Strong persistence index and fluctuations in colon powers of monomial ideals

TL;DR

This paper introduces the concepts of strong persistence index and fluctuation in colon powers for monomial ideals, analyzing their properties and occurrences in commutative algebra.

Contribution

It defines and studies the strong persistence index and fluctuation phenomena specifically for monomial ideals in Noetherian rings.

Findings

Characterization of the strong persistence index for monomial ideals.

Identification of conditions under which fluctuations in colon powers occur.

Insights into the behavior of colon ideals in monomial ideal theory.

Abstract

Let $I$ be an ideal in a commutative Noetherian ring $R$. We say that a positive integer $\ell_0$ is the strong persistence index of $I$ if $\ell_0$ is the smallest integer such that $(I^{\ell+1} :_R I) = I^{\ell}$ for all $\ell \geq \ell_0$. The first aim of this paper is to study this notion for monomial ideals. We also say that $I$ has the phenomenon of fluctuation in colon powers if there exist positive integers $a < b < c$ such that at least one of the following cases occurs: (i) $(I^{a} : I) = I^{a-1}$, $(I^{b} : I) \neq I^{b-1}$, but $(I^{c} : I) = I^{c-1}$. (ii) $(I^{a} : I) \neq I^{a-1}$, $(I^{b} : I) = I^{b-1}$, but $(I^{c} : I) \neq I^{c-1}$. The second purpose of this work is to explore this phenomenon for monomial ideals.