DUALITIES AND EQUIVALENCES OF THE CATEGORY OF RELATIVE COHEN–MACAULAY MODULES

Abstract

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.