Noncommutative Deformations of Sheaves of Modules

Let k be an algebraically closed field, let X be a topological space, let A be a sheaf of associative k-algebras on X, and let F = {F1, . . . ,Fp} be a finite family of sheaves of left A-modules on X. In this paper, we develop a noncommutative deformation theory for the family F . This theory generalizes the noncommutative deformation theory for finite families of left modules over an associative k-algebra, due to Laudal. More precisely, we construct a deformation functor DefF : ap → Sets, and give sufficient conditions for DefF to have an obstruction theory with cohomology (H(U ,Fj ,Fi)). It seems essential that the ringed space (X,A) and the family F of left A-modules have good properties under localization for these conditions to be satisfied. This happens in many interesting cases, including all cases where (X,A) is a quasi-coherent ringed scheme over k and F is a family of quasi-coherent left A-modules. We also show that DefF has a pro-representing hull H(F) whenever DefF has an obstruction theory with finite cohomology. The global Hochschild cohomology groups H(U ,Fj ,Fi) are essential in the construction of the hull H(F). Locally, these groups are given in terms of extensions, and we find conditions for this to hold globally as well. We prove that for all integers n, qExt A (Fj ,Fi) ∼= H (U ,Fj ,Fi) for 1 ≤ i, j ≤ p when X is a separated, locally Noetherian scheme over k, U is an open affine cover of X which is closed under finite intersections, and F is a family of quasi-coherent OX -modules. More generally, we prove that there is such an isomorphism for n = 1 and n = 2 when (X,A) is a quasi-coherent ringed scheme over k such that X is an integral, quasi-projective scheme over k, U is an open affine localizing cover of X which is closed under finite intersections, and F is a family of quasi-coherent left A-modules which are OX -torsionfree.