Reducing system of parameters and the Cohen--Macaulay property

2007/07/14 by Björn Mäurer, Bjorn Maurer, Maurer, Bjorn +3
Mathematics · #Advanced Optimization Algorithms Research #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Graph theory and applications #math.AC

paper · pdf · doi:10.48550/arxiv.0707.2136

7 pages

arxiv created 2007/07/14 · openalex publication_date 2007/07/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let R be a local ring and let (x₁\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 (y₁\biss y_r) is part of a reducing system of parameters of M with (y₁\biss y_r)M=(x₁\biss x_r)M then (x₁\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∈ \supp M ∩ V_R(x₁\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/(y₁\biss y_d-1)M, where (y₁\biss y_d) is a reducing system of parameters of M (d := dim_R M).

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