  We present an in-depth exploration of the module structures of local
(co)homology modules (moreover, for complexes) over the completion $\widehat
R^{\mathfrak a}$ of a commutative noetherian ring $R$ with respect to a proper
ideal $\mathfrak a$. In particular, we extend Greenlees-May Duality and MGM
Equivalence to track behavior over $\widehat R^{\mathfrak a}$, 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.
