Reduction Numbers and Balanced Ideals

TL;DR

This paper generalizes the concept of balanced ideals in Noetherian local rings, linking the independence of certain colon ideals to the reduction number and providing new conditions under which this independence holds.

Contribution

It extends the notion of balanced ideals by relating colon ideal independence to the reduction number in broader settings, including one-dimensional rings and Cohen-Macaulay associated graded rings.

Findings

Independence of $J^{n+1}:I^n$ characterizes the reduction number bound

Generalization applies to one-dimensional rings and Cohen-Macaulay graded rings

Provides criteria for when colon ideals are independent of minimal reductions

Abstract

Let $R$ be a Noetherian local ring and let $I$ be an ideal in $R$. The ideal $I$ is called balanced if the colon ideal $J:I$ is independent of the choice of the minimal reduction $J$ of $I$. Under suitable assumptions, Ulrich showed that $I$ is balanced if and only if the reduction number, $r(I)$, of $I$ is at most the `expected' one, namely $\ell(I)- \height I+1$, where $\ell(I)$ is the analytic spread of $I$. In this article we propose a generalization of balanced. We prove under suitable assumptions that if either $R$ is one-dimensional or the associated graded ring of $I$ is Cohen-Macaulay, then $J^{n+1}:I^n$ is independent of the choice of the minimal reduction $J$ of $I$ if and only if $r(I) \leq \ell(I)-\height I+n$.