Duality and equivalence of module categories in noncommutative geometry I

This is the first in a series of papers that deals with duality statements such as Mukai-duality (T-duality, from algebraic geometry) and the Baum-Connes conjecture (from operator $K$-theory). These dualities are expressed in terms of categories of modules. In this paper, we develop a general framework needed to describe these dualities. In various geometric contexts, e.g. complex geometry, generalized complex geometry, and noncommutative geometry, the geometric structure is encoded in a certain differential graded algebra. We develop the module theory of such differential graded algebras in such a way that we can recover the derived category of coherent sheaves on a complex manifold. In this paper and ones to follow we apply this to stating and proving the duality statements mentioned above. After developing the general framework, we look at a (complex) Lie algebroid $\A\to T_\cx X$. One can then consider our analogue of the derived category of coherent sheaves, integrable with respect to the Lie algebroid. We then establish a (Serre) duality theorem for "elliptic" Lie algebroids and for noncommutative tori.