Generalized Cohen–Macaulay Ring

In algebra, a generalized Cohen–Macaulay ring is a commutative Noetherian local ring $(A, \mathfrak)$ 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:
*:$\operatorname_A(\operatorname^i_(A)) < \infty$.
*$\sup_Q (\operatorname_A(A/Q) - e(Q)) < \infty$ where the sup is over all parameter ideals $Q$ and $e(Q)$ is the multiplicity of $Q$.
*There is an $\mathfrak$-primary ideal $Q$ such that for each system of parameters $x_1, \dots, x_d$ in $Q$, $(x_1, \dots, x_) : x_d = (x_1, \dots, x_) : Q.$
*For each prime ideal $\mathfrak$ of $\widehat$ that is not $\mathfrak \widehat$, $\dim \widehat_ + \dim \widehat/\mathfrak = d$ and $\widehat_$ is Cohen–Macaulay.

The last condition implies that the localization $A_\mathfrak$ is Cohen–Macaulay for each prime ideal $\mathfrak \ne \mathfrak$.

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 $\operatorname_A(A/Q) - e(Q)$ is constant for $\mathfrak$-primary ideals $Q$; see the introduction of.

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Ring theory

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