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]

  • 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.[3]

Notes

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References

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  • Herrmann, Manfred; Orbanz, Ulrich; Ikeda, Shin (1988), Equimultiplicity and Blowing Up : an Algebraic Study with an Appendix by B. Moonen, Berlin: Springer Verlag, ISBN 3-642-61349-7, OCLC 1120850112
  • Trung, Ngô Viêt (1986). "Toward a theory of generalized Cohen-Macaulay modules". Duke University Press. OCLC 670639276.