Generalized Cohen–Macaulay ring

Template:Short description In algebra, a generalized Cohen–Macaulay ring is a commutative Noetherian local ring $(A,𝔪)$ 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:

  ${\mathrm{length}}_A({\mathrm{H}}_{𝔪}^i(A))<\mathrm{\infty }$.

• $\underset{Q}{\sup }({\mathrm{length}}_A(A/Q)-e(Q))<\mathrm{\infty }$ where the sup is over all parameter ideals $Q$ and $e(Q)$ is the multiplicity of $Q$.
• There is an $𝔪$-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 $𝔭$ of $\hat{A}$ that is not $𝔪\hat{A}$, $\dim {\hat{A}}_{𝔭}+\dim \hat{A}/𝔭=d$ and ${\hat{A}}_{𝔭}$ is Cohen–Macaulay.

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

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

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