Uniform equivalence of symbolic and adic topologies
type: Item
instance of: scholarly article
title: Uniform equivalence of symbolic and adic topologies
zbMATH Open document ID: 1200.13007
author: Craig Huneke
author: Daniel Katz
author: Javid Validashti
published in: Illinois Journal of Mathematics
publication date: 2010-03-01T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ijm/1264170853
review text: Let \((R,\mathfrak{m})\) denote a local ring. For two ideals \(I, J \subset R\) denote by \(I : \langle J\rangle = \bigcup_{m \geq 1} I : J^m.\) Very often it is of some interest whether the family of ideals \(\{I^n :\langle J\rangle\}_{n \geq 1}\) is equivalent to the \(I\)-adic topology, i.e., for each \(n \geq 1\) there is an \(m(n) \geq 1\) such that \(I^{m(n)}:\langle J\rangle \subseteq I^n.\) The situation when \(m(n) = n +k\) for a certain integer \(k \geq 1\) was characterized by the reviewer [Math. Nachr. 129, 123--148 (1986; Zbl 0606.13001)]. He posed the problem whether the equivalence of \(\{I^n :\langle J\rangle\}_{n \geq 1}\) to the \(I\)-adic topology implies that one might choose \(m(n) = nk\) for a certain integer \(k \geq 1,\) now called linear equivalence. This problem was solved by \textit{I. Swanson} [Math. Z. 234, No. 4, 755--775 (2000; Zbl 1010.13015)]. For symbolic powers of a prime ideal \(\mathfrak{p} \subset R\) this implies that \(\mathfrak{p}^{(nk)} \subseteq \mathfrak{p}^n\) for all \(n \geq 1.\) While a priori \(k\) depends on the ideal \(\mathfrak{p},\) in their paper \textit{M. Hochster} and \textit{C. Huneke} [Invent. Math. 147, No. 2, 349--369 (2002; Zbl 1061.13005)] have shown that \(\mathfrak{p}^{(nd)} \subseteq \mathfrak{p}^n\) for all \(n \geq 1\) and \(\dim R = d\) for a prime ideal \(\mathfrak{p}\) of a regular local ring \(R.\) In the paper under review the authors study when there is a uniform \(k \geq 1\) such that \(I^{nk}:\langle J\rangle \subseteq I^n\) for all \(n \geq 1\) and all ideals \(I,\) provided \(\{I^n :\langle J\rangle\}_{n \geq1}\) is equivalent to the \(I\)-adic topology. As the main result of the paper the authors prove the following Theorem. Let \(R\) be an equicharacteristic, local domain such that \(R\) is an isolated singularity. Assume that \(R\) is either essentially of finite type over a field of characteristic zero or \(R\) has positive characteristic and is \(F\)-finite. Then there exists an integ…
reviewed by: Peter Schenzel
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13A35
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13P99
Mathematics Subject Classification ID: 14Q20
zbMATH DE Number: 5676323
zbMATH Keywords: equivalence of ideal topologies
zbMATH Keywords: uniform equivalence
zbMATH Keywords: uniform Artin-Rees Lemma
zbMATH Keywords: Chevalley's Theorem
MaRDI profile type: Publication
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description: scientific article