  In this paper, we establish the global analogues of some dualities and
equivalences in local algebra by developing the theory of relative
Cohen-Macaulay modules. Let R be a commutative Noetherian ring (not necessarily
local) with identity and a a proper ideal of R. The notions of a-relative
dualizing modules and a-relative big Cohen-Macaulay modules are introduced.
With the help of a-relative dualizing modules, we establish the global analogue
of the duality on the subcategory of Cohen-Macaulay modules in local algebra.
Lastly, we investigate the behavior of the subcategory of a-relative
Cohen-Macaulay modules and a-relative generalized Cohen-Macaulay modules under
Foxby equivalence.
