• Reducing System of Parameters and the Cohen–Macaulay Property

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/117/02/0159-0165

• # Keywords

Systems of parameters; Cohen–Macaulay modules.

• # Abstract

Let 𝑅 be a local ring and let $(x_1,\ldots,x_r)$ be part of a system of parameters of a finitely generated 𝑅-module 𝑀, where $r &lt; \dim_R M$. We will show that if $(y_1,\ldots,y_r)$ is part of a reducing system of parameters of 𝑀 with $(y_1,\ldots,y_r)M=(x_1,\ldots,x_r)M$ then $(x_1,\ldots,x_r)$ is already reducing. Moreover, there is such a part of a reducing system of parameters of 𝑀 iff for all primes $P\in \mathrm{Supp} M \cap V_R(x_1,\ldots,x_r)$ with $\dim_R R/P = \dim_R M-r$ the localization $M_P$ of 𝑀 at 𝑃 is an 𝑟-dimensional Cohen–Macaulay module over $R_P$.

Furthermore, we will show that 𝑀 is a Cohen–Macaulay module iff $y_d$ is a non zero divisor on $M/(y_1,\ldots,y_{d-1})M$, where $(y_1,\ldots,y_d)$ is a reducing system of parameters of $M(d:=\dim_R M)$.

• # Author Affiliations

1. Fakultät für Mathematik und Informatik, Universität Leipzig, Augustus Platz 10/11, D-04109 Leipzig, Germany

• # Proceedings – Mathematical Sciences

Volume 131, 2021
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019