Stability properties of powers of ideals in regular local rings of small dimension

Jürgen Herzog and Amir Mafi

Vol. 295 (2018), No. 1, 31–41

DOI: 10.2140/pjm.2018.295.31

Abstract

Let $\left(R,\mathfrak{m}\right)$ 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 $astabI$, $\overline{astab}I$ and $dstabI$, respectively, be the smallest integers $n$ for which $Ass{I}^n$, $Ass{\overline{I}}^n$ and $depth{I}^n$ stabilize. Here ${\overline{I}}^n$ denotes the integral closure of $I^n$.

We show that $astabI=\overline{astab}I=dstabI$ if $dimR\le 2$, while already in dimension three, $astabI$ and $\overline{astab}I$ may differ by any amount. Moreover, we show that if $dimR=4$, there exist ideals $I$ and $J$ such that for any positive integer $c$ one has $astabI-dstabI\ge c$ and $dstabJ-astabJ\ge c$.

Keywords

associated primes, depth stability number

Mathematical Subject Classification 2010

Primary: 13A15, 13A30, 13C15

Milestones

Received: 11 July 2017

Revised: 15 November 2017

Accepted: 18 November 2017

Published: 13 March 2018