Cohen-Macaulay modules over Noetherian local rings

摘要

Let $(R,mathfrak m)$ 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 ${m depth}(M/I^nM)=d$ for $ngg0$. Also we show that, if $dim(R)=d$ and $I_1subsetcdotssubset I_n$ is a chain of ideals of $R$ such that $R/I_k$ is maximal Cohen-Macaulay for all $k$, then $nleq ell_R(R/(a_1,ldots,a_d)R)$ for every system of parameters $a_1,ldots,a_d$ of $R$. Also, in the case where $dim(R)=2$, we prove that the ideal transform $m D_{mathfrak m}(R/mathfrak p)$ is minimax balanced big Cohen-Macaulay, for every $mathfrak p in {m Assh}_R(R)$, and we give some equivalent conditions for this ideal transform being maximal Cohen-Macaulay.