Extension of weakly and strongly F-regular rings by flat maps
type: Item
instance of: scholarly article
title: Extension of weakly and strongly F-regular rings by flat maps
zbMATH Open document ID: 1007.13002
author: Ian M. Aberbach
published in: Journal of Algebra
publication date: 2003-03-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0209251
reviewed by: Irena Swanson
Mathematics Subject Classification ID: 13A35
Mathematics Subject Classification ID: 13B02
Mathematics Subject Classification ID: 13B22
zbMATH DE Number: 1657187
zbMATH Keywords: tight closure
zbMATH Keywords: characteristic \(p\)
zbMATH Keywords: flat extension
zbMATH Keywords: F-regularity
MaRDI profile type: Publication
OpenAlex ID: W2009708294
arXiv ID: math/0209251
cites work: On the behavior of F-rational rings under flat base change
cites work: Q3973359
cites work: Cohen-Macaulay \(F\)-injective homomorphisms.
cites work: Contracted ideals from integral extensions of regular rings
cites work: Cyclic Purity Versus Purity in Excellent Noetherian Rings
cites work: Tight closure and strong F-regularity
cites work: Tight Closure, Invariant Theory, and the Briancon-Skoda Theorem
cites work: Tight closure and elements of small order in integral extensions
cites work: F-Regularity, Test Elements, and Smooth Base Change
cites work: Infinite integral extensions and big Cohen-Macaulay algebras
cites work: On Noetherian Rings of Characteristic p
cites work: On the commutation of the test ideal with localization and completion
cites work: Q3995023
cites work: F-regularity does not deform
DOI: 10.1006/JABR.2001.8785
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review text: This paper is about the theory of tight closure in positive prime characteristic \(p\), and has ramifications for the existence of test elements. The main results are about the passage of weak/strong F-regularity under a flat map. By Singh's counterexample [\textit{A. K. Singh}, Am. J. Math. 121, No. 4, 919-929 (1999; Zbl 0946.13002)] it is known that flatness is an insufficient assumption. The author's paper is a considerable generalization of the previously known cases of when F-regularity lifts. In particular, he proves the following: NEWLINENEWLINENEWLINELet \((R,m) \to (S,n)\) be a flat local map of Noetherian rings of characteristic \(p\) and with Gorenstein closed fiber. If \(R\) is weakly F-regular and Cohen-Macaulay, then \(S\) is weakly F-regular if one of the following conditions is satisfied: NEWLINENEWLINENEWLINE1. \(R\) and \(S\) have a common test element and \(S/mS\) is F-injective; NEWLINENEWLINENEWLINE2. \(S/mS\) is F-rational and \(S \setminus mS\) contains a test element for \(S\); NEWLINENEWLINENEWLINE3. \(R\) is excellent and \(S/mS\) is F-rational.
description: scientific article; zbMATH DE number 1657187