Generalized Cohen–Macaulay ring

In algebra, a generalized Cohen–Macaulay ring is a commutative Noetherian local ring of Krull dimension d > 0 that satisfies any of the following equivalent conditions:

For each integer, the length of the i-th local cohomology of A is finite:
:.
where the sup is over all parameter ideals and is the multiplicity of.
There is an -primary ideal such that for each system of parameters in,
For each prime ideal of that is not, and is Cohen–Macaulay.

The last condition implies that the localization is Cohen–Macaulay for each prime ideal.
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 is constant for -primary ideals ; see the introduction of.