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
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