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:[1][2] 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

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