Duality and Equivalence of Module Categories in Noncommutative Geometry Ii: Mukai Duality for Holomorphic Noncommutative Tori

This is the second in a series of papers intended to set up a framework to study categories of modules in the context of non-commutative geometries. In [3] we introduced the basic DG category PA• , the perfect category of A•, which corresponded to the category of coherent sheaves on a complex manifold. In this paper we enlarge this category to include objects which correspond to quasi-coherent sheaves. We then apply this framework to proving an equivalence of categories between derived categories on the noncommutative complex torus and on a holomorphic gerbe on the dual complex torus.