Tight closure of parameter ideals
type: Item
instance of: scholarly article
title: Tight closure of parameter ideals
zbMATH Open document ID: 0820.13007
DOI: 10.1007/BF01231753
published in: Inventiones Mathematicae
publication date: 1995-09-17T00:00:00Z
full work available at URL: https://eudml.org/doc/144162
review text: Let \(R\) be a locally excellent Noetherian domain of positive prime characteristic, and let \(R^ +\) be the integral closure of \(R\) in an algebraic closure of its fraction field. The main result of the paper under review (theorem 5.1) answers a question of \textit{M. Hochster} and \textit{C. Huneke}; it states that the tight closure of a parameter ideal \(I\) in \(R\) coincides with the contracted expansion \(IR^ + \cap R\) of \(I\) from \(R^ +\) [Hochster and Huneke could only handle the case when \(I\) was generated by at most three parameters (preprint)]. (Theorem 5.1 support the underlying conjecture that the description of the tight closure as contracted expansion from \(R^ +\) holds for any ideal in \(R\). The proof of theorem 5.1 depends heavily on the result of \textit{M. Hochster} and \textit{C. Huneke} that \(R^ +\) is a balanced big Cohen- Macaulay module over \(R\) [Ann. Math., II. Ser. 135, No. 1, 53-89 (1992; Zbl 0753.13003)] (see the proof of the key-lemma 5.2). In the final section 6, applications to \(F\)-rational rings and pseudo-rational singularities are the main aim.
reviewed by: Jürgen Ribbe
Mathematics Subject Classification ID: 13B22
Mathematics Subject Classification ID: 13A35
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 13C14
Mathematics Subject Classification ID: 13F40
zbMATH DE Number: 549838
zbMATH Keywords: \(F\)-rational ring
zbMATH Keywords: prime characteristic
zbMATH Keywords: tight closure
zbMATH Keywords: balanced big Cohen-Macaulay module
zbMATH Keywords: pseudo-rational singularities
author: K. E. Smith
MaRDI profile type: Publication
cites work: Algebraic approximation of structures over complete local rings
cites work: On the joints of Hensel rings
cites work: Localization of tight closure and modules of finite phantom projective dimension.
cites work: Q3973359
cites work: Tight Closure, Invariant Theory, and the Briancon-Skoda Theorem
cites work: Infinite integral extensions and big Cohen-Macaulay algebras
cites work: Q4318628
cites work: F-Regularity, Test Elements, and Smooth Base Change
cites work: Applications of the existence of big Cohen-Macaulay algebras
cites work: Pseudo-rational local rings and a theorem of Briancon-Skoda about integral closures of ideals
cites work: General Néron desingularization based on the idea of Popescu
OpenAlex ID: W2064715723
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description: scientific article; zbMATH DE number 549838