Duality, localization and completion

AbstractIn this paper we study relative duality theory, with respect to an idempotent kernel functor σ over some commutative ring R and prove that σ-dualizing R-modules are not only locally injective, but (somewhat surprisingly) globally injective. Using a relative version of completion, we show that the endomorphism ring of a σ-dualizing module coincides with the completion of R with respect to σ. In the final part of the paper we consider relative Gorenstein rings, giving an explicit calculation of their generalized local cohomology groups