Reducing system of parameters and the Cohen--Macaulay property

Abstract

Let $R$ be a local ring and let (x_1\biss x_r) be part of a system of parameters of a finitely generated $R$-module $M,$ where $r < \dim_R M$. We will show that if (y_1\biss y_r) is part of a reducing system of parameters of $M$ with (y_1\biss y_r)M=(x_1\biss x_r)M then (x_1\biss x_r) is already reducing. Moreover, there is such a part of a reducing system of parameters of $M$ iff for all primes P\in \supp M \cap V_R(x_1\biss x_r) with $\dim_R R/P = \dim_R M -r$ the localization $M_P$ of $M$ at $P$ is an $r$-dimensional \cm\ module over $R_P$. Furthermore, we will show that $M$ is a \cm module iff $y_d$ is a non zero divisor on M/(y_1\biss y_{d-1})M, where (y_1\biss y_d) is a reducing system of parameters of $M$ ($d := \dim_R M$).Comment: 7 page