Generalized Cohen–Macaulay ring

In algebra, a generalized Cohen–Macaulay ring is a commutative Noetherian local ring ${\displaystyle (A,{\mathfrak {m}})}$ of Krull dimension d > 0 that satisfies any of the following equivalent conditions:^[1][2]

• For each integer ${\displaystyle i=0,\dots ,d-1}$, the length of the i-th local cohomology of A is finite:

  ${\displaystyle \operatorname {length} _{A}(\operatorname {H} _{\mathfrak {m}}^{i}(A))<\infty }$.

• ${\displaystyle \sup _{Q}(\operatorname {length} _{A}(A/Q)-e(Q))<\infty }$ where the sup is over all parameter ideals ${\displaystyle Q}$ and ${\displaystyle e(Q)}$ is the multiplicity of ${\displaystyle Q}$.
• There is an ${\displaystyle {\mathfrak {m}}}$-primary ideal ${\displaystyle Q}$ such that for each system of parameters ${\displaystyle x_{1},\dots ,x_{d}}$ in ${\displaystyle Q}$, ${\displaystyle (x_{1},\dots ,x_{d-1}):x_{d}=(x_{1},\dots ,x_{d-1}):Q.}$
• For each prime ideal ${\displaystyle {\mathfrak {p}}}$ of ${\displaystyle {\widehat {A}}}$ that is not ${\displaystyle {\mathfrak {m}}{\widehat {A}}}$, ${\displaystyle \dim {\widehat {A}}_{\mathfrak {p}}+\dim {\widehat {A}}/{\mathfrak {p}}=d}$ and ${\displaystyle {\widehat {A}}_{\mathfrak {p}}}$ is Cohen–Macaulay.

The last condition implies that the localization ${\displaystyle A_{\mathfrak {p}}}$ is Cohen–Macaulay for each prime ideal ${\displaystyle {\mathfrak {p}}\neq {\mathfrak {m}}}$.

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 ${\displaystyle \operatorname {length} _{A}(A/Q)-e(Q)}$ is constant for ${\displaystyle {\mathfrak {m}}}$-primary ideals ${\displaystyle Q}$; see the introduction of.^[3]