Extended Local Cohomology and Local Homology

Abstract

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.