I try to extend here some results of the paper *Moduli
spaces of decomposable morphisms of sheaves and quotients by non-reductive
groups*. Let *E*, *F* be decomposable sheaves on a smooth
projective variety. I find more linearizations of the natural action of
the group
on
such that a good quotient of the open set of semi-stable morphisms exists.
To obtain this, one associates to
another space of morphisms
in such a way that there is a natural bijection between the set of orbits
of an open subset of
and the set of orbits of an open subset of .
We can in this case deduce the existence of good quotients of some open
subsets of
from the existence of good quotients of the corresponding open subsets
of .