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

Abstract

Let $M$ be a finitely generated module of dimension $d$ over a Noetherian local ring $(R,\m)$ and $\q $ the parameter ideal generated by a system of parameters $\x = (x_1,..., x_d)$ of $M$. For each positive integer $n$, set $${\Lambda }_{d,n}=\{\alpha =({\alpha }_1,...,{\alpha }_d)\in {\mathbb{Z}}^d|{\alpha }_i⩾1,\forall 1⩽i⩽d\text{and}\sum _{i=1}^d{\alpha }_i=d+n-1\}$$ and $\qa = (x_1^{α_1},...,x_d^{α_d})$. Then we prove in this note that $M$ is a sequentially Cohen-Macaulay module if and only if there exists a certain system of parameters $\x$ such that the equality $\q^nM=\pd$ holds true for all $n$. As an application of this result, we can compute the Hilbert-Samuel polynomial of a sequentially Cohen-Macaulay module with respect to certain parameter ideals

10 pages

References in corpus (1)

• On Sequentially Cohen-Macaulay Modules