INTRODUCTION T O SOME METHODS O F ALGEBRAIC #-THEOR Y
11
Thus the stabilit y result s above have the effec t o f reducin g Theorem 0 to a question abou t
the behavio r of Kx unde r a polynomial extension . Thi s will be take n up in the following §3 .
(4) Th e conclusion o f Theorem (2.7 ) applies , more generally, to an y yl-algebr a finitely
generated a s an A -module, e.g. a group algebr a An wit h n a finite group . Th e argumen t
in tha t generalit y i s somewhat mor e technica l tha n tha t i n the commutativ e case which follows.
PROOF O F THEORE M
(2.7) . Suppos e n d + 1 an d letJ
C
= (a Q, av ••• , a n) b e a
unimodular elemen t in A
n + l.
W e seek b
v
••• , b
n
EA suc h that, if d
i
= a{ + btaQ
(1 i «), the n (a
v
••• , a
n
) i s unimodular.
Given m G I = ma x (A) w e can certainl y fin d a linear combinatio n c
Q
a0 + +
cn-ian-i o f QQ *"* » Qn~i wruc h adde d t o a
n
, yields an a"
n
= c
0
a0 + ••• +
cn_lan_l + an whic h is not zer o modulo m ; this is becauseJ C ^ 0 mod m. Usin g the
Chinese Remainde r Theorem we can even fin d c 0, •••, c n_t whic h accomplis h thi s simul-
taneously fo r al l m i n a given finite subse t S of X. W e do this for a n S larg e enoug h
to meet eac h o f th e irreducible component s of X (whic h ar e finite i n numbe r becaus e X
is Noetherian).
If d = 0 the n X i s finite, S = X, an d a"n i s invertible. I t follows tha t (a
l9
•••,
a
n-\-
a
n "* "
c oflo)
*
s
unimodular, s o the proof i s completed i n this case, and, more generally ,
whenever a
n
is invertible.
If a"
n
i s not invertibl e put A = A/Aa„ an d conside r the unimodular elemen t
x~0*o •" ^n-i)
intiie
^-modul e A
n,
wher e a denote s a modAa^ fo r aEA.
If di m max (A ) d - 1 the n we can appl y inductio n o n d t o obtain b t, - * •, b n_1 GA
so that (a
x
+ ^ ^ Q, ••• , ^n-\ + ^w -i^o) * s unimodular i n th e A -module A n~l. It
follows the n tha t (a
x
+ bxa09 ••• , *„_ ! + ^w_iflo flD is unimodular i n ^4" . Pu t AJ =
fli + ** flo 0 i n). Recal l then tha t
= * * +C
0
*
0
+ ( C 1«1 + •• * +^-lfln-l)-(Cl^lflO + * " + *I,-1*I,-1«O)
= *n + Vo + ( C1* 1 + ' " + *n-l*li-l )
where &
n
= c
0
- ( c^ + ••• 4 - cn__lbn_l). I t follows then , a s desired, tha t (a
x
+ b
x
a0,
,#* » cn + M o ) = (ai "" a «-1 - fl« + M o ) i s unimodular.
It remain s only t o justify th e inductio n ste p by showin g tha t di m max (A) d. But
max (A ) ca n be identified wit h th e (closed ) se t of al l m E X = max (A) whic h contai n
an. Sinc e a
f
n^ m fo r al l m e S i t follows fro m th e choic e of S tha t ma x (A ) con -
tains no irreducible componen t of X. Henc e a chain of irreducible close d set s in ma x (A )
can always be lengthened i n X, s o indeed di m max (A ) dim X d
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