In algebra, a generalized Cohen–Macaulay ring is a commutative Noetherian local ring ( A , m ) {\displaystyle (A,{\mathfrak {m}})} of Krull dimension d > 0 that satisfies any of the following equivalent conditions:
For each integer i = 0 , … , d − 1 {\displaystyle i=0,\dots ,d-1} , the length of the i-th local cohomology of A is finite:
length A ( H m i ( A ) ) < ∞ {\displaystyle \operatorname {length} _{A}(\operatorname {H} _{\mathfrak {m}}^{i}(A))<\infty } .
sup Q ( length A ( A / Q ) − e ( Q ) ) < ∞ {\displaystyle \sup _{Q}(\operatorname {length} _{A}(A/Q)-e(Q))<\infty } where the sup is over all parameter ideals Q {\displaystyle Q} and e ( Q ) {\displaystyle e(Q)} is the multiplicity of Q {\displaystyle Q} . There is an m {\displaystyle {\mathfrak {m}}} -primary ideal Q {\displaystyle Q} such that for each system of parameters x 1 , … , x d {\displaystyle x_{1},\dots ,x_{d}} in Q {\displaystyle Q} , ( x 1 , … , x d − 1 ) : x d = ( x 1 , … , x d − 1 ) : Q . {\displaystyle (x_{1},\dots ,x_{d-1}):x_{d}=(x_{1},\dots ,x_{d-1}):Q.}
For each prime ideal p {\displaystyle {\mathfrak {p}}} of A ^ {\displaystyle {\widehat {A}}} that is not m A ^ {\displaystyle {\mathfrak {m}}{\widehat {A}}} , dim A ^ p + dim A ^ / p = d {\displaystyle \dim {\widehat {A}}_{\mathfrak {p}}+\dim {\widehat {A}}/{\mathfrak {p}}=d} and A ^ p {\displaystyle {\widehat {A}}_{\mathfrak {p}}} is Cohen–Macaulay. The last condition implies that the localization A p {\displaystyle A_{\mathfrak {p}}} is Cohen–Macaulay for each prime ideal p ≠ m {\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 A ( A / Q ) − e ( Q ) {\displaystyle \operatorname {length} _{A}(A/Q)-e(Q)} is constant for m {\displaystyle {\mathfrak {m}}} -primary ideals Q {\displaystyle Q} ; see the introduction of.
Notes
References Herrmann, Manfred; Orbanz, Ulrich; Ikeda, Shin (1988), Equimultiplicity and Blowing Up : an Algebraic Study with an Appendix by B. Moonen, Berlin: Springer Verlag, ISBN 3-642-61349-7, OCLC 1120850112 Trung, Ngô Viêt (1986). "Toward a theory of generalized Cohen-Macaulay modules". Nagoya Mathematical Journal. 102 (none). Duke University Press: 1–49. doi:10.1017/S0027763000000416. OCLC 670639276.