Deformation of F-purity and F-regularity
type: Item
instance of: scholarly article
title: Deformation of F-purity and F-regularity
zbMATH Open document ID: 0946.13001
DOI: 10.1016/S0022-4049(98)00014-0
author: Anurag K. Singh
published in: Journal of Pure and Applied Algebra
publication date: 2000-10-18T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0209241
review text: At the time this paper was written, it was an open question whether F-regularity deforms. Since then it has been answered in the negative by \textit{A. G. Singh} [Am. J. Math. 121 No. 4, 919-929 (1999; see the following review Zbl 0946.13002)]. However, the paper under review proves that F-regularity deforms for normal local rings whenever for any ideal \(I\) of pure height one representing the inverse of the canonical module in the class group, the symbolic powers \(I^{(i)}\) all satisfy Serre's condition (S\(_3\)) and the anti-canonical cover \(\bigoplus_{i \geq 0} I^{(i)}\) is a Noetherian ring. The deformation of F-regularity had previously been shown for more restrictive classes by \textit{I. Aberbach, M. Katzman} and \textit{B. MacCrimmon} [J. Algebra 204, No. 1, 281-285 (1998; Zbl 0915.13003)], \textit{M. Hochster} and \textit{C. Huneke} [Trans. Am. Math. Soc. 346, No. 1, 1-62 (1994; Zbl 0844.13002)]\ and \textit{K. E. Smith} [``F-regularity deforms for \(\mathbb Q\)-Gorenstein rings'' (preprint)]. The author also shows examples for which F-purity does not deform. Earlier examples had been constructed by \textit{R. Fedder} [Trans. Am. Math. Soc. 278, 461-480 (1983; Zbl 0519.13017)]. Unlike Fedder's, Singh's examples are integral domains.
reviewed by: Irena Swanson
Mathematics Subject Classification ID: 13A35
Mathematics Subject Classification ID: 13D10
Mathematics Subject Classification ID: 13B40
Mathematics Subject Classification ID: 13C20
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 1330819
zbMATH Keywords: tight closure
zbMATH Keywords: F-regular ring
zbMATH Keywords: F-pure ring
zbMATH Keywords: Weil divisor
zbMATH Keywords: cyclic cover
zbMATH Keywords: canonical module
MaRDI profile type: Publication
arXiv ID: math/0209241
description: scientific article