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 $(R,\mathfrak {m})$ and $\mathfrak {q}$ an ideal generated by a system of parameters $\underline {x} = (x_1,\ldots , x_d)$ of $M$. For each positive integer $n$, set \[ \Lambda _{d,n}=\{ \alpha =(\alpha _1,\ldots ,\alpha _d)\in \mathbb {Z}^d|\alpha _i\geqslant 1, 1\leqslant i\leqslant d \text { and } \sum \limits _{i=1}^d\alpha _i=d+n-1\}\] and $\mathfrak {q}(\alpha )=(x_1^{\alpha _1},\ldots ,x_d^{\alpha _d})$ 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 $\underline {x}$ such that the equality $\mathfrak {q}^nM=\bigcap \limits _{\alpha \in \Lambda _{d,n}}\mathfrak {q}(\alpha )M$ holds true for all $n\ge 1$. 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.