Vanishing of differentials along ideals and non-archimedean approximation
type: Item
instance of: scholarly article
title: Vanishing of differentials along ideals and non-archimedean approximation
zbMATH Open document ID: 0984.13017
published in: Mathematische Annalen
publication date: 2002-05-14T00:00:00Z
review text: Let \(R\) be a regular local ring, containing a perfect field \(k\), over which \(R\) is essentially of finite type, and let \(C(R/k)=\{x\in R:\delta (x)=0\) for all \(\delta\in \text{Der}_k (R)\}\). The authors prove that if \(R\) is a semi-local regular excellent and irreducible \(k\)-algebra such that \(R/R^p\) is finite (where \(p=\text{char}(k)>0)\), and \(I\subseteq R\) is an ideal, then there exists an integer \(l\) with the property that if \(x\in R\) with \(\delta(x)\in I^{n+l}\) for all \(\delta\in \text{Der}_k(R)\), then there exists a \(c\in C(R/k)\) with \(x-c\in I^n\). This provides a positive answer to a question raised by C. Huneke [see \textit{R. Fedder}, \textit{C. Huneke} and \textit{R. Hübl}, Proc. Am. Math. Soc. 108, No. 2, 319-325 (1990; Zbl 0691.13028)]. The solution for characteristic 0 was given by \textit{R. Hübl} [Proc. Am. Math. Soc. 127, No. 12, 3503-3511 (1999; Zbl 0938.13008)]. In the characteristic 0 case \(l\) can be bounded by a constant depending only on \(R\). But the situation in the present work is more difficult and the authors are able to find a uniform bound only with restrictions. The goal is to enable the use of these techniques to clarify or provide alternate proofs of known results and extend work to more general problems. See for example the algebraic formulation of the Kodaria vanishing theorem by \textit{C. Huneke} and \textit{K. Smith} [J. Reine Angew. Math. 484, 127-152 (1997; Zbl 0913.13003)].
reviewed by: James G. Timourian
Mathematics Subject Classification ID: 13N05
Mathematics Subject Classification ID: 14F10
Mathematics Subject Classification ID: 13A35
Mathematics Subject Classification ID: 14F17
zbMATH DE Number: 1488549
zbMATH Keywords: tight closure
zbMATH Keywords: vanishing theorem
zbMATH Keywords: rational singularities
zbMATH Keywords: \(F\)-rationality
author: Steven Dale Cutkosky
author: Reinhold Hübl
MaRDI profile type: Publication
full work available at URL: s002080000095
OpenAlex ID: W1976034802
DOI: 10.1007/S002080000095
description: scientific article; zbMATH DE number 1488549