Remarks on a conjecture of Nakai
type: Item
instance of: scholarly article
title: Remarks on a conjecture of Nakai
zbMATH Open document ID: 0592.13005
DOI: 10.1016/0021-8693(85)90090-0
author: Yasunori Ishibashi
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: Let R be a geometric local domain over a field k of characteristic zero, Der(R/k) the algebra of high-order derivations of R/k and der(R/k) the subalgebra, generated by the first-order derivations. Nakai's problem asks, whether \(Der(R/k)=der(R/k)\) implies the regularity of R. The author investigates conditions under which every high-order derivation can be extended to the integral closure of R. From these he can prove that the condition \(Der(R/k)=der(R/k)\) implies regularity for the localizations of R with respect to the prime ideals of height one. Generalizing results of Bernstein et al. on derivations of graded rings he then proves a graded version of Nakai's conjecture: Let R be a two-dimensional weighted complete intersection over an algebraically closed field k of characteristic zero. If \(Der(R/k)=der(R/k)\) then R is a polynomial ring over k.
Mathematics Subject Classification ID: 13B10
Mathematics Subject Classification ID: 13N05
Mathematics Subject Classification ID: 13H05
zbMATH DE Number: 3950672
zbMATH Keywords: Nakai conjecture
zbMATH Keywords: high-order derivations
zbMATH Keywords: first-order derivations
zbMATH Keywords: regularity for the localizations
reviewed by: Robert W. Berger
MaRDI profile type: Publication
full work available at URL: 0021-8693(85)90090-0
OpenAlex ID: W2022061185
Wikidata QID: Q123101678
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description: scientific article; zbMATH DE number 3950672