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

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 ${\rm dstab}(J)-{\rm astab}(J)\geq c$.