Derived equivalences of functor categories

Let \Mod \CS denote the category of \CS-modules, where \CS is a small category. Using the notion of relative derived categories of functor categories, we generalize Rickard's theorem on derived equivalences of module categories over rings to \Mod \CS. Several interesting applications will be provided. In particular, it will be shown that derived equivalence of two coherent rings not only implies the equivalence of their homotopy categories of projective modules, but also implies that they are Gorenstein derived equivalent. As another application, it is shown that a good tilting module produces an equivalence between the unbounded derived category of the module category of the ring and the relative derived category of the module category of the endomorphism ring of it