  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
\cite{mem} we introduced the basic DG category $\Pc_{\A^\bullet}$, the perfect
category of $\A^\bullet$, 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.
