Deformations of morphisms of coherent sheaves

We generalise Hinich’s Theorem of descent of Deligne groupoids to the case where the dgLas involved have no negative cohomology. We apply this result to study the infinitesimal deformations of a morphism $\alpha :\mathcal{F}\to \mathcal{G}$ of coherent sheaves, where both the sheaves $\mathcal{F}$ and $\mathcal{G}$ and the map $\alpha$ can be deformed, on a smooth variety over a field of characteristic zero. In particular, we provide an explicit dgLa that controls these deformations via the Deligne functor, applying the Thom-Whitney totalisation to a specific semicosimplicial dgLa, constructed from geometrical data.