In algebra, a generalized Cohen–Macaulay ring is a commutative Noetherian local ring ( A , m ) {\displaystyle (A,{\mathfrak {m}})} (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:
length \( {\displaystyle \operatorname {length} _{A}(\operatorname {H} _{\mathfrak {m}}^{i}(A))<\infty }. \)
\( {\displaystyle \sup _{Q}(\operatorname {length} _{A}(A/Q)-e(Q))<\infty } \) where the sup is over all parameter ideals Q and \( {\displaystyle e(Q)} \) is the multiplicity of Q .
There is an \( {\mathfrak {m}}\)-primary ideal Q such that for each system of parameters \( {\displaystyle x_{1},\dots ,x_{d}} \) in Q} \( {\displaystyle (x_{1},\dots ,x_{d-1}):x_{d}=(x_{1},\dots ,x_{d-1}):Q.} \)
For each prime ideal \( {\mathfrak {p}} \) of \( \widehat {A} \) 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 \( A_{{\mathfrak {p}}} \) 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 length \( {\displaystyle \operatorname {length} _{A}(A/Q)-e(Q)} \( is constant for \( {\mathfrak {m}} \)-primary ideals Q; see the introduction of (Trung 1986).
References
Herrmann–Ikeda–Orbanz, Theorem 37.4.
Herrmann–Ikeda–Orbanz, Theorem 37.10.
Herrmann, M., S. Ikeda, and U. Orbanz: Equimultiplicity and Blowing Up. An Algebraic Study with an Appendix by B. Moonen. Springer Verlag, Berlin Heidelberg New-York, 1988.
N. V. Trung, Towards a theory of generalized Cohen-Macaulay modules, Nagoya Math. J. 102, 1 – 49(1986)
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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