Noncommutative Deformations of Sheaves and Presheaves of Modules

We work over an algebraically closed field k, and consider the noncommutative deformation functor DefF of a finite family F of presheaves of modules defined over a presheaf of k-algebras A on a small category c. We develop an obstruction theory for DefF , with certain global Hochschild cohomology groups as the natural cohomology. In particular, we show how to calculate the prorepresenting hull H(DefF ) in concrete terms. When (X,A) is a ringed space over k, we also consider the noncommutative deformation functor DefF of a finite family F of quasi-coherent sheaves of left A-modules on X. We give conditions for this deformation functor to be isomorphic to the corresponding deformation functor of presheaves on U, where U is some open cover of X considered as a small category, and show that these conditions are satisfied in many interesting examples. In these cases, it follows that we can calculate the pro-representing hull H(DefF ) in concrete terms using presheaf techniques. Finally, we show that n-fold extensions in the category of quasi-coherent sheaves can be calculated using global Hochschild cohomology in many cases.