About: Generalized Cohen–Macaulay ring

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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 .

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  • 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. (en)
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  • 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 . (en)
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  • Generalized Cohen–Macaulay ring (en)
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