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:

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

• $\underset{Q}{\sup }({\mathrm{length}}_A(A/Q)-e(Q))<\mathrm{\infty }$ 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 $\hat{A}$ that is not $\mathfrak{𝔪}\hat{A}$, $\dim {\hat{A}}_{\mathfrak{𝔭}}+\dim \hat{A}/\mathfrak{𝔭}=d$ and ${\hat{A}}_{\mathfrak{𝔭}}$ is Cohen–Macaulay.

The last condition implies that the localization $A_{\mathfrak{𝔭}}$ is Cohen–Macaulay for each prime ideal $\mathfrak{𝔭}\ne \mathfrak{𝔪}$. 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 ${\mathrm{length}}_A(A/Q)-e(Q)$ 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.