Parametric Decomposition of Powers of Parameter Ideals and Sequentially Cohen-Macaulay Modules

• Published in: Proceedings of the American Mathematical Society 2009-01, Vol.137 (1), p.19-26
• Main Authors: Cuong, Nguyen Tu, Le Truong, Hoang
• Format: Article
• Language: English
• Subjects: Algebra; Combinatorics; Exact sciences and technology; General mathematics; General, history and biography; Integers; Mathematical independent variables; Mathematical induction; Mathematical rings; Mathematical sequences; Mathematical theorems; Mathematics; Polynomials; Sciences and techniques of general use
• ISSN: 0002-9939; 1088-6826
• DOI: 10.1090/s0002-9939-08-09437-9

Let M be a finitely generated module of dimension d over a Noetherian local ring (R, m) and q an ideal generated by a system of parameters x̱ = $x_{1},\ldots ,x_{d}$ of M. For each positive integer n, set $\Lambda _{d,n}=\{\alpha =(\alpha _{1},\ldots,\alpha _{d})\in {\Bbb Z}^{d}|\alpha _{i}\geq 1,1\leq i\leq d\ \text{and}\ \sum_{i=1}^{d}\alpha _{i}=d+n-1\}$ and $q(\alpha)=(x_{1}^{\alpha _{1}},\ldots ,x_{d}^{\alpha _{d}})$ for each $\alpha \in \Lambda _{d,n}$. Then we prove in this note that M is a sequentially Cohen-Macaulay module if and only if there exists a good system of parameters x̱ such that the equality $\germ{q}^{n}M=\underset \alpha \in \Lambda _{d,n}\to{\bigcap}\germ{q}(\alpha )M$ holds true for all n ≥ 1. As an application, we show that the sequentially Cohen-Macaulayness of a module can be characterized by a very special expression of the Hilbert-Samuel polynomial of a good parameter ideal.