c) i-> (a+ec, b+ec, c) >-> (a+ec, b, c) or by (a, b—a, c)i->(a+ec, b—a, c )^ (a + e c , b—a, —ea)i->(a+ec, b, —e a )^ (a + e c , b, c ) . Hence (A, \i) can be reduced if either (9)

|b+ec| < |b|,

(1 0 )

|c| > !a|.

The inequality

or

> cr +1 in this ca s e im plies that !a | > |a+ec |

so that a and e c have opposite sign, and Ic | < 2 1a |„ Therefore

b and e c

which im plies (9), or

have opposite sign

(by (1)), so either

|c| < 2|b|

|c| > 2|b| which together with (1) im plies

(10).

com pletes the proof of S ilv e ste r’s lemma, as The next step in the proof of 10.1 and 10.5 is the following. L e m m a 10.7.

For n > 2

c/>: St(n,Z ) -» E (n ,Z )

the k e r n e l o f the n atural hom om orphism

is c o n ta in ed in the su bgrou p Wn.

T h is

COM PUTATION O F K 2 Z

§10.

91

P r o o f by in du ction on n. Since the statem ent is certainly true for n = 1,

we may assume n > 2. As standard vector /3 in Z n ,

n-th b a sis vector (0, 0, ..., 0,

1). By S ilv e ste r’s lemma,

take the anygiven elem ent

in the kernel of can be written as a product g j. . . gfw with 1 < lt/3gx II < The equation

< ... < l|/3g1 ... grw|| = 1.

|jiS g 1 1| = 1 im plies that the Steinberg generator g1 must

leave

/S fixed, and it follow s inductively that every g^ must leave (3

fixed.

Further, sin ce 0 ( g 1 ... gfw) = 1 the elem ent w e Wn must also

leave /3 fixed. £ Thus the word gj ... gr cannot contain any Steinberg generator x £ with i = n. It may contain some x - • with j = n, but if so, using the

£

Steinberg relation s, we can push all such x^n to the left. Setting the £ product of all th ese x^n equal to x, this means that we can write g^ ... gfw as a product xt(y)w where t denotes the natural homomorphism i : S t(n -1 , Z) -> St(n, Z). Taking the ca s e

n = 4 for illu strativ e purposes, th is means that the two

m atrices K 2 (Z / m Z )

K 1(mZ)

K ^ Z -> ... .

Using the Mennicke, B a s s , Lazard, Serre theorem that S K 1 (m Z ) = 1, we conclude that the group K 2 (Z / m Z ) is also generated by 1—1, —1

If m

is a power of an odd prime, it follows from §9.9 that K2(Z/mZ) = 1. More generally, using the identity K 2 (A x AO s K2A x K2A" and the C hinese remainder theorem, this im plies the following. C o r o l l a r y 10.8. For m ee 0 mod 4, order 2.

// m 4 0 mod 4 then K2(Z/mZ) - 1. K. Dennis has recently shown that K 2(Z/mZ) has

§11.

Matsumoto’s Computation of

For any field F ,

of a Field

the group K2F has been described by generators

and relation s in the th e sis of H. Matsumoto, based upon earlier work by C. Moore. (See the lis t of references in the P re fa ce .) T h is sectio n w ill sta te Matsumoto’s theorem, and then derive some consequences from it.

In particular, following a letter from John T a te, the

group K2Q w ill be determined com pletely. Here Q denotes the field of rational numbers. If F

is a field, then we have seen in §9 that the group K 2F

ated by certain symbols {x,y

is gener­

Here x and y range over the m ultiplica­

tive group F® = F — lO l T H E O R E M 1 1 . 1 (M atsu m oto).

T h e a b e lia n group K 2 F

has a

p rese n ta tio n , in term s o f g en era to rs an d r e la tio n s , a s fo llo w s . T h e g iv en g en era to rs {x,y S, w ith x a n d y in F #, a r e s u b ­ j e c t on ly to the fo llo w in g r e la tio n s an d th eir c o n s e q u e n c e s : (1) { x , l - x } = 1 fo r x ^ 0 ,1 , (2)

{ x 1x 2 ,y} = { x x ,y} { x 2 ,y},

an d (3) \x,y1y 2 \ = \x,y1 \\x,y2 \. The proof w ill be given in §12. Since the kernel Cn of the central extension S t(n ,F ) -» S L (n ,F ) is generated by corresponding symbols { x , y } which s a tis fy a ll of th ese re­ la tio n s ,* there is a can o n ical su rjectio n K2F immediate consequence.

See §9. 7, 9. 8, 9. 11, and 9. 12.

93

Cn< The following is an

94

A L G E B R A IC K -T H E O R Y C O R O L L A R Y 11.2.

F o r an y f i e l d F , C3 -

C4 -

C5

the g rou p s ' -

a re c a n o n ic a lly iso m o rp h ic to e a c h other, an d to th eir d ir e c t lim it K2F . It follow s that the Schur multiplier H2S L (n ,F ) is also isomorphic to K2F ,

for any n > 3, providing that we exclude the exceptional ca s e

n = 3 and

|F| = 2 or 4, and n = 4,

|F| = 2.

(Compare p. 4 8 .)

Steinberg Symbols Here is a reformulation of Theorem 11.1. C O R O L L A R Y 11.3.

G iven any b im u ltip lic a tiv e sy m b ol x,y

c(x ,y )

on F®, w ith v a lu e s in a m u ltip lic a tiv e a b e lia n group A,

s a tis ­

fyin g the id en tity c (x ,1— -x) = 1, th ere e x is t s on e an d on ly on e hom om orphism from K2F

to A

w hich c a r r ie s th e sy m b o l |x,y| to c(x ,y ) fo r a l l x an d y. Any such bim ultiplicative mapping c : F * x F * -> A to an abelian group, satisfyin g c ( x ,l —x) = 1, will be called a S tein b erg sy m b o l on the field F . A c la s s ic a l example of such a Steinberg symbol is the Hilbert quad­ ratic residue symbol in a lo c a l field, which can be defined by setting c(a ,b ) equal to +1 or —1 according as the equation ax2 + by2 = 1 does or does not have a solution x,y in the field.

(See for example O ’Meara,

In trodu ction to Q u adratic F orm s, p. 164.) More generally, if F

is a lo ca l

field containing the n-th roots of unity, then the n-th power norm residue symbol, with values in the group of n-th roots of unity, can be defined. (See §1 5 .9 .) Note that a Steinberg symbol is n ecessa rily skew-symmetric c(x ,y ) = c (y ,x )“ 1,

S T E IN B E R G SYM BOLS

95

and n ecessa rily s a tis fie s the identity c(x, —x) = 1. In fact, sin ce —x = (1—x )/ (l—x - 1 ), the identity c ( x ,—x) = 1 can be proved by dividing the equation c ( x ,l —x) = 1 by c ( x , l —x ”"1) = c(x

1,1 —x 1) 1 = 1.

Now multiplying c(x ,y ) by c (x ,—x), dividing by

c (x y ,—xy), and then multiplying by c(y - 1 ,—y - 1 ), we obtain the equation c(x ,y ) = c ( x ,- x y ) = c(y _ 1 ,- x y ) = c (y - 1 ,x), which proves skew-symmetry. We w ill make frequent use of th ese fa c ts in

§ 12. REMARK. An important consequence of 11.2 and 11.3 is the following. T h ere is a o n e-to -o n e c o r r e s p o n d e n c e b etw ee n S tein b erg s y m b o ls on F w ith v a lu e s in A a n d c en tra l e x t e n s io n s o f S L (n ,F ) w ith k e r n e l A. (Here we assume that n > 3, and exclude the three excep tion al c a s e s SL (3, F 2)> SL (3, F 4),

and SL (4, F 2).) In fact any central extension 1 -» A

G t SL(n, F ) -» 1

determines a Steinberg symbol

where y denotes the unique homomorphism from S t(n ,F ) to G over S L (n ,F ).

(Compare p. 4 8 .)

C onversely, given a Steinberg symbol c,

N C Cn x A C S t(n ,F ) x A be the graph of the homomorphism {u,v} h» c(u,v) 1 from Cn to A. Then G = (S t(n ,F ) x. A)/N is the required central extension, with 1 -» A -> G -> S L (n ,F ) -> 1.

let

96

A L G E B R A IC K -T H E O R Y

Expressed in terms of homological algebra, the co llectio n of a ll isomor­ phism c la s s e s of central extensions with kernel A forms a group H2(S L (n ,F ); A) as Hom(H2S L (n ,F ), A) as Hom(K2F , A) which is isomorphic to the group of Steinberg symbols with values in A. If the field F has a topology, so that. S L (n ,F ) is a topological group, it is natural to look for central extensions which are also topological groups. D E F IN ITIO N .

If S is a Hausdorff to p o lo g ica l group, then a t o p o lo g i­

c a l group e x ten s io n will mean an e x a c t s e q u e n c e

1~>A^G^S->1 where A and G are Hausdorff topological groups, the homomorphism t being continuous and clo sed , and the homomorphism i[j being continuous and open. A

(T h ese la s t conditions are equivalent to the requirement that

embeds homeomorphically as a closed subgroup of G,

quotient G / A

and that the

maps homeomorphically onto S.)

Matsumoto proves the following.

We assume that F

is Hausdorff,

with continuous addition, m ultiplication, and division, and that A is a commutative Hausdorff topological group. A SSE R TIO N 11.4.

A S tein b erg sy m b o l c on F

w ith v a lu e s in

A g iv e s r is e to a t o p o lo g ic a l c en tra l e x te n s io n o f S L (n ,F )

if

an d on ly i f c is con tin u ou s a s a fu n ction o f two v a r ia b le s an d s a t i s f i e s the co n d itio n lima,b^Oc (a ’ 1+ab) = L For the proof that th ese conditions are su fficien t, we refer the reader to Matsumoto. lows.

The proof that they are n ecessary can be sketched as fol­

Given such a topological central extension, let y be the unique

homomorphism from S t(n ,F ) to G over S L (n ,F ). the homomorphism b i-> y(xb2)

We w ill first show that

97

S T E IN B E R G SYM BOLS

from

F to

G is continuous.

Clearly it su ffic e s to prove continuity as b

tends to 0. The function g from

h> [ y (x j3),

G to its e lf is certainly continuous.

g] H ence, given a neighborhood

U

of the identity in G, there e x is ts a neighborhood V of the identity so that [y (x *3), g] e U whenever g e V. Since the function b H e j 2 = Min(v(x), v(y)).

The asso ciated valuation ring A C F

of all x with v(x) > 0, together with the zero elem ent of F .

co n sis ts There is a

unique maximal ideal SJJ C A; and the quotient A/^J is called the residu e cla ss field F . L e m m a 11.5.

The formula dy(x, y) = ( ~ l ) v(x)v(y)xv(yV y V^

d efin es a continuous Steinberg symbol dy on F

mod ^5

with values

in the d iscrete group F * = (A/^5)e. (Compare Serre, Corps locaux, p. 217.) T h is dy is called the tame symbol asso ciate d with the valuation v. Evidently dy gives rise to a homomorphism from K2 F Proof of 11.5.

onto the group F * = K 1(F ).

The element ±xv^ V y V^

is a unit of A, sin ce both

xv(y) and yv(x) have the same image (namely v(x)v(y)) under v. It is clear that dy is bim ultiplicative, and continuous in the v-topology. proof that dy( l —x, x) = 1 w ill be divided into sev eral c a s e s . then x

hence l —x = 1 mod

If v(x) > 0,

and v ( l—x) = 0, so that

( _ 1 ) V ( 1 - X ) V ( X ) ( 1_ X)V(X)/ X V ( 1 - X) = ( 1 _ x ) v ( x ) s

The proof when v ( l—x) > 0 is sim ilar. x _1 e

The

J m od

Now suppose that v(x) < 0.

Then

hence the quotient ( l —x)/x = —1 + x - 1 = —1 mod ^

is a unit.

Therefore v ( l—x) = v(x), and ( l _ x )v (x )/ x v ( l~ x ) = ((1 _ x ) / x ) v (x )

s (_ 1}v(x) mod sp.

Multiplying by the sign (—l ) v( 1—x )v(x)= ( _ l ) v(x ), we obtain 1 mod $J3, as required.

The ca s e v ( l—x) < 0 is sim ilar.

v(x) = v ( l—x) = 0 is trivial, this proves 11.5. »

Since the remaining c a s e

GAUSS AND Q U A D RA TIC R E C IP R O C IT Y

99

G auss and Quadratic R eciprocity To illu strate th ese concepts let us look at the field Q of rational num­ bers.

What Steinberg symbols c(x ,y ) can be defined on the field Q?

For any prime p, the p-adic valuation berg symbol

d

(x,y)

on Q gives rise to a Stein ­

with values in the c y c lic group (Z/pZ)® of order

P p—1. If p is odd we w ill denote this symbol briefly by (x,y)p, and its target group (Z/pZ)® by A . For p = 2 this construction is u se le ss.

However a 2-adic symbol

(x ,y )2 can be defined as follow s. Any non-zero rational can be written uniquely as a product of the form ±2^5^u, where k equals 0 or 1, and where u is a quotient of integers congruent to 1 modulo 8. Now if X =

(—1 ) ^ 5 ^ ,

y

=

( - 1 ) ^ 5 ^ ;

then se t ( x , y )2 = ( _ l ) i I + j K + k J

Thus the target group A2 is the c y c lic group S± 1 \. The verification that this is a w ell defined Steinberg symbol w ill be le ft as an e x e rcise . REM ARK.

T h e follow in g a s s e r t io n may help to motivate the definition

of (x .y )pF o r any prim e p s u p p o s e that a S tein b erg sy m b o l c : Q®xQ® -> A, w ith v a lu e s in a H a u sd o rff t o p o lo g ic a l group A, w ith r e s p e c t to the p -a d ic to p o lo g y on Q*.

is con tin u ou s

T h en th ere is on e

an d on ly on e hom om orphism from Ap to A w h ich c a r r ie s the sy m b o l (x,y)p

to c(x ,y ) for ev ery x an d y.

B riefly speaking,

(x,y)p is the “ universal continuous Steinberg sym bol”

for the p-adic topology on Q*. T h is statem ent is a sp e cia l ca s e of a much more general theorem, due to Calvin Moore, which is proved in the Appendix. Here is an outline of the proof. L e t pn be any prime power which is greater than 2.

Then the congruence

100

A L G E B R A IC K -T H E O R Y

(4)

(1—rpn)p = l - r p n+1

follow s ea sily from the binomial theorem.

(mod pn+2) Now suppose that p is odd, and

that r is prime to p. L et u 1 denote any quotient of the form s/t with s = t == 1 (mod p).

Using (4), we note that u1 can be approximated arbi­

trarily clo sely , in the p-adic topology, by a power of 1-rp.

In fact we can

first choose i so that (1—rp/t

ee s

(mod p2),

then choose j so that (l-r p )^ P t ~ s

(mod p3),

and so on. Since c(rp, (1—rp)1) = 1 for every exponent i, that c(rp, U j) = 1 fo r every such Uj = s/t.

it follow s by continuity

But the entire m ultiplicative

group Q® is generated by such products rp, with r relatively prime to the fixed prime p. Thus we have proved that (5)

c(x ,u 1) = 1 If r and t

for a ll x in Q*.

denote integers prime to p, then it follow s immediately

from (5) that c ( r ,r ') depends only on the residue c la s s e s of r and r/ modulo p. But, applying Steinberg’s theorem that every symbol on a finite field must be trivial (§ 9 .9 ), this proves that (6)

c (r,r') = 1. L e t A denote a primitive root modulo p.

Then any x and y in Q®

can be written more or le s s uniquely in the form x = p ^ U j,

y = p V u ';

and it follows that c(x ,y ) = c C p ^ c ^ p y 1-1-*. Since the eq u alities c (^.P)P_1 = c(Ap _ 1 ,p) = 1 and c(p,p) =

C (- I,p )

= c(\;p)(P —1) /2

follow from. (5), the proof for p odd can now e a sily be completed.

GAUSS AND Q U A D RA TIC R E C IP R O C IT Y

101

For p = 2 a sim ilar argument shows that every number u which can be expressed as a quotient s/t with s == t = 1 (mod 8) can be approxi­ mated arbitrarily clo se ly , in the 2-adic topology, by a power of 9.

Using

the eq u alities c ( 9 ,—1) = c (3 ,—l ) 2 = c (3 ,(—l ) 2) =

1,

c ( 9 ,—2) = c ( 3 , - 2 ) 2 = 1, and c (9 ,3 )

= c (—3 ,3 )2 = 1,

it follow s by continuity that c(u ,—1) = c(u ,—2) = c(u ,3) = 1 for every such

u. Since —1, —2, and 3 generate a subgroup of Q® which

is everywhere dense, this proves that (7)

c(u ,x) = 1

for a ll x.

As an example, taking u = —5/3, it follow s that c(5 ,x ) = c (—3,x). Taking x = 4, we se e that c (5 ,4 ) = 1, hence c ( 5 ,—1) = c ( 5 ,—4) = 1, and therefore (8)

c (5 ,5 ) = c (5 ,—1) = 1.

Similarly the equation c ( - 5 , - l ) = c (3 ,x ) for x = - 2

im plies that c ( - 5 , - 2 )

1, and hence (9)

c (5 ,2 ) = c ( - l , - l ) .

Now combining (7), (8), and (9) with the evident equation c (2 ,2 ) - c ( 2 , - l ) = 1, we se e that

( - l ^ J s V ) = c ( - l , - l ) u + jK + kJ; which clearly com pletes the proof. ■ Using th ese Steinberg symbols (x,y )p, we are now ready to compute the group K2Q. Th

eo r em

11.6 ( T a t e ) .

T h e group K2Q is c a n o n ic a lly iso m o r­

p h ic to th e d ir e c t sum A2©A3©A5© ...,

w h ere

is th e c y c l ic

group 5± 1 1, an d w h ere Ap = (Z/pZ)® fo r p odd. In fact the isomorphism w ill be given by the correspondence

102

A L G E B R A IC K -T H E O R Y

Sx,yS

h

(x ,y )2 © (x ,y )3 © (x ,y )s © ...

for all x and y in Q®. T ate remarks that his proof of this theorem is lifted directly from the argument which was used by Gauss in his first proof of the quadratic re­ ciprocity law. (Compare G auss, D is q u is itio n e s A rith m etica e, Y ale Univ. P re ss 1966, pp. 84-98.) To start the proof, for each positive integer m le t L m denote the sub­ group of K2Q generated by a ll symbols Sx,y} where x and y are in­ tegers of absolute value < m. Then clearly L i C L 2 C L 3 C ...

with union K2Q. Note that L m = L m_ 1 if m is not a prime number. L EMMA 11.7.

F o r e a c h prim e p the q u o tien t group L /L

j

is c y c l ic o f order p—1. In particular the quotient L 2/ L 1 is trivial.

Assuming this lemma for

the moment, the proof proceeds e a sily as follow s. For each prime p the correspondence {x,y S h> (x,y)p defines a homo­ morphism from K2Q to Ap. If p is odd, it is clea r that this homomor­ phism annihilates Lp__i > but maps Lp onto the c y c lic group Ap = (Z/pZ)*. Hence it induces an isomorphism ^ p A 'p - i ^ Ap. On the other hand, for p = 2, this homomorphism maps the generator {—1 ,- 1 \ of L 1 onto the element (—1 ,~ 1 )2 = —1, and hence induces an isomorphism from h 1 = L 2 to A2> An easy induction now shows that, for each prime p, the corre­ spondence {x,y} y> (x ,y )2 © (x ,y )3 ® ... © (x,y )p maps the group Lp isom orphically onto the d irect sum A2 © A3© ... © Ap. Taking the direct limit as p -> oo, the Theorem follow s. To prove Lemma 11.7, consider the correspondence 0 : (Z/pZ) -> L p / L p ^ defined by the formula x

h> jx ,p !

modulo L p _ j .

103

GAUSS AND Q U A D RA TIC R E C IP R O C IT Y

Here x is to vary over a ll non-zero integers of absolute value le s s than p. To show that cj) is w ell defined, and a homomorphism, we suppose that xy = z mod p, where x, y and z are a ll non-zero integers of absolute value le s s than p. Then xy = z+pr with

|pr| < |xy| + |z| < (p—I ) 2 + p—1, hence

Now

|r| < p.

1 = z/ xy + pr/xy

30

1 = Sz/ xy,pr/ xyS = Jz/ xy ,p S

Therefore so that 0

\z,p\ = {xy,pj

mod L

mod L p l .

1?

is a homomorphism, and (taking y = 1) is w ell defined.

To prove that

is su rje ctiv e , note that Lp is generated by the sym­

bols {x,±p}, S±p,xi, and i±p,±pl, together with L p_ 1. Hence the identitle s

f -p ,- p ! =

= ^ (-l)

|±p,xS_1 = lx,±p! = 0 (x ) and

S-p»p!

mod L p_ 1, mod L

x,

= ip ,-p != l»

show that

c/> is indeed su rjectiv e.

most p—1

elem ents. Sin ce we already know, using the symbol (x,y)p,

that L p / L p ^ j

T h is proves that L / L p ^ has at

has at le a s t p—1 elem ents, this com pletes the proof. ■

Another way of stating our conclusion is the following. C O R O L L A R Y 11.8.

G iv e n a n y S t e in b e r g s y m b o l

r a t i o n a l n u m b e r s , w it h v a l u e s in a n a b e l i a n g r o u p

c(x ,y ) o n t h e A,

th er e

e x i s t u n iq u e h o m o m o r p h i s m s

: AP -* A s o th a t

c(x ,y ) = J J 9^p((x .y)p). t h e p r o d u c t b e i n g t a k e n o v e r a l l p r im e n u m b e r s

p.

In this formulation, the result could have been proved d irectly, without ever mentioning K 2 To illu strate this corollary, consider the lo ca l symbol ( x ^ ) ^ , fined by

de­

104

A L G E B R A IC K -T H E O R Y

1 4-1

if x > 0

or y

>

0

(x >y)oo = \

I —1 if x , y < 0,

which is asso ciated with the embedding of the rational numbers in the real numbers. (Compare § 8 .4 .) This is the “ universal continuous Steinberg sym bol” for the archimedean topology of Q. According to 11.8 there must be a relation of the form

(x>yL=II ^P((x>yV• In fa ct one has the following. Qu a d r a t i c R e c

ipr o c it y

L

aw.

T h e sy m b o l ( x . y ) ^

e q u a l to the product, o v er a l l p rim es p, th e H ilbert sy m b o l ( ( x ,y ) ) p = ±1

is

o f ( ( x , y ) ) n , w h ere

is d e fin e d to b e (x ,y ) 2

if p = 2 an d is d e fin e d by the co n d itio n ((x ,y ))p = (x,y ) p ^p “

1 )/ 2

mod p

if p is odd. P ro o f.

It is clear from the Corollary that there e x is ts some relation of

the form

(x,y)oo=II ((x>y))p pwhere the exponents e 2 , £ 3 , £ 5 ^-x = y = —1

must be either

we s e e that the exponent e 2

0

or

must be 1.

1

. Taking

If p is a prime of

the form 8k±3, then sin ce ( 2 ,^

= 1, (2,p )2 = - 1 ,

we must have

(2,p))pEP = -1, so that Sp cannot be zero.

Sim ilarly, if p is a prime of the form* 8k+7

(or 8k+3), then the equations

(-1.PL = 1 imply that

C-1.P)2 = -!

cannot be zero.

There remains only the ca s e of a prime of the form Gauss we prove the following.

8

k + l.

Follow ing

105

GAUSS AND Q U A D RA TIC R E C IP R O C IT Y L EMMA 11.9.

I f p is a prim e o f the form 8k + l,

e x is t s a prim e q
/p. Then we w ill prove that N

ee

0 mod (m+1)! ,

thus yielding a contradiction. We w ill u se the notation [f] for the largest integer F ir s t note, following G auss, that in order to prove a congruence of the form a 1a 2 ...a ^ = 0 mod n! it su ffic e s to prove, for each prime power qs < n, that at le a s t [n/qs ] of the factors aj are d iv isib le by qs . The congruence then follow s e a sily , using the identity n!= T T Thus in our c a s e , for each prime power q

s

c

q M i s d e fin e d by the form u la p(tmt') = m. (Of course p is not a homomorphism.) P r o o f o f 12.6.

The first statem ent follow s immediately from 9 .1 5 , or

can be proved by using su itab le elementary row and column operations.

To

prove the second, suppose that t xmt2 = t 3rn t4 . Multiplying on the le ft by t j 1 and on the right by t j 1, it follow s that tm = nTT for su itab le t ,t ' Now examine the standard formula which e x p resses the determinant of the matrix tm as a sum of n! monomials. Inspection shows that p re cise ­ ly one of th ese monomials is non-zero: namely the monomial whose n factors are p recisely the n non-zero entries of the matrix m. Since a sim ilar statem ent holds for the product nTT, this proves that m = nT. m

114

A L G E B R A IC K -T H E O R Y

T o simplify the n o tation , it will be convenient to use a sing le letter

to denote a pair of con secu tive ind ices ( i,i+ l). will be denoted by —a.

a

The reversed pair ( i+ l,i)

Thus da (u) stands for the diagonal matrix with

entry u in the i-th p lace, u™1 in the (i+ l)-s t p lace and l\s along the d iag o n a l otherw ise.

Similarly ma (u) stands for the monomial matrix with

entry u in the a-th p lace,

—u” 1 in the - a -th p lace, and 1 ’s along the

diagonal otherw ise. Although the function p : S L (n ,F )

M is not a homomorphism, note

that it does sa tisfy the condition p (d s) - dp(s), p(sd) = p(s)d for any diagonal matrix d in S L (n ,F ).

The following properties of p

will be particularly important. LEMMA 12.7.

p(ma ( l) s )

F or each

s

in 3 L (n ,F )

is e q u a l e ith e r to ma (l)p (s ),

th e e x p r e s s io n or to da (u)“ 1p(s)

for

so m e u n iqu ely d eterm in ed ele m en t u o f F @. S im ilarly p(sm ^(—l) )

is eq u a l e ith e r to p(s)m^g(--l) or to p (s)d^(v)

for so m e v. Rem

ark

.

We are being careful to write th ese formulas in such a man­

ner that they remain true in the Steinberg group, with wa in place of ma and ha in p lace of dfl. P r o o f o f 12.7.

(Compare § 9 .1 5 .) In the expression s = tmt/ we can

write t' uniquely as a product ejg t^g, where ejg denotes the elementary matrix with /3-th entry equal to v, and where tg

stands for any matrix

in T whose /3-th entry is zero.

Then

U)

= tmejg i n g (—1) tjg

for some tg .

If v happens to be zero, this shows already that

p (sm p (~ 1 ) ) = p ( s )m ^ (- l) . Suppose then that v ^ 0.

L e t rr be the permutation which is a s s o c i­

ated with the monomial matrix m, and let /3 = (j,j+ 1 ).

If

ttQ)

< rrQ+D,

§ 12 .

115

P R O O F O F MATSUMOTO’S THEO REM

so that the conjugate mejg m“

belongs to T a lso , then we can push the

factor ejg to the left in equation (1) and conclude again that p(sm^g(—1)) = p(s) m^g(—l) . If 7r(j) > 7r(j+l), then substitu te the identity v v - 1 —v /\ e£ = e - £ e £ m/3 -1

into (1), and push the factor e__^g

to the le ft and the factor e^g

to the

right. It follow s that sm^g(—1) = Since m^g(v)m^g(-l) = dg(v), this proves that p ( s tn g (- l) ) = p(s)d^g(v) in this ca s e . The computation of p(ma ( l ) s )

is sim ilar.

L e t s = ta e “ umt/, so that

ma d ) s = Then if

77

1 (i)
7r 1(i-bl), then substituting e ~ u = ma (“ u) e^e ~a

, and pushing

the two elementary m atrices to the le ft and right, we obtain ma ( l ) s = t #ma (l)m a ( - u ) m r . Since ma ( l ) ma (—u) = da (u)“ 1, this com pletes the proof of 12.7. « Rem

a rk

.

For later u se, we w ill need the equations which were used

above in a slig htly more e x p licit form. To each monomial matrix m and each index pair /3 a sso c ia te the field element f(/3,m) which is defined by the equation

Then if (w) = m, it follow s that wh^g(v) w- 1 = c(f(/3,m), v)h^(^)(v). (Compare §§9.4, 1 2 .5 .) Note that f(~/3,m) = Now if (2)

s = tme^ tg ,

.

with v ^ 0, then either

s m ^ ( - l ) = t e ^ ’^ v m m ^ (~ -l)r

116

A L G E B R A IC K -T H E O R Y

(3)

s m ^ - l ) = t e ^ )m>V_1 md ( v ) t"

is the required equation.

Similarly, if s = ta e ” umt with u ^ 0, and if

a = rr(y)} then either (4)

ma ( l ) s = t'ma (l)m e y- uf(y-n,) ' 1t

or (5)

ma ( l ) s = t ,da (u)“ 1m e~y^ ,m^u

t

is the required equation. We are now ready to give Matsumoto’s ingeneous construction of a cen ­ tral extension of S L (n ,F ).

L et X C S L (n ,F ) x W

be the se t of a ll pairs (s,w ) satisfy in g the condition p(s) - 0(w ). Since there is no obvious way of making this s e t X into a group, we do something e ls e instead.

L e t G be the group of permutations of X which

is generated by certain permutations A(h), p(t),

and r\a , defined as follow s.

For each h e H let A(h) denote the permutation A(h) (s,w ) = (c/>(h)s,h w ) of the s e t X. It is clear that this transformation preserves the condition p (s) = 0(w ),

and that A maps the group H in je ctiv ely into the group of

a ll permutations of X. For each t e T let p(t) denote the permutation p (t)(s,w ) = (ts,w ). C learly p maps T in je ctiv ely into the group of a ll permutations of X. For each a = (i,i+ l) le t r\a denote the permutation defined by settin g ?7a (s,w ) equal to either (ma ( l ) s,w a ( l ) w) or (ma ( l ) s,h a (u)~~1 w) accord­ ing as p(ma ( l ) s )

is equal to m (l)p (s ) or da (u)~"1p(s).

Clearly this

definition is concocted so as to preserve the condition p(s) = pS(w). It is not d ifficu lt to ch eck that W a = which shows that rj is used here.)

is indeed a permutation.

(The identity c(u ,—u) = 1

§12.

117

P R O O F O F MATSUMOTO’S THEO REM

T h ese permutations \(h), p(t), and rj

must certainly generate some

subgroup G of the group of a ll permutations of X.

The key lemma is now

the following. L EMMA 12.8.

T h is group G o p e r a t e s in a sim p ly tra n sitiv e

m anner on X. In other words, given any (s,w ) and ( s 'w ') in X, there is one and only one g e G with g(s,w ) = ( s 'w ') . The proof of transitivity is comparatively easy .

As in the proof of 9 .1 5 ,

we note that S L (n ,F ) is generated by T and the elem ents ma ( l) .

So

operating on (s,w ) by some sequence of the permutations p(t) and rja we can certainly transform the first component s of (s,w ) to s'. is we can find a gQ e G with g0(s,w ) = (s',w *).

That

Now sin c e both (s',w ')

and (s',w *) belong to X, we conclude that w'== w* modulo the subgroup A of W. Hence operating on ( s 'w * ) by a su itab le A(a) we obtain (s',w '). T h is proves the e x iste n ce of g with g(s,w ) = (s',w ').

The proof of

uniqueness is more d ifficu lt, and will be given later. Assuming the e x iste n ce and uniqueness of such an elem ent g for the moment, we e a sily prove the following. T H E O R E M 1 2.9.

T h e group G i s a cen tra l e x te n s io n o f

S L (n ,F ) w ith k e r n e l A(A) ^ A. P roo f.

F ir s t note that the action of any group elem ent g on the first

component of any pair (s,w ) e X ment W(g) of S L (n ,F ).

is ju st le ft m ultiplication by some e le ­

T h is fact is true for the generators of G, and

hence is true for arbitrary elem ents of G. T h is defines a homomorphism W: G

S L (n ,F ).

Since G acts transitively on X, it follow s that W is

su rjectiv e. The kernel of T can be computed as follow s. If W(g) = l f then g(s,w ) must be a pair (s,w ').

The equation

that w '= aw for some a e A. Thus g(s,w ) = A(a) (s,w ).

p(s) = 0(w ) = (w') im plies

118

A L G E B R A IC K -T H E O R Y

Using the simple transitivity of the action of G, this proves that g = A(a). Therefore the sequence 1

A ip A -> G -> S L (n ,F )

1

is ex act. Since inspection shows that each A(a) commutes with each generator of G, this proves Theorem 12.9, modulo the Lemma. But to complete the proof we must s till verify Lemma 12.8. fication will be based on the following construction. of permutations, acting on the right of the se t X, certain permutations A*(h), /i*(t), and

The veri­

L e t G* be the group

which is generated by

constructed as follow s.

For

each h in H let (s,w)A.*(h) = (s(h),wh). For each t in T le t (s,w )/i*(t) = (st,w ). F in ally , for each /3 = ( j,j+ l) , w ^ D )

define ( s , w ) t o be either (sin g (--l),

or (sm^g(™l), wh^(v)) according as p(sm ^(—1)) is equal to

p ( s ) o r A (h), (i (t),

p (s)d ^ (v).

(Compare 1 2 .7 .) Clearly th ese permutations

and rj^ generate a transitive group

G of permutations

which operates on the right of the se t X. LEMMA 12.10.

E a c h e l e m e n t o f t h e p e r m u t a t io n g r o u p

«

•

m u t e s w it h e a c h e l e m e n t o f t h e p e r m u t a t io n g r o u p

G com -

^

G .

In other words the “ a sso cia tiv e law ” (g x )g * = g(xg*) is valid for every g e G, x e X,

and g* e G*.

To prove this

s u ffices to consider the sp e cia l c a s e where g is one

of the

law, it clearly generators of

G and g* is one of the generators of G*. If g is a generator of type A(h) or type A*(h) or //*(t), difficulty.

or if g* is a generator of

then this a sso cia tiv e law can be verified without

So we will concentrate on the c a s e

g=

Note also that the first component of rja (s,w )

77

*

and g * =

77*

P with either placem ent

119

P R O O F O F MATSUMOTO’S THEO REM

§12.

of parentheses, is clearly equal to ma ( l ) sm ^(—1).

So we need only con­

centrate on the second component of t? (s,w )t 7 q. a JO L et x = (s,w ), let s = ta e ” um e^t^, and le t n be the permutation which is a sso ciate d with the monomial matrix m =

0

(w).

If n{( 3) ^ ±a, then the second component of

Case 1.

77

(s,w ) 77*

p

is

equal either to wa ( l) w w ^ ( - l ) or ha (u)_ 1 w w ^ (-l), if v = 0 or

7r(j)




77O +I).

In each of th ese c a s e s , the result does not

depend on the placement of parentheses! C ase 2 . If 77(78 ) = a,

then it follow s from equation ( 2 ) that the second

component of Va (xr/ ^ ) (6 )

equal to ha (u - f(/3,m)v)—I ww^(—1 ),

providing that u ^ f(/3,m )v. ponent of

(77

a

x) 77*

p

On the other hand, using (4), the second com­

is equal to

(7)

wH(l)w h ^ (v - f(/3,m)“ 1 u),

providing that u ^ f(/3,m)v. V erification of the identity ( 6 ) = (7) in the group W w ill be left to the intrepid reader. If u = f(jS,m)v, then the second component of rja x r]^ is equal to wa ( l ) ww^(—1 ) for either placem ent of parentheses! Case 3.

Suppose that

second component of (8 )

77

a

77( 78 )

= —a. Using equation (3) we s e e that the

(x 77* ) is equal to p

ha (u - f(—/3,m)v~1 r

1 w hg(v),

providing that uv ^ f(—78 ,m) and that v ^ 0. second component of (rja x) 77^ (9)

Sim ilarly, using (5), the

is equal to .

ha ( u ) - 1 wh/g(v - fC -jS.n O u "1),

120

A L G E B R A IC K -T H E O R Y

providing that u / 0 and uv / f(~-/3,m). In order to prove that (8) = (9), we push the factor w to the right in both expression s using 12.5, and sub­ stitu te h__a for h^"1 in the first factor of each expression. £= f ( ~ i t

Setting

therefore su ffices to esta b lish the equation

h _ a (u - f/v)h^a ( v ) c ( r \ v ) = h _ a (u )h _ a (v - f / u ) c ( r \ v - f/u), or in other words c(u — f / v ,v ) c ( r \ v ) = c(u,v — f/ u )c(f““1 ?v — f/u). Substituting z = f/uv, and can cellin g factors of c(u, v) and c(f~~'1,v) from both sid e s, it su ffices to prove that c ( l —z,v) = c ( u ,l—z) cCf”"1,1 —z). But this equality follow s e a sily from the equation c ( z , l —z) = 1. Rem ark.

T his is the only place in the entire proof where the full

force of the equation c(z,l~~z) = 1 is needed. F in ally we must se e what happens if u = 0 or v = 0 or uv = f(—j8,m). In order to take care of th ese three c a s e s , we must show that ha ( - f ( - , 8 , m) v

~

1) ~ 1w

h g (v )

= wa (l)w w ^ (~ l),

Of wa ( l ) v W g ( - l ) = ha (u )~ i w li^ (-f(~ ^ ,m )u “ i ), or wa (l)w h ^ (v ) = ha (u)” 1w w ^ (-l), respectively.

The proofs, although tedious, present no particular difficulty.

T his com pletes the proof of 12.10. ■ P r o o f that the a c tio n o f G on X i s sim p ly tra n sitiv e

(Lemma 12.8).

If gxx = g2x, then gX(xg *) = g2(xg *) for every g* in G *.

But G* acts transitively, so this proves that

g | = g2x / for every x ' in X.

Therefore g j = g2, which proves 12.8,

and com pletes the proof of 12.9. m

§12.

P R O O F O F MATSUMOTO’S THEO REM

Now we are ready to prove Matsumoto’s theorem 11.1. be the u n iv er sa l S tein b erg sy m b o l on the field

F.

121

L e t c : P®xF®

A

In other words le t A

be the abelian group which is defined by generators c(u,v) s u b je ct only to the relations c(u 1u2,v) = c(u 1,v )c (u 2 ,v), c(u ,v 1v2) = c(u ,v 1) c (u ,v 2), and

c ( u ,l—u) = 1,

and to the consequences of th ese relations. Let

5u,vS e K 2 F be the symbol of Sectio n s 8 and 9.

bim ultiplicative and s a tis fie s

Since |u#v| is

{ u ,l—u j = 1, there is one and only one homo­

morphism rj : A -> K2 F which carries c(u,v) to {u,vi for all u and v in F .

We must prove that

j] is an isomorphism. Let

1

It is not

A -> Gn-> S L (n ,F ) -» 1 be the central extension of Theorem 12.9.

d ifficu lt top ass to the d irect limit as n

oo, thus obtaining a

corresponding central extension 1 -> A

G -> S L (F ) -> 1.

But according to §5.10 the extension 1 -> K2 F -> S t(F ) -> S L (F ) -> 1 is the universal central extension of S L (F ).

So there is one and only one

homomorphism £ : S t(F ) - G which covers the identity map of S L (F ).

Clearly f

maps K 2F

into A.

Comparing §8.3 with §12.1, and recallin g that H s X(H) C G, we s e e that f

carries

{ufv} to c(u,v) for all u and v. But rj carries

{u,vi for a ll u and v. Since K 2F

c(u,v) to

is generated by the symbols {u,v|,

and A is generated by the symbols c(u,v), this com pletes the proof that K 2F s

A. m

§13.

More about Dedekind Domains

L et A be a Dedekind domain with quotient field F . TH EO R EM 13.1 ( B a s s , T a t e ) .

We w ill prove

T h ere is an e x a c t s e q u e n c e

K 2P -> © KjA/t) -> K XA -» K jF -» © KQA/p -> KQA

KQF -> 0,

w h ere both d ir e c t su m s e x te n d o v er a ll n on -zero prim e id e a ls p

o f A.

Compare B a s s , A lg e b r a ic K-T h eory , pp. 702, 323. Most of the proof will be given below; but one key step w ill depend on a reference to B a s s . REMARK.

There is of course a natural homomorphism K2A

the domain A has only countably many id eals, then B a s s

If

has recently

shown that the extended sequence K2A -» K2F -> © K ^ K /p One would like to push even further to the left.

K2 F .

is a lso ex act.

For example, if F

is a

number field, so that K 2A/p = 0, one would conjectu re that K2A in je cts into K2 F .

But so far this is known only for the sp e cia l c a s e A = Z.

(Compare §10.2 and § 8 .4 .) For a general Dedekind domain, no one has suggested a su itab le homomorphism from © K 2A/p to K 2A. To begin the proof of 13.1, let K QA/p

KqA

be the homomorphism which sends the standard generator [A/p] to the difference [A1] — [pi K j F -> 0

KQA/p -* K qA

K QF -> 0

is ea sily verified. ( REMARK.

T h e group

group of ^d ivisors” of A;

0

KQA /p

can of course be identified with the

and is canonically isomorphic to the group of

fractional id eals of A .) In order to define the homomorphism i^A/p

K jA

we use a construction due to Mennicke.

L et a and b be relatively prime

elem ents of A: aA + bA = A.. Then there e x ist elem ents c and d so that ad — be = 1.

Consider the

matrix a

b

c

d/

6 SL(2,A ) C G L (A ).

If we work modulo the normal subgroup E(A ), then this matrix depends only on a and b. For if ad' — b c/ is also equal to 1,

then computation

shows that a

c

b d

for su itable x; so that 'a c De f i n

it io n

.

b\ / a ■j * —a , d/ \c

b

,,, d

The Mennicke symbol

mod E(A) .

. aJ

I is defined to be the element

of the subgroup SL(A)/E(A) C K j A which i s represented by the unimodular matrix ^

jjY

§13,

MORE A B O U T D ED EKIN D DOMAINS

125

"b~ Kj A, w h ich is d e fin e d w hen_a_ e v e r a an d b a r e r e la t iv e ly prim e, is sym m etric, b im u ltip lic a ­

L EMMA 13.2.

T h is sy m b o l

tive, an d is not a lt e r e d if w e a d d a m u ltip le o f a to b or a m u ltip le o f b to a. Here we are thinking of K jA P ro o f.

as a m ultiplicative group.

The properties

r a - m

-

r a ■ u

]

are clear sin ce elementary column operations on a matrix correspond to right m ultiplication by elementary m atrices. It follow s that

K

H

. y - & ] - [ - : ] •

Furthermore, if u is a unit of A, then

tt-jSLCR) s Z/2Z. Cy

But the matrix

x ) co rresPon(^s to a generator of 7t1SL (2, R), and hence maps to the

non-trivial element of n-^ShiR).

§13.

129

MORE A B O U T D ED EKIN D DOMAINS

Thus the Mennicke symbol J jQ

is non-trivial.

Since the principal

ideal Ax sp lits as a product bQ> where y = 1 mod p, y = —1 mod q, it follows that -1 9

1 ^ (Here p

[respectively

q] is the ideal co n sistin g of polynomial functions

which vanish at the point (0,1) [respectively

(0, —1)] of the unit c ir c le .)

Therefore the elem ent

[i]

( - 1 ) ( 1 - [q]) € SK 1A

is non-trivial. (In fact it is not d ifficu lt to show that SK jA order 2 with generator

is c y c lic of

Compare B a s s , A lg e b r a ic K -T h eory , p. 7 1 4 .)

If R (x,y) denotes the quotient field of A, it is amusing to sp ecu late about the kernel of the natural homomorphism K2A -> K2R (x,y ), and as to whether it equals the image of a su itab le homomorphism from © K ^ A /q.

If

such a homomorphism e x is ts , then the composition K2A -» K2A/ q -» K2A should be m ultiplication by 1 -- [q]. Hence the elem ent {—1, —1} e K2A/ q should map to the product { - 1 , - 1 1 ( 1 - [q]) = (~ 1 )2(1 - [q]) = (“ l ) [ y in the kernel of K2A -» K2R (x,y ). ( - 1)

The question as to whether or not

is trivial seem s very difficult.

We return to the proof of Theorem 13.1. main, and let b be a non-zero prime id eal.

L e t A be any Dedekind do­ Then the correspondence

PI gives rise to a homomorphism from the group (A/b)* = K ^ A /p

to K 1A .

Forming the d irect sum over a ll non-zero primes, we obtain the required homomorphism

© K ^A /p

K XA .

In order to prove that the resulting sequence © K xA/b -> K t A ^ K 1F is ex act, we w ill need the following:

130

A L G E B R A IC K -T H E O R Y

Every matrix in SL(A) can be redu ced

LEMMA 13.6 (B a s s).

by elementary row and column operations to a matrix in the subgroup SL (2,A ). The proof works for any commutative A sa tisfy in g the condition that a non-zero element is contained in only finitely many maximal id eals. Proof of 13.6.

L et (a 1?. .. ,a n) be the la s t row of an arbitrary matrix

A in SL(n,A ), with n > 3.

C learly

A ax + ... + Aan = A . C ase 1. If the ideal generated by

is already equal to

the entire ring A, then elementary column operations can be used to re­ place an by 1.

Suitable row and column operations w ill then replace the

matrix A by a matrix of the form d iag (A ',l),

lying in the subgroup

SL (n—l,A ). C ase 2.

If a 2 = 0,

then elementary column operations w ill replace

a2 by 1, and we can proceed as in C ase 1. C ase 3. id eals, say

If a 2 ^ 0,

then there can only be finitely many maximal which contain the elem ents a 2 ,a 3 , . . . ,

and an -j,.

Of these id eals, suppose that the first r contain a 1? but that the remain­ ing s —r do not contain a 1 . Choose an element e of A so that e EE 1

mod m v

e

mod mr+1, . . . , mg .

ee

0

m r,

Now adding e times the la s t cok in g to the first, we replace a^ by a i + ean“ Clearly the ideal generated by the elem ents a l +ean' a2' a 3*

an - l

is equal to the entire ring A. Hence we can proceed as in C ase 1. Thus the given matrix is congruent modulo E(A) to a matrix in SL(n—l,A ), SL(2,A ).

and continuing inductively, it is congruent to a matrix in

T h is proves 13.6. m

§ 13 .

131

MORE A B O U T D ED EKIN D DOMAINS

P r o o f o f e x a c t n e s s o f the s e q u e n c e © K^A /b -> K^A -» K-^F. The image of the first homomorphism is clearly the subgroup of K j A generated by a ll Mennicke symbols K jA .

and hence is equal to the image of SL (2,A ) in

On the other hand the kernel of the second homomorphism is clearly

the subgroup SL(A)/E(A) of K 1A. But by 13.6 the image of SL (2,A ) is p recisely equal to SL(A)/E(A).

T h is proves that the above sequence is

ex act. Next we w ill prove the following. Again let A be a Dedekind domain with quotient field F . L EMMA 13.7 (T ate).

T h e group K^F

is g e n e r a t e d by th o s e sy m ­

b o ls {a, b! fo r w h ich a an d b a r e r e la t iv e ly prim e e le m e n ts o f the dom ain A. L e t L be the subgroup of K2F

P ro o f.

generated by a ll symbols

{a, bS, with a and b relatively prime in A. For any x and y we must prove that {x, y] e L.

in

F@

The proof w ill be by induction on the num­

ber of maximal id eals p for which both v^(x) / 0 and v^(y) ^ 0 (where v^ is the ^)-adic valuation). C a s e 1.

Suppose there are no primes p with v^(x) v^(y) / 0. In

the

group of fractional id eals of A, we can express the fractional ideal Ax uniquely as the quotient ct/b A.

of two relatively prime id eals contained in

Similarly we can write Ay = c / b ;

where ah

is prime to

cb by hy­

p oth esis. According to Lemma 1.8, there e x ists an ideal e which belongs to the ideal c la s s of b""1, and is relatively prime to

ch.

Setting

b e = Ab, and a = bx, we have expressed x as a quotient x = a/b with a and b relatively prime to

cb .

Similarly we can write y as a quotient c/d with c and d

relatively prime to both a and b.

Then

Sx, y| = ja , c| {b, d| {a, d !“ X {b, c } - 1 , which proves that jx , y! e L.

132

A L G E B R A IC K -T H E O R Y

C a s e 2.

Suppose that there is ju st one prime f) with v^(x) v^(y) ^ 0.

Choose an element z of P® so that

v^(z) = -1, but so that Vq ( z ) >

q ^ p.

for e v e r y prim e e v e ry

q with

0

v

and

( z ) vq ( y )

=

0

(In f a c t , l e t Az = f) * £ w h ere £ e If)! i s prim e to Setting

v ^ (y) ^ 0 .)

i = v^(x), j = v«p(y)? n o te th a t

|z1x, y| f L by C a s e 1.

(1 )

H ence

{z, y }" " 1 mod L .

lx , y } -

B u t a s h o r t argum en t s h o w s th a t th e e le m e n t d itio n

y ( l —z)**

s a t i s f i e s th e c o n ­

•

vq(z ) V q(y(l—z) )) = 0 fo r e v ery prim e

q.

T h e r e fo r e

\z, a ls o by C ase 1.

S in c e

y ( l —z )^ } f L ,

|z, 1 —z i = 1,

th e c o n g ru e n c e (1 ), it fo llo w s th a t

C a se 3.

th is p ro v e s th a t

|z, y ! e L .

U s in g

|x, y ! £ L .

F i n a l l y s u p p o s e th a t th e re a re

n d is t in c t p rim e s

bp

•••> f>n

fo r w h ich

vp .(x) vpX y) ^ w ith

n > 1.

C h o o s e an e le m e n t

w of

F® s o th a t

v ^ (w ) = - v ^ x ) , bu t s o th a t V q (w ) V q (y )

=

0

for

i]

T hen |xw, y! e L

and |w, yi e L

by th e in d u c tio n h y p o t h e s is ; s o it fo llo w s a g a in th a t p l e t e s th e p ro o f o f 13.7. m Now d e fin e a hom om orphism d : K2F

-> ®

K xA/p

|x, y ! e L .

T h i s com -

§ 13 .

as follow s.

MORE A B O U T D ED EKIN D DOMAINS

E ach generator {x, yS of K2F

133

is to map to the element in

the direct sum whose p-th coordinate is the tame symbol d^(x, y) of §11.5. We will prove the following. L EMMA 13.8.

T h e co m p o s itio n o f the hom om orphism s K 2F i

0

KjA/p -» K jA

is z ero. P ro o f.

By 13.7, it su ffic e s to prove that the element {a, bi in K2F

maps to zero in K jA

whenever

a and b are relatively prime elem ents

m1

n1 n and Ab = q x ... qg s ,

of A. Setting Aa = p x

... p f

m

we see that the image of ia, b| in KwV/cj • is equal to the residue c la s s nJ of a J modulo q-. Similarly the image of {a, b| in K,A/p- is the reJ mciprocal of the residue c la s s of b modulo p -. Therefore the image of {a, bl in K-j^A is equal to the quotient of

by

Since

by 1 3.2, this com pletes the proof of Lemma 13.8. ■

The proof that the sequence of 13.8 is actually ex a ct w ill be based on a very pretty result which we sta te without proof.

F irs t a definition..

Our

previous concept of “ Mennicke sym bol’ ’ can be generalized as follow s. L et

C A x A be the se t co n sistin g of a ll pairs (x, y) of relatively

prime elem ents of A.

134

A L G E B R A IC K -T H E O R Y

D E F IN IT IO N .

A M en n icke fu n ction on A with values in a commutative

group C will mean a bim ultiplicative function H : WA -> C such that /i(x, y) is unchanged if we add a multiple of x to y or a multi­ ple of y to x. L e t N denote the kernel of the natural su rjectio n G L(2, A) -» K XA. (Compare 1 3 .6 .) If p :

KUBOTA-BASS THEOREM.

tion on the D ed e k in d dom ain A,

(l

i s a M en n icke fu n c­

C

then the c o r r e s p o n d e n c e

S) -

b>

d e fin e s a hom om orphism from G L(2, A) to C w h ich a n n ih i­ la t e s N. Hence there e x ists one and only one homomorphism p from K jA ~ G L(2, A)/N to C so that the following diagram commutes: G L(2, A)

I

KXA

first.,r°.w- » W*

-

I"

-——JL

c

For the proof of this theorem, we refer the reader to B a s s , A lg e b r a ic K~ T heory, p. 298, or to B a s s , Milnor, and Serre, Publ. Math. I.H .E .S . 33, §§6-9. The K ubota-Bass Theorem will be applied as follow s. C = (©

L et

KjA/p)/d(K2F )

be the cokernel of the homomorphism d.

Then we will construct a Mennicke

function on A with values in C. For each (a, b) e W^, let ijj(a, b) e © K-j^A/b be the element whose b~th coordinate is equal to v. (a) (b mod b) ^

§13.

135

MORE A B O U T D ED EKIN D DOMAINS

if p divides a, and to 1 otherw ise. Evidently the function if/ is b i­ m ultiplicative, and if/( a, b) is unchanged if we add a multiple of a to b. Furthermore note the congruence if/( a, b) ee if/(bf a) mod d(K2F ).

In fact direct computation shows that */r(b, a)/if/(a, b) - d\a, b j. Therefore, defining fi(a, b) e C as the residue c la s s of iff (a, b) modulo d(K2F ), it follows that fi :

-» C is a Mennicke function.

Thus, according to

Kubota and B a s s , there is one and only one homomorphism JI : K XA -> C which carries the c la s s of ^

^

Next note that the composition

to fi(a, b). ® K 1A/p -> K XA

is equal to the

natural projection homomorphism (reduction modulo d(K2F ).

In fa ct the

image of a given element (b mod p) in the p-th summand can be evaluated as follow s. By definition the image b in K jA is equal to j j ^ j where LbJ

p £ = Aa, b' = b mod p, b ' ee 1 mod £ for some ideal £. V A

Then V"

= ^ a

//(a, b ') = (iff(a, by) mod d(K2F )).

Inspection shows that the p-th component of iff (a, bO is equal to (b mod p), and that a ll other components are trivial. Thus if an element in ® KjA/p maps to the identity in Kj A, hence maps to the identity in C, then it must belong to d(K2F ).

and T h is

shows that the sequence of 13.8 is ex act, and com pletes the proof of 13.1. ■

§14.

The Transfer Homomorphism

The resu lts of this sectio n are largely due to B a s s and T ate. L e t A be a ring (alw ays a sso cia tiv e with 1), larger ring such that F

is finitely generated and projective when con­

sidered as left A-module. by f : A -» F .

and let V I) A be a

The inclusion homomorphism w ill be denoted

We assume of course that f ( l ) = 1.

T h is sectio n w ill define the tra n sfer hom om orphism f* : I ^ r -> K j A for i = 0, 1, 2,

and prove the following property.

T H E O R E M 14.1.

If F

is com m u tativ e, an d i, j < 2,

then the

id en tity f*(x • f*(y )) = (f*x ) • y i s v a lid for e v ery x e K-F an d y e KjA. In other words the transfer homomorphism to K*A is K *A -linear. an example, taking x to be the identity elem ent 1 = [F ] e K QF ,

As

we obtain

the formula f*(f*(y )) = f * ( l ) • y • (Compare § 1 3 .4 .) R E M A R K 1.

B a s s defines the transfer homomorphisms K QF -> KQA

and K jF -» K^A assuming only that there e x is ts a finite A-linear resolu­ tion ° - P n - > Pn - l - " - P0 - r - 0 ’ where the

are finitely generated and projective over A. (A lg e b r a ic

K -T h eory , p. 451.) In this generality, the transfer homomorphism includes the homomorphisms KQA/a -» K QA and K jA / a -> K jA

of §13.

It would

be of great in terest to know whether or not the transfer homomorphism for K2

can also be defined in th is more general context.

137

138

A L G E B R A IC K -T H E O R Y

REMARK 2.

The terms “ norm” or “ restrictio n of s c a la r s ” are both

sometimes used for the transfer homomorphism. quite descriptive.

Certainly both terms are

I feel that the term transfer is useful sin c e it can also

be used for related homomorphisms in hom ological algebra and in topology. Thus, if II is a group and IT a subgroup of fin ite index, then the c l a s s i­ ca l transfer homomorphism Hi (H) -> H ^IT) is related to our f * : K y z n ) -> K j( z i r ) ; and is a sp e cia l c a s e of the “ topological transfer homomorphism” from the homology of a sp ace B

to the homology of a fin ite covering sp ace E .

The

analogous homomorphism K *(E ) -» K *(B ) of topological K-theory is clo sely related to our transfer K *(C

E

) -> K *(C

B

), where C

continuous complex valued functions on E .

F

denotes the ring of

(Compare §7.)

D efin itio n o f th e tra n sfer. Every finitely generated p rojective P F

over

can also be considered as a finitely generated projective over the sub­

ring A. We w ill denote the resulting A-module by P ^ ,

whenever it is

n ecessary to make the d istinction. There is clearly one and only one homomorphism f * : Kor

- K 0A

which carries each generator [P] of KQF To define the transfer on

to the generator [P/y] of K QA.

and K2 , we first define an embedding

f # : G L (n ,F ) -> GL(A). Any matrix X e G L (n ,F ) gives rise to a F -lin ea r automorphism of the free module F n, which will be denoted by X also.

Choose a p rojectiv e Q

over A so that the direct sum I 1 © Q is free, say on r generators, over A. Then F n © Qn is also A-free, and we can consider the A-linear auto­ morphism X © (identity map of Qn) of F n © Qn. Choosing a A -basis for this d irect sum, the automorphism X © (identity) is represented by a matrix which we denote by

§14.

139

TH E T R A N S F E R HOMOMORPHISM

f#(X) e GL(nr,A) C G L (A ). Proceeding as in §3.2, this homomorphism f# is w ell defined up to inner automorphism of GL(A). Now abelianizing, and taking the d irect limit as n -> *>, we obtain the transfer homomorphism from

to K jA . Sim ilarly, taking the Schur

multiplier of the induced homomorphism E (n ,D

E(A ),

and then p assing to the lim it as n -> oo, we obtain the transfer from K2F to K2A. D etails will be left to the reader. D E FIN IT IO N 14.2.

If A is commutative, then composing the natural

homomorphisms

r* _ Kxr -fH, KjA -iet A» we obtain the norm hom om orphism norm: F * -* A*.

As an example, if F

free over A with b a sis b ^ . .. , bn, then settin g yb- =

is

A - bj the norm

of y is clearly equal to the determinant of the matrix (A -).

Compare van

der Waerden, Modern A lg eb ra I. The proof of Theorem 14.1 w ill be based on the following a sso cia tiv e law.

L et M be a r-m odule and N a A~module, with F

d

A commutative.

Then M ® N - (M ® F ) ® N

r

a

a

is can onically isomorphic to M ® (F ® N).

r

a

To simplify the notation, we w ill denote either tensor product simply by M N. Thus the symbol ® stands for the tensor product over A; but we note that M ® N has the structure of a F-module whenever M is a F-module. P r o o f o f 14.1 for the c a s e i = j = 1. Given x e K xr ,

y e K XA.

choose a representative X e G L (m ,F) for x, and a representative Y e G L(n,A) for y. We w ill think of X and Y as automorphisms of the

140

A L G E B R A IC K -T H E O R Y

free modules P m and An resp ectively. sponding identity automorphisms. pm 0

L e t I and V denote the corre­

Form the direct sum of three copies of

and consider the P -lin ear automorphisms £ = ( x ® r ) © ( x ^ r ) ” 1 © (i® r )

and T) = (I ® Y) © ( I ® r ) © ( I ® Y ) - 1

of this direct sum.

L iftin g cf and 7] to the Steinberg group, we can form

the commutator £ * 1 e K 2r . By definition, £ *7 ] is equal to the product x-f*y. In order to transfer to K2A, it evidently su ffic e s to apply the homo­ morphism f # : G L (3m n ,r) -> GL(A) to f

and rj. Choose Q so that P © Q is A-free, and let i denote the Qmn ^ identity map of Q . Thus f (x-f*y ) is equal to the commutator (£ © 0 ★ (77 ©t) e K2A . Now we must compute f (x) •y.

The elem ent f*(x ) e K j A is repre­

sented by the A-linear automorphism X © I" of the direct sum P m © Q m; where I" denotes the identity map of Q m. Form the direct sum of three copies of (P m ©Qm) ©An, Setting = ( x ® n ® r ® ( x ® r ) - 1®i ' ® ( i ® i " ) ® r

and rl l = ( i ® n ® Y © ( i ® r ) ® r ® ( i © n ® Y - 1 ,

it follows that f*(x )-y =

* ii! t K2A .

In order to transform th ese exp ressions into something more amenable, we replace each (P m ©Qm)® A n by (P m®An)© Q mn, and then sh ift all of the Qmn summands to the right. Thus tjj

is transformed into something of the form

is transformed into f © i, 77 ©£.

Hence

f*(x)-y = (f ®0 * 0?®O e K2r •

and

§ 14 .

141

T H E T R A N S F E R HOMOMORPHISM

Sin ce the commutator is bim ultiplicative (§ 8 .1 ), this expression is equal to the product of (£ * 0 * (y © 0 = f (x •f*y ) and of the element (xy;

where 1 denotes the identity element of A. Thus the monomials xxy^ with 0 < i < n, 0 < j < n are to form a b a sis for A over F . The proof that such an algebra e x is ts is not d ifficu lt, and w ill be left to the reader. Now recall the definition of the Brauer group of F . F

An algebra A over

(alw ays a sso cia tiv e with 1) is s im p le if it has no two-sided id eals other

than 0 and A; and c en tra l if its center is equal to F I ^ F .

Wedder-

burn’s theorem a sse rts that every finite dimensional simple algebra over F is isomorphic to an algebra M^(D) co n sistin g of a ll kxk m atrices over

143

144

A L G E B R A IC K -T H E O R Y

some division algebra* D D F .

Furthermore, the integer k > 1 and the

isomorphism c la s s of D are uniquely determined by A. So we can define two such algebras to be sim ila r if their asso cia te d division algebras are isomorphic over F .

The B rau er group B r(F ) is the abelian group con­

sistin g of a ll sim ilarity c la s s e s of fin ite dim ensional central simple alg e­ bras over F ,

using the tensor product over F

as composition operation.

For further d etails, se e O ’Meara, or van der Waerden, Modern A lg eb ra , or Weil, B a s i c Number T h eo r y . T HE O R EM 15.1.

T h e a lg e b r a A = A^(a,/3) is c en tra l s im p le ,

an d h e n c e r e p r e s e n ts an ele m en t a ^ (a 9/3) of th e B rau er group B r(F ).

T h e resu ltin g fu n ction ua ) : F*x F *

B r(F )

is a S tein b erg sy m b o l on F . P r o o f. Consider the F -lin ear transformations T x(a) = xax- 1, T y(a) = y a y " 1

More g e n e r a lly , any sim p le artin ia n rin g is a m atrix a lg e b ra ov er a s k e w -fie ld . H ere Is a b r ie f proof. L e t tit C A b e a m in im al rig h t id e a l. S in c e A i s sim p le, the tw o -sid ed id e a l Attt m ust be eq u al to A; h e n c e the e le m e n t 1 ca n be re p re ­ se n te d a s a sum 1

€

a-jITt + a2ttt + ... a^ n t .

C h o o se su ch a re p re s e n ta tio n w ith k a s sm a ll a s p o s s ib le . T h en the k -fo ld d i ­ re c t sum tit© ... ©Tit is i so m o rph ic a s right A -m o d u le to A i t s e l f . F o r the c o rre ­ sp o n d e n ce

(m1,...,m k)

ajrrij + ... + akmk

is c e r ta in ly rig h t A -lin e a r and s u r je c t iv e . a l ml © ••• © a k mk ~

w ith sa y

m^ £ 0

If th ere w ere a re la tio n of the form and h e n c e

m^A = HT,

then the in c lu s io n

a k ttt~ a^m^A C a-jttt + ... + ak _^Ttl would imply th a t

1 e a^tlt + ... -f- a^_ ^Itt;

c o n tra d ic tin g the c h o ic e of k.

It fo llo w s th a t the ring End^(JXt©... ©IB), c o n s is t in g of a ll rig h t A -lin e a r m ap­ p in g s from TTt©... © ttl to i t s e l f , i s iso m o rp h ic to the ring E n d ^ (A ) A. B u t End^(1Tt©... ©HI) can be id e n tifie d w ith the ring o f kxk m a tr ic e s ov er the sk ew fie ld E n d ^ (m ) . T h is co m p le te s the proof. H

from A to its e lf.

145

POW ER NORM R E SID U E SYM BOLS

§15.

E ach b a sis vector x*y^ is an eigenvector both of T x

and Ty, with eigenvalu es

and co1 resp ectiv ely.

ately that the center of A is spanned over F 0 0 -i x y =1.

It follow s immedi­

by the sin g le element

To prove that A is sim ple, consider a non-zero two-sided ideal b, and choose an element b0 = X with say

^ 0.

^ i jx V

f 6

Then the element bi = x" Pb0y ~ q

of b can be written as

lP - x 1y^ with W00 £ 0.

Next consider the

element b2 = (T x“ "> ( T x " " 2) ••• of b.

Computation shows that b2 is equal to (1 - « 2 ) ... ( l - w 11- 1) ^

’V

-

Similarly the element b3 = (T y -fi,) (T y- « 2) ... (T y—con - 1 ) b 2 of b is equal to (1—a>)2 ( l —n - 1 ) 2n 1 = 0 , but no eq u a tio n o f s m a lle r d eg ree . I f f(x) s p li t s in to d is tin c t lin ea r fa c t o r s o v er F ,

then A is iso m o rp h ic to th e m atrix

a lg e b r a Mn(F ). P ro o f.

The subalgebra of A spanned by the powers of x is clearly

isomorphic to the quotient ring F[x]/(f(x)).

By the C hinese Remainder

Theorem, this sp lits as the cartesian product of n co p ies of F . it contains n mutually annihilating idempotents

Hence

146

A L G E B R A IC K -T H E O R Y

e je j = ei? e je j = 0 with e j + ... + en = 1.

for i ^ j,

Therefore A sp lits as the d irect sum e 1A ®...©enA

of right id eals, at le a s t one of which has dimension < n over F .

Com­

paring any proof of Wedderburn's theorem, it follows easily that A is a matrix algebra over F . m Assume once more that F

E X A M P L E 15, 3.

root of unity.

If a has an n-th root

F,

in

contains a primitive n-th

then the polynomial x n~a

splits into distinct linear factors, so it follows that a^ (a,/3) = 1.

In par­

ticular, this proves that a ^ (l,/3 ) = 1. P r o o f that the sy m b o l a (a,j8) i s b im u ltip lic a tiv e . The tensor product B = Aw( a ,0 ) ®Aa (a,y) r

has generators x, y, X,

and Y, su b je ct to the relations = a l,

(1)

xn

(2)

Xn = a l,

(3)

xX = Xx,

yn = j8 l,

Y n = y l,

yx = coxy,

YX = a)XY,

xY = Yx, yX = Xy, yY = Yy .

L et B / be the subalgebra generated by x and yY, subalgebra generated by x “"1X and Y. B'~

and

A^ a ^

and let B " be the

Then clearly

’ B" -

Aw U , y ) .

Note also that each generator of B ' commutes with each generator of B" Since B'® B^ is sim ple, it follow s that the natural map B ' ® B ' -> B is an isomorphism.

This proves that

A