Non-commutative deformations of simple objects in a category of perverse coherent sheaves

Abstract

We define a category of perverse coherent sheaves as the abelian category corresponding to the category of modules under Bondal-Rickard equivalence which arises from a tilting bundle for a projective morphism. The purpose of this paper is to determine versal non-commutative deformations of simple collections in the categories of perverse coherent sheaves in some cases. In general we prove that the non-commutative structure algebra is recovered as the parameter algebra of the versal non-commutative deformation of the simple collection consisting of all simple objects over a closed point of the base space. In the case where the fiber dimensions are at most 1 and the structure sheaf is relatively acyclic, we determine the versal deformations of some partial simple collections consisting of vanishing simple objects. In particular it is proved that the parameter algebra of the versal non-commutative deformation is isomorphic to its opposite algebra in this case.