2. Stabilit y

Let A be B. ring . I f M i s a (right) ^4-module , the content o f a n elementJ C G M i s

defined t o be the left idea l

cont (JC ) = {/(x ) | / € Hom^(M , ,4)}.

We callJ C unimodular in M i f con t (JC ) = A; thi s is equivalent t o J C bein g a basis for a

free direc t summan d o f M. I f M i s the fre e modul e A

n

an d J C = (a x, •• • , a„) the n i t

is easily seen that con t (JC ) = Aax + •• • + Aa

n

.

EXAMPLE.

I f a G GLn(A) the n th e equatio n a"

1

a- I show s that th e column s of

a ar e unimodular i n A

n.

It i s natural t o as k whether, conversely , ever y unimodular elemen t o f A

n

i s a column

of an element o f GL

n

(A). Thi s is equivalent t o asking , for a given A -module P, whethe r

P®A^An implie s P^A n~~l.

The answe r is trivially affirmativ e i f 72=1. I f n - 2 and if A i s commutative, th e

answer i s again affirmative . Fo r i f J C = (£ ) i s unimodular w e ca n solve ad - be = 1,

whenceJ C i s a column o f ( a

b

c

d

) E SL

2

(A).

For « = 3 an d A th e coordinat e rin g R[x , y, z] o f th e rea l 2-sphere , x 2 + y2 +

z2

= 1, th e unimodular elemen t

(?)

of ^4

3

furnishe s a counterexample. Fo r if

is an element o f GL

3

(A) the n th e ma p t * ( / ( 4 £(0 A(0) f rom S 2 t o R 3 define s a

continuous vector field which is nowhere norma l t o S

2

(i t i s linearly independen t o f

(x(t),y(t), z(t)), s o its projection o n th e tangen t plan e i s a continuous nonvanishin g tangen t

vector field on S

2;

suc h is well known no t t o exist. (Thi s exampl e wa s first pointed ou t

by Kaplansky. )

PROBLEM (SERRE) .

Let A = F[r

x

, •*• , t

d

], a polynomial ring in d variables over

a field F . Given n 1, is it true that every unimodular element in A

n

is a column of

an element of GL n(A)l

7