Generalized Cohen–Macaulay ring

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

• For each integer $i=0,\dots ,d-1$, the length of the i-th local cohomology of A is finite:

  $\operatorname {length} _{A}(\operatorname {H} _{\mathfrak {m}}^{i}(A))<\infty$.

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

The last condition implies that the localization $A_{\mathfrak {p}}$ is Cohen–Macaulay for each prime ideal ${\mathfrak {p}}\neq {\mathfrak {m}}$.

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 $\operatorname {length} _{A}(A/Q)-e(Q)$ is constant for ${\mathfrak {m}}$-primary ideals $Q$; 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". Nagoya Mathematical Journal. 102 (none). Duke University Press: 1–49. doi:10.1017/S0027763000000416. OCLC 670639276.