Symbolic powers and generalized-parametric decomposition of monomial ideals on regular sequences

Abstract

Let $R$ be a commutative Noetherian ring and let $\mathbf{x}:=x_1,\dots ,x_d$ be a regular $R$-sequence contained in the Jacobson radical of $R$. An ideal $I$ of $R$ is said to be a monomial ideal with respect to $\mathbf{x}$ if it is generated by a set of monomials $x_1^{e_1}\dots x_d^{e_d}$. It is shown that, if $\mathbf{x}R$ is a prime ideal of $R$, then each monomial ideal $I$ has a canonical and unique decomposition as an irredundant finite intersection of primary ideals of the form $x_{\tau (1)}^{e_1}R+\cdots +x_{\tau (s)}^{e_s}R$, where $\tau$ is a permutation of $\{1,\dots ,d\}$, $s\in \{1,\dots ,d\}$ and $e_1,\dots ,e_s$ are the positive integers. This generalizes and provides a short proof of the main results of \cite{HMRS, HRS}. Also, we show that for every integer $k\ge 1$, $I^{(k)}=I^k$, if and only if $\Ass_R R/{I^k }\subseteq \Ass_R R/{I}$, whenever $I$ is a squarefree monomial ideal, where $I^{(k)}$ is the $k$th symbolic power of $I$. Moreover, in this circumstance it is shown that all powers of $I$ are integrally closed.

10 pages