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 \({\text {D}}_{qc}(X)\) 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 \(\mathcal {F}\) on a finite-dimensional Noetherian separated scheme (even if \(\mathcal {F}\) 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 \(\mathcal {F}\).