CHAPTER I

INTRODUCTION

This brief serie s of lectures o n algebraic ^-theor y i s addressed t o a n audience no t pre-

sumed t o be familiar wit h th e subject . Ther e is perhaps some merit therefor e i n beginning

with a n interesting theore m (Theore m 0 below) whose statemen t make s no reference t o

K-theory, bu t whose proof (th e onl y proof I know) invokes several basic techniques o f alge-

braic K- theory. Th e cours e will then evolv e a s an elaboration o f th e idea s introduced i n

this proof.

The important recen t wor k o f Quillen, Karoubi, Gersten an d other s o n higher ^-theor y

(cf. Proc . Battelle Conf . on Algebraic AT-theory , Springer Lectur e Notes , pp. 341, 342, 343 )

will not be treated du e t o limited time , and th e formidabl e technica l preliminaries necessar y

for thi s material. Nevertheles s th e present note s can serve a s some historical motivatio n fo r

the latter, in particular fo r Quillen' s construction o f ^-group s fo r categorie s with exact se-

quences.

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We shall use th e followin g notatio n throughout . Fo r a ring A, it s group o f units is

denoted A\ th e rin g o f n X n matrice s ove r A i s denoted M n(A), an d w e put

GLn(A) = Mn(A)\ th e genera l linear group o f invertibl e n X n matrice s ove r A. Th e

letters Z , Q, R, C, Fq denote , as usual, the integers , rational numbers , real numbers, com-

plex numbers, and a finite field wit h q elements , respectively.

Our first main objectiv e wil l be th e proof o f

THEOREM

0 . Let R be either Z or F [t] where t is an indeterminate. Let

A = R [t2, • ••, t

a

], a polynomial ring in d - 1 variables over R (where d 1). Then

GLn(A) is a finitely generated group for all n d + 2 .

Several comments abou t thi s theorem ar e in order here . Th e first is that it s statemen t

would have been understoo d an d appreciate d b y Frobenius , so that i t hardly give s the appear -

ance o f a theorem o f algebrai c £-theory .

The theore m admit s various natural generalisations. First o f al l it remain s valid if R

is allowed t o be th e rin g of algebrai c integers of a number field, or , more generally , a ring

of "^-integers " of a global field F ( = a finite extension o f Q o r o f F^(f)) . I n additio n

we may localise A b y invertin g an y number o f th e ^.'s , t o obtai n a ring of polynomial s

in som e variables, and "Lauren t polynomials " in th e others , and th e theore m remain s valid.

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I n fact a n introduction t o Quillen ^ theor y wa s actuall y presente d i n th e lectures , i n plac e o f th e

material prepare d i n Chapte r 2 o f thes e notes .

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http://dx.doi.org/10.1090/cbms/020/01