Abstract: | A functorial filtration GL
n
=S–1L
n
S0L
n
S
i
L
n
E
n of the general linear group GL
n, n 3, is defined and it is shown for any algebra A, which is a direct limit of module finite algebras, that S–1 L
n
(A)/S0L
n
(A) is abelian, that S0L
n
(A)
S1L
n
(A)
is a descending central series, and that S
i
L
n
(A) = E
n(A) whenever i the Bass-Serre dimension of A. In particular, the K-functors k
1 S
i
L
n
=S
i
L
n
/E
n are nilpotent for all i 0 over algebras of finite Bass-Serre dimension. Furthermore, without dimension assumptions, the canonical homomorphism S
i
L
n
(A)/S
i+1 L
n
(A)S
i
L
n+ 1(A)/S
i+1
L
n + 1 (A) is injective whenever n i + 3, so that one has stability results without stability conditions, and if A is commutative then S0L
n
(A) agrees with the special linear group SL
n
(A), so that the functor S0L
n
generalizes the functor SL
n
to noncommutative rings. Applying the above to subgroups H of GL
n
(A), which are normalized by E
n(A), one obtains that each is contained in a sandwich GL
n
(A, )
H
E
n(A, ) for a unique two-sided ideal of A and there is a descending S0L
n
(A)-central series GL
n
(A, )
S0L
n
(A, )
S1L
n
(A, )
S
i
L
n
(A, )
E
n(A, ) such that S
i
L
n
(A, )=E
n(A, ) whenever i Bass-Serre dimension of A.Dedicated to Alexander Grothendieck on his sixtieth birthday |