We extend the methods of geometric invariant theory to
actions of non-reductive groups in the case of homomorphisms between decomposable
sheaves whose automorphism groups are non-reductive. Given a linearization
of the natural action of the group on ,
a homomorphism is called stable if its orbit with respect to the unipotent
radical is contained in the stable locus with respect to the natural reductive
subgroup of the automorphism group. We encounter effective numerical conditions
for a linearization such that the corresponding open set of semi-stable
homomorphisms admits a good and projective quotient in the sense of geometric
invariant theory, and that this quotient is in addition a geometric quotient
on the set of stable homomorphisms. In particular we study morphisms of
type
where *E*_{1}, *E*_{2} are simple sheaves with
.