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- W3100115247 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$,Rleq 2$, 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 ${rm dstab}(J)-{rm astab}(J)geq c$." @default.
- W3100115247 created "2020-11-23" @default.
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- W3100115247 date "2018-03-13" @default.
- W3100115247 modified "2023-09-26" @default.
- W3100115247 title "Stability properties of powers of ideals in regular local rings of small dimension" @default.
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- W3100115247 doi "https://doi.org/10.2140/pjm.2018.295.31" @default.
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