# Stability properties of powers of ideals over regular local rings of small dimension

## TL;DR

This paper investigates the stability of associated primes and depth of powers of ideals in regular local rings, revealing that these properties coincide in low dimensions but can diverge arbitrarily in higher dimensions.

## Contribution

It establishes the equality of stability indices for associated primes and depth in dimensions up to 2, and constructs examples showing divergence in higher dimensions.

## Key findings

- In dimension ≤ 2, astab(I)=astab̄(I)=dstab(I).
- In dimension 3, astab(I) and astab̄(I) can differ arbitrarily.
- In dimension 4, there are ideals with arbitrarily large differences between astab and dstab.

## Abstract

Let $(R,\mathfrak{m})$ be a regular local ring or a polynomial ring over a field, and let $I$ be an ideal of $R$ which we assume to be graded if $R$ is a polynomial ring. Let astab$(I)$ resp. $\overline{\rm astab}(I)$ be the smallest integer $n$ for which Ass$(I^n)$ resp. Ass$(\overline{I^n})$ stabilize, and dstab$(I)$ be the smallest integer $n$ for which depth$(I^n)$ stabilizes. Here $\overline{I^n}$ denotes the integral closure of $I^n$. We show that astab$(I)=\overline{\rm astab}(I)={\rm dstab}(I)$ if dim$\,R\leq 2$, while already in dimension $3$, astab$(I)$ and $\overline{\rm astab}(I)$ may differ by any amount. Moreover, we show that if dim$\,R=4$, then there exist ideals $I$ and $J$ such that for any positive integer $c$ one has ${\rm astab}(I)-{\rm dstab}(I)\geq c$ and ${\rm dstab}(J)-{\rm astab}(J)\geq c$.