  It is well-known that DG-enhancements of D(QCoh(X)) are all equivalent to
each other, see [23]. Here we present an explicit model which leads to
applications in deformation theory. In particular, we shall describe three
models for derived endomorphisms of a quasi-coherent sheaf F on a
finite-dimensional Noetherian separated scheme (even if F does not admit a
locally free resolution). Moreover, these complexes are endowed with DG-Lie
algebra structures, which we prove to control infinitesimal deformations of F.
