On derived categories of differential complexes

This paper is devoted to the comparison of different localized categories of differential complexes. The first result is that the canonical functor from the category of complexes of differential operators of order one (defined by Herrera and Lieberman) to the category of differential complexes (of any order, defined by M. Saito), both localized with respect to a suitable notion of quasi-isomorphism, is an equivalence of categories. Then we prove a similar result for a filtered version of the previous categories (defined respectively by Du Bois and M.Saito), localized with respect to graded-quasi-isomorphisms, thus answering a question posed by M. Saito.