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

Table of contents :
CONTENTS
Preface and Guide to the Literature
§1. Projective Modules and K0Λ
§2. Constructing Projective Modules
§3. The Whitehead Group K1Λ
§4. The Exact Sequence Associated with an Ideal
§5. Steinberg Groups and the Functor K2
§6. Extending the Exact Sequences
§7. The Case of a Commutative Banach Algebra
§8. The Product K1Λ ⊗ K1Λ → K2Λ
§9. Computations in the Steinberg Group
§10. Computation of K2Z
§11. Matsumoto’s Computation of K2 of a Field
§12. Proof of Matsumoto’s Theorem
§13. More about Dedekind Domains
§14. The Transfer Homomorphism
§15. Power Norm Residue Symbols
§16. Number Fields
Appendix — Continuous Steinberg Symbols
Index

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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