Generalized Cohen–Macaulay ring explained

(A,ak{m})

of Krull dimension d > 0 that satisfies any of the following equivalent conditions:

For each integer

i=0,...,d-1

, the length of the i-th local cohomology of A is finite:

	i
\operatorname{length}
	ak{m

}(A)) < \infty.

\sup_Q(\operatorname{length}_A(A/Q)-e(Q))<infty

where the sup is over all parameter ideals

and

e(Q)

is the multiplicity of

There is an

ak{m}

-primary ideal

such that for each system of parameters

x_1,...,x_d

(x_1,...,x_d-1):x_d=(x_1,...,x_d-1):Q.

For each prime ideal

ak{p}

\widehat{A}

that is not

ak{m}\widehat{A}

\dim\widehat{A}_ak{p

} + \dim \widehat/\mathfrak = d and

\widehat{A}_ak{p

} is Cohen–Macaulay.

The last condition implies that the localization

A_ak{p}

is Cohen–Macaulay for each prime ideal

ak{p}\neak{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

ak{m}

-primary ideals

; see the introduction of.

References

- Ngô Viêt . Trung . Toward a theory of generalized Cohen-Macaulay modules . Nagoya Mathematical Journal . Duke University Press . 1986 . 102 . none . 1–49 . 10.1017/S0027763000000416 . 670639276 .