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 let $\mathfrak{𝔞}$ be a proper ideal of $R$. The notions of $\mathfrak{𝔞}$-relative dualizing modules and $\mathfrak{𝔞}$-relative big Cohen–Macaulay modules are introduced. With the help of $\mathfrak{𝔞}$-relative dualizing modules, we establish the global analogue of duality on the subcategory of Cohen–Macaulay modules in local algebra. Lastly, we investigate the behavior of the subcategory of $\mathfrak{𝔞}$-relative Cohen–Macaulay modules and $\mathfrak{𝔞}$-relative generalized Cohen–Macaulay modules under Foxby equivalence.