#### Vol. 295, No. 1, 2018

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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
##### 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 $astabI$, $\overline{astab}\phantom{\rule{0.3em}{0ex}}I$ 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 $astabI=\overline{astab}\phantom{\rule{0.3em}{0ex}}I=dstabI$ if $dimR\le 2$, while already in dimension three, $astabI$ and $\overline{astab}\phantom{\rule{0.3em}{0ex}}I$ 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