Openness of the F-rational locus and smooth base change
type: Item
instance of: scholarly article
title: Openness of the F-rational locus and smooth base change
zbMATH Open document ID: 0826.13009
DOI: 10.1016/S0021-8693(05)80010-9
published in: Journal of Algebra
publication date: 1995-07-05T00:00:00Z
review text: Throughout, for simplicity, \(R\) will denote a commutative noetherian reduced ring of prime characteristic \(p\). We say that \(R\) is \(F\)-rational if every ideal generated by parameters in \(R\) is tightly closed, and that \(R\) is strongly \(F\)-rational if for each nonzero divisor \(x\) in \(R\) there exists an integer \(q_0\) such that the \(R\)-linear map \(R \otimes R/ I\to R^{1/q} \otimes R/I\), \(1\otimes 1 \mapsto x^{1/q} \otimes 1\), is injective for any ideal \(I\) and for any \(q = p^e \geq q_0\). The main purpose of the paper is to prove the openness of the \(F\)-rational locus in the case that \(R\) is finitely generated over an excellent local ring. For this, the author studies strongly \(F\)-rational rings in section 1, develops the theory of \(\Gamma\)-construction due to Hochster and Huneke in section 2, and proves that \(F\)-rationality is preserved under smooth base change (theorem 3.1). -- The author also shows that certain \(\Gamma\)-constructions of an \(F\)-rational ring are \(F\)-rational, that \(F\)-rationality implies strong \(F\)-rationality, and that a test element for strong parameter ideals exists, under the same condition in the main result. (By parameters in \(R\) we mean a sequence of elements \(x_1, \ldots, x_k\) in \(R\) such that for each prime ideal \(P\) of \(R\) containing them, their images in \(R_P\) form part of a system of parameters in \(R_P\). If this condition holds modulo every minimal prime ideal of \(R\), then we say \(x_1, \ldots, x_k\) form a sequence of strong parameters in \(R\) (section 0). In general, strong \(F\)-rationality implies \(F\)-rationality (proposition 1.4)).
reviewed by: Yoichi Aoyama
Mathematics Subject Classification ID: 13B22
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13A35
Mathematics Subject Classification ID: 13H05
zbMATH DE Number: 763201
zbMATH Keywords: tight closure
zbMATH Keywords: \(F\)-rational ring
zbMATH Keywords: openness of the \(F\)-rational locus
zbMATH Keywords: \(\Gamma\)-constructions
zbMATH Keywords: prime characteristic
zbMATH Keywords: system of parameters
author: Juan Diego Vélez
MaRDI profile type: Publication
cites work: Localization of tight closure and modules of finite phantom projective dimension.
cites work: Singularités rationnelles et quotients par les groupes réductifs. (Rational singularities and quotients by reductive groups)
cites work: Q4064285
cites work: On the Canonical Element Conjecture
cites work: Q3973359
cites work: Local cohomology. A seminar given by A. Grothendieck, Harvard University, Fall 1961. Notes by R. Hartshorne
cites work: Tight Closure, Invariant Theory, and the Briancon-Skoda Theorem
cites work: F-Regularity, Test Elements, and Smooth Base Change
cites work: Q4318628
cites work: Infinite integral extensions and big Cohen-Macaulay algebras
cites work: Tight closure and strong F-regularity
cites work: Phantom homology
cites work: Contracted ideals from integral extensions of regular rings
cites work: Q4055756
cites work: Cohen-Macaulay Rings, Invariant Theory, and the Generic Perfection of Determinantal Loci
cites work: Splitting theorems for quadratic ring extensions
cites work: The purity of the Frobenius and local cohomology
cites work: Jacobian ideals and a theorem of Briancon-Skoda
cites work: Pseudo-rational local rings and a theorem of Briancon-Skoda about integral closures of ideals
cites work: Q3739243
cites work: General Néron desingularization and approximation
cites work: Dimension projective finie et cohomologie locale. Applications à la demonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck
cites work: General Néron desingularization based on the idea of Popescu
cites work: The vanishing of intersection multiplicities of perfect complexes
cites work: Q3749157
cites work: Q3360991
cites work: Q3795766
cites work: History of the `modularity conjecture'.
cites work: Tight closure of parameter ideals
full work available at URL: s0021-8693(05)80010-9
OpenAlex ID: W1995736956
description: scientific article; zbMATH DE number 763201