An explicit isomorphism of different representations of the Ext functor using residue currents

Let F be a coherent Ox-module over a complex manifold X, and let G be a vector bundle on X. We describe an explicit isomorphism between two different representations of the global Ext groups Extk(F,G). The first representation is given by the cohomology of a twisted complex in the sense of "'bled() and Tong, and the second one is obtained from the Dolbeault complex associated with G. A key tool that we introduce for explicitly describing this isomorphism is a residue current associated with a twisted resolution of F.