A DG-Enhancement of ${\text {D}}_{qc}(X)$ with Applications in Deformation Theory

It is well-known that DG-enhancements of the unbounded derived category D_qc(X) of quasi-coherent sheaves on a scheme X are all equivalent to each other. 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.