Extended Local Cohomology and Local Homology

We present an in-depth exploration of the module structures of local (co)homology modules (moreover, for complexes) over the completion (widehat {R}^{mathcal {a}}) of a commutative noetherian ring R with respect to a proper ideal (mathcal {a}). In particular, we extend Greenlees-May Duality and MGM Equivalence to track behavior over (widehat {R}^{mathcal {a}}), not just over R. We apply this to the study of two recent versions of homological finiteness for complexes, and to certain isomorphisms, with a view toward further applications. We also discuss subtleties and simplifications in the computations of these functors.