Tight closure and strong test ideals
type: Item
instance of: scholarly article
title: Tight closure and strong test ideals
zbMATH Open document ID: 0898.13003
DOI: 10.1016/S0022-4049(97)00053-4
author: Craig Huneke
published in: Journal of Pure and Applied Algebra
publication date: 1998-10-25T00:00:00Z
review text: This is in the spirit of the author's work on the uniform Artin-Rees theorem [see \textit{C. Huneke}, Invent. Math. 107, No. 1, 203-223 (1992; Zbl 0756.13001)]), in which he proved some uniform properties of tight closure, integral closure and powers of ideals. The main result in this paper is that in reasonably good rings, there exists a positive integer \(M\) such that for every ideal \(I\) and every element \(x\) in the tight closure of \(I\), \(x\) satisfies an integral equation over \(I\) of degree at most \(M\). The good rings for which the author proves the existence of this uniform bound \(M\) on the integral degrees contain what the author terms strong test ideals. The author proves the existence of strong test ideals in certain rings with desingularizations. The paper ends with a section of related and interesting results.
Mathematics Subject Classification ID: 13B22
Mathematics Subject Classification ID: 13A35
zbMATH DE Number: 1097606
zbMATH Keywords: integral closure
zbMATH Keywords: tight closure
zbMATH Keywords: test elements
zbMATH Keywords: strong test ideals
reviewed by: Irena Swanson
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1097606