ALMOST COHEN–MACAULAY MODULES

Abstract

A commutative noetherian ring R is called an almost Cohen–Macaulay ring if depth(P, R) = depth(P R P , R P ) for every P ∈ SpecR. [It was called a D-ring by Han (Acta Math. Sinica 1998, 4, 1047–1052).] Several fundamental properties of almost Cohen–Macaulay rings were estabished by Han. In this note, a new characterization is proved: R is an almost Cohen–Macaulay ring if and only if height P ≤ 1 + depth(P, R) for every P ∈ Spec(R). By this characterization, we settle an unsolved problem in Han's paper: R is an almost Cohen–Macaulay ring if and only if so is the power series ring R[[X 1, …, X n ]]. The notion of an almost Cohen–Macaulay ring is generalized to that of an almost Cohen–Macaulay module in this note.