Big tight closure test elements for some non-reduced excellent rings
type: Item
instance of: scholarly article
title: Big tight closure test elements for some non-reduced excellent rings
zbMATH Open document ID: 1256.13004
author: R. Y. Sharp
published in: Journal of Algebra
publication date: 2012-05-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/1108.1660
reviewed by: Karl Schwede
Mathematics Subject Classification ID: 13A35
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 13D45
Mathematics Subject Classification ID: 13E05
zbMATH DE Number: 6037578
zbMATH Keywords: tight closure
zbMATH Keywords: test element
zbMATH Keywords: commutative Noetherian Ring
zbMATH Keywords: Frobenius homomorphism
zbMATH Keywords: excellent ring
zbMATH Keywords: generically reduced
zbMATH Keywords: Frobenius skew polynomial ring
zbMATH Keywords: Gorenstein ring
zbMATH Keywords: weakly \(F\)-regular ring
MaRDI profile type: Publication
OpenAlex ID: W2963261524
arXiv ID: 1108.1660
cites work: Extension of weakly and strongly F-regular rings by flat maps
cites work: Discreteness and rationality of \(F\)-thresholds
cites work: Discreteness and rationality of \(F\)-jumping numbers on singular varieties
cites work: F-Purity and Rational Singularity
cites work: Q3154990
cites work: On a generalization of test ideals
cites work: A generalization of tight closure and multiplier ideals
cites work: Local cohomological dimension in characteristic p
cites work: Tight Closure, Invariant Theory, and the Briancon-Skoda Theorem
cites work: F-Regularity, Test Elements, and Smooth Base Change
cites work: Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay
cites work: Q4871734
cites work: Q4251932
cites work: Q5603302
cites work: Parameter-test-ideals of Cohen–Macaulay rings
cites work: On Noetherian Rings of Characteristic p
cites work: F-modules: applications to local cohomology and D-modules in characteristic p>0.
cites work: On the commutation of the test ideal with localization and completion
cites work: Q5611879
cites work: Q3739243
cites work: General Néron desingularization
cites work: General Néron desingularization and approximation
cites work: Q4013102
cites work: \(F\)-adjunction
cites work: On the Hartshorne–Speiser–Lyubeznik theorem about Artinian modules with a Frobenius action
cites work: Graded annihilators of modules over the Frobenius skew polynomial ring, and tight closure
cites work: Graded annihilators and tight closure test ideals
cites work: An excellent $F$-pure ring of prime characteristic has a big tight closure test element
cites work: Right and left modules over the Frobenius skew polynomial ring in theF-finite case
cites work: Q5656860
cites work: F-rational rings have rational singularities
cites work: Q4938825
cites work: Adjoint ideals along closed subvarieties of higher codimension
DOI: 10.1016/J.JALGEBRA.2011.08.009
Recommended article: Tight closure test exponents for certain parameter ideals
Recommended article: Relative test elements for tight closure
Recommended article: Q4664235
Recommended article: On the commutation of the test ideal with localization and completion
Recommended article: F-Regularity, Test Elements, and Smooth Base Change
Recommended article: Tight closure with respect to a multiplicatively closed subset of an \(F\)-pure local ring
Recommended article: Strong test ideals
Recommended article: Some results on test elements
Recommended article: Strong test modules and multiplier ideals
Recommended article: Phantom depth and flat base change
review text: In the theory of tight closure for commutative Noetherian rings of characteristic \(p > 0\), test elements play a central role. Recall that \(z\) is in the tight closure of an ideal \(I\) if there exists \(c \in R\), not in any minimal prime, such that \(c z^{p^e} \in I^{[p^e]}\) for all \(e \gg 0\) (here \(I^{[p^e]}\) is the ideal generated by the \(p^e\)th powers of elements of \(I\)). A test element is a choice of \(c\) that works for all \(z\) in the tight closure of \(I\).NEWLINENEWLINEIn the paper under review the author proves the following theorem, generalizing results of \textit{M. Hochster} and \textit{C. Huneke} [Trans. Am. Math. Soc. 346, No. 1, 1--62 (1994; Zbl 0844.13002)] to the non-reduced setting.NEWLINENEWLINETheorem. Suppose that \((R, \mathfrak{m})\) is an excellent local ring that is regular in codimension 0 (i.e., generically reduced). Then \(R\) has a test element. More generally if \(c\) is an element not in any minimal prime such that \(R_c\) is Gorenstein and weakly \(F\)-regular, then some power of \(c\) is a big test element.NEWLINENEWLINEIn fact, the author proves an even stronger statement. He shows that the test elements that are constructed are \textit{big test elements} (test elements that work for modules, including not necessarily finitely generated modules).
description: scientific article; zbMATH DE number 6037578