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      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 $r < \dim_R M$. 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

       

      Björn Mäurer1 Jürgen Stückrad1

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

       
  • Proceedings – Mathematical Sciences | News

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