On the Parametric Decomposition of Powers of Parameter Ideals in a Noetherian Local Ring

Abstract

There is given a characterization of Noetherian local rings $A$ with $d = \dim A \geq 2$, in which the equality $(a_i \mid 1 \leq i \leq d)^n=\underset{\alpha}{\bigcap} (a_1^{\alpha_1}, a_2^{\alpha_2}, \cdots, a_d^{\alpha_d})$ holds true for all systems $a_1,a_2, \cdots, a_d$ of parameters and integers $n \geq 1$, where the suffix $\alpha$ runs over $\alpha = (\alpha _1,\alpha_2,\cdots,\alpha _d) \in \mathbf{Z}^d$ such that $\alpha _i \geq 1\ \text{for all} \ 1\leq i \leq d$ and $\sum_{i=1}^d \alpha _i =d+n-1$.