Quotients algébriques par des groupes non réductifs et variétés de modules de complexes. International Journal of Mathematics Vol. 9 No 7 (1998), 769-819

This paper is a continuation of the papers Moduli spaces of decomposable morphims of sheaves and quotients by non-reductive groups and Espace abstraits de morphismes et mutations. I show here that the method used in the first paper to construct moduli spaces of morphisms and the one used in the second paper to extend the results of the first are particular cases of a more general method of construction of algebraic quotients. It is based on the notion of quasi-isomorphism. Let G, G' algebraic groups acting on algebraic varieties X, X' respectively. A quasi-morphism is a map    that can be everywhere locally lifted to morphisms from open subsets of X to X' (plus some additional technical conditions...). A quasi-isomorphism is a quasi-morphism which is bijective and whose inverse is also a quasi-morphism. If there is a quasi-isomorphism between X and X'it is possible to deduce the existence of algebraic quotients of open subsets of X from the existence of  quotients of the corresponding open subsets of X'. The method used to construct moduli spaces of morphisms is extended in this paper to complexes of decomposables sheaves.