Espaces abstraits de morphismes et mutations.   Journal für die reine und angewandte Mathematik 518 (2000) , 41-93.

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 .