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:
 * $$\operatorname{length}_A(\operatorname{H}^i_{\mathfrak{m}}(A)) < \infty$$.
 * $$\sup_Q (\operatorname{length}_A(A/Q) - e(Q)) < \infty$$ 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 $$\mathfrak{p} \ne \mathfrak{m}$$.

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