Generalized Cohen–Macaulay ring
In algebra, a generalized Cohen–Macaulay ring is a commutative Noetherian local ring of Krull dimension d > 0 that satisfies any of the following equivalent conditions:
The last condition implies that the localization is Cohen–Macaulay for each prime ideal.
A standard example is the local ring at the vertex of an affine cone over a smooth projective variety. Historically, the notion grew up out of the study of a Buchsbaum ring, a Noetherian local ring A in which is constant for -primary ideals ; see the introduction of.
