Generalized Cohen–Macaulay ring

Short description: Local ring in mathematics

In algebra, a generalized Cohen–Macaulay ring is a commutative Noetherian local ring [math]\displaystyle{ (A, \mathfrak{m}) }[/math] of Krull dimension d > 0 that satisfies any of the following equivalent conditions:

• For each integer [math]\displaystyle{ i = 0, \dots, d - 1 }[/math], the length of the i-th local cohomology of A is finite:

  [math]\displaystyle{ \operatorname{length}_A(\operatorname{H}^i_{\mathfrak{m}}(A)) \lt \infty }[/math].

• [math]\displaystyle{ \sup_Q (\operatorname{length}_A(A/Q) - e(Q)) \lt \infty }[/math] where the sup is over all parameter ideals [math]\displaystyle{ Q }[/math] and [math]\displaystyle{ e(Q) }[/math] is the multiplicity of [math]\displaystyle{ Q }[/math].
• There is an [math]\displaystyle{ \mathfrak{m} }[/math]-primary ideal [math]\displaystyle{ Q }[/math] such that for each system of parameters [math]\displaystyle{ x_1, \dots, x_d }[/math] in [math]\displaystyle{ Q }[/math], [math]\displaystyle{ (x_1, \dots, x_{d-1}) : x_d = (x_1, \dots, x_{d-1}) : Q. }[/math]
• For each prime ideal [math]\displaystyle{ \mathfrak{p} }[/math] of [math]\displaystyle{ \widehat{A} }[/math] that is not [math]\displaystyle{ \mathfrak{m} \widehat{A} }[/math], [math]\displaystyle{ \dim \widehat{A}_{\mathfrak{p}} + \dim \widehat{A}/\mathfrak{p} = d }[/math] and [math]\displaystyle{ \widehat{A}_{\mathfrak{p}} }[/math] is Cohen–Macaulay.

The last condition implies that the localization [math]\displaystyle{ A_\mathfrak{p} }[/math] is Cohen–Macaulay for each prime ideal [math]\displaystyle{ \mathfrak{p} \ne \mathfrak{m} }[/math].

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 [math]\displaystyle{ \operatorname{length}_A(A/Q) - e(Q) }[/math] is constant for [math]\displaystyle{ \mathfrak{m} }[/math]-primary ideals [math]\displaystyle{ Q }[/math]; see the introduction of.

References

• 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. http://projecteuclid.org/euclid.nmj/1118780407.