Cohen-Macaulay modules over Noetherian local rings
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Let (R,m) be a commutative Noetherian local ring. In this paper we show that a finitely generated R-module M of dimension d is Cohen-Macaulay if and only if there exists a proper ideal I of R such that depth(M/InM) = d for n ≫ 0. Also we show that, if dim(R) = d and I1 ⊂ · · · ⊂ In is a chain of ideals of R such that R/Ik is maximal Cohen-Macaulay for all k, then n ≤ ℓR(R/(a1, . . . , ad)R) for every system of parameters a1, . . . , ad of R. Also, in the case where dim(R) = 2, we prove that the ideal transform Dm(R/ p) is minimax balanced big Cohen-Macaulay, for every p ∈ AsshR(R), and we give some equivalent conditions for this ideal transform being maximal Cohen-Macaulay.
Abstract
Let (R,m) be a commutative Noetherian local ring. In this paper we show that a finitely generated R-module M of dimension d is Cohen-Macaulay if and only if there exists a proper ideal I of R such that depth(M/InM) = d for n ≫ 0. Also we show that, if dim(R) = d and I1 ⊂ · · · ⊂ In is a chain of ideals of R such that R/Ik is maximal Cohen-Macaulay for all k, then n ≤ ℓR(R/(a1, . . . , ad)R) for every system of parameters a1, . . . , ad of R. Also, in the case where dim(R) = 2, we prove that the ideal transform Dm(R/ p) is minimax balanced big Cohen-Macaulay, for every p ∈ AsshR(R), and we give some equivalent conditions for this ideal transform being maximal Cohen-Macaulay.
참고문헌 (12)
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