A differential condition on the normal cone
type: Item
instance of: scholarly article
title: A differential condition on the normal cone
zbMATH Open document ID: 0594.13025
DOI: 10.1016/0021-8693(84)90158-3
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: Let R be a ring, P a proper ideal of R. P is said to be maximally \({\mathcal D}\)-differential for \({\mathcal D}\subset Der(P)\) if d(P)\(\subset P\) for \(d\in {\mathcal D}\) and if Q is any ideal such that \(P\subsetneqq Q\subsetneqq R\) then there exists \(d\in {\mathcal D}\) such hat d(Q)\(\not\subset Q\). A graded ring \(A=\oplus A_ n\) satisfies condition (C) iff \(A_+\) is maximally D(A)-differential. If P is a maximal differential ideal of R then \(gr_ P(R)\) satisfies condition (C). The author considers for a noetherian local ring R containing a field of characteristic zero and having the property to be a G-ring the following conditions: (i) P is maximally differential in R; (ii) \(gr_ P(R)\) satisfies condition (C); ((iii) P is permissible in R. Then: \[ (i)\to (ii)\to (iii)\nrightarrow (ii)\nrightarrow (i). \] This result shows that a criterion given by Seibt \(((iii)\to (i))\) [\textit{P. Seibt}, J. Algebra 66, 484-491 (1980; Zbl 0469.13002)] is wrong. Giving a structure theorem for noetherian graded rings A such that \(A_ 0\) is a complete local ring containing a field of characteristic zero which satisfies condition (C) (Theorem 2.8) the author shows: If R/P is complete then condition (C) on \(gr_ P(R)\) is equivalent to saying that Spec\((gr_ P(R))\) is the product of an affine algebroid space and a suitable tangent cone.
Mathematics Subject Classification ID: 13N05
Mathematics Subject Classification ID: 14A05
Mathematics Subject Classification ID: 13H99
Mathematics Subject Classification ID: 13E05
zbMATH DE Number: 3955081
zbMATH Keywords: normal cone
zbMATH Keywords: maximally differential ideal
zbMATH Keywords: noetherian local ring
zbMATH Keywords: Spec
author: Dilip P. Patil
author: Balwant Singh
reviewed by: Karl-Heinz Kiyek
MaRDI profile type: Publication
full work available at URL: 0021-8693(84)90158-3
OpenAlex ID: W2047187465
cites work: Derivations on Commutative Rings
cites work: Resolution of singularities of an algebraic variety over a field of characteristic zero. I
cites work: Q5611879
cites work: Differential criteria on the permissibility of a blowing-up
cites work: Differential Ideals in Rings of Finitely Generated Type
cites work: Maximally differential prime ideals in a complete local ring
cites work: Permissibility of a maximally differential prime ideal
cites work: Studies in Equisingularity I Equivalent Singularities of Plane Algebroid Curves
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description: scientific article; zbMATH DE number 3955081