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: For each integer , the length of the i-th local cohomology of A is finite: . where the sup is over all parameter ideals and is the multiplicity of . There is an -primary ideal such that for each system of parameters in , For each prime ideal of that is not , and is Cohen–Macaulay.
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.
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