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 (R,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 astabI, astab¯I and dstabI, respectively, be the smallest integers n for which AssIn, Ass I¯n and depthIn stabilize. Here I¯n denotes the integral closure of In.

We show that astabI = astab¯I = dstabI if dimR 2, while already in dimension three, astabI and 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 c and dstabJ astabJ 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
Authors
Jürgen Herzog
Fachbereich Mathematik
Universität Duisburg-Essen
Essen
Germany
Amir Mafi
University of Kurdistan
Sanandaj 416
Iran