Generalized Cohen–Macaulay ring

{{Short description|Local ring in mathematics}}

In algebra, a generalized Cohen–Macaulay ring is a commutative Noetherian local ring $(A,\; \backslash mathfrak\{m\})$ of Krull dimension d > 0 that satisfies any of the following equivalent conditions:{{harvnb|Herrmann|Orbanz|Ikeda|1988|loc=Theorem 37.4.}}{{harvnb|Herrmann|Orbanz|Ikeda|1988|loc=Theorem 37.10.}}

• For each integer $i\; =\; 0,\; \backslash dots,\; d\; -\; 1$, the length of the i-th local cohomology of A is finite:
• :.
• where the sup is over all parameter ideals $Q$ and $e(Q)$ is the multiplicity of $Q$.
• There is an $\backslash mathfrak\{m\}$-primary ideal $Q$ such that for each system of parameters $x\_1,\; \backslash dots,\; x\_d$ in $Q$, $(x\_1,\; \backslash dots,\; x\_\{d-1\})\; :\; x\_d\; =\; (x\_1,\; \backslash dots,\; x\_\{d-1\})\; :\; Q.$
• For each prime ideal $\backslash mathfrak\{p\}$ of $\backslash widehat\{A\}$ that is not $\backslash mathfrak\{m\}\; \backslash widehat\{A\}$, $\backslash dim\; \backslash widehat\{A\}\_\{\backslash mathfrak\{p\}\}\; +\; \backslash dim\; \backslash widehat\{A\}/\backslash mathfrak\{p\}\; =\; d$ and $\backslash widehat\{A\}\_\{\backslash mathfrak\{p\}\}$ is Cohen–Macaulay.

The last condition implies that the localization $A\_\backslash mathfrak\{p\}$ is Cohen–Macaulay for each prime ideal $\backslash mathfrak\{p\}\; \backslash ne\; \backslash mathfrak\{m\}$.

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 $\backslash operatorname\{length\}\_A(A/Q)\; -\; e(Q)$ is constant for $\backslash mathfrak\{m\}$-primary ideals $Q$; see the introduction of.{{harvnb|Trung|1986}}