Tight closure in non-excellent rings
type: Item
instance of: scholarly article
title: Tight closure in non-excellent rings
zbMATH Open document ID: 1202.13003
author: Raymond C. Heitmann
published in: Journal of Algebra
publication date: 2010-08-11T00:00:00Z
review text: Introduced by Hochster and Huneke, the tight closure theory is an important tool for studying excellent rings of equal characteristic. The paper under review is a study of the tight closure in non-excellent local rings of prime characteristic \(p\). The author provides examples showing that important properties such as ``colon capturing'' and ``persistence'', which make tight closure such a powerful tool in excellent rings, do not necessarily hold in the non-excellent case. If \(I^{\ast}\) denotes the tight closure of the ideal \(I\), the colon capturing property means that for \(x_1,\dots, x_n\) parameters in \(R\), we have \(((x_1,\dots, x_{n-1}): x_n) \subseteq (x_1,\dots, x_{n-1})^{\ast}\), while persistence means that \(\phi(I^{\ast}) \subseteq (\phi(I)S)^{\ast}\) whenever \(\phi:R \to S\) is a ring homomorphism. First it is noted that in the ring from Example 2 in the Appendix of \textit{M. Nagata} [Local rings. New York and London: Interscience Publishers (1962; Zbl 0123.03402)], which is not universally catenarian, the colon capturing of tight closure fails. Moreover, the author provides examples of universally catenarian rings \(R\) where the same property fails. As a consequence, this shows that the plus closure \(R^{+}\) need not be Cohen-Macaulay. Finally, the author shows the failure of the persistence of tight closure in non-excellent rings by giving an example of a four dimensional unique factorization domain and a prime element \(x \in R\) such that \(\pi(I^{\ast}) \not\subseteq (\pi(I))^{\ast}\), where \(\pi : R \to R/xR\) is the natural projection. To overcome the difficulties caused by the failure of persistence and colon capturing in the non-excellent case, the author introduces a new definition of tight closure for local rings \(R\) of prime characteristic \(p\). More precisely, if \(N\) is a submodule of an \(R\)-module \(M\) and \(x \in M\), then \(x\) is said to be in the general tight closure of \(N\) if \(x \otimes 1 \in (N \otimes \widehat…
Mathematics Subject Classification ID: 13A35
zbMATH DE Number: 5768892
zbMATH Keywords: tight closure
zbMATH Keywords: local cohomology
zbMATH Keywords: non-excellent rings
reviewed by: Catalin Ciuperca
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2009.12.018
OpenAlex ID: W2025214233
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DOI: 10.1016/J.JALGEBRA.2009.12.018
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description: scientific article; zbMATH DE number 5768892