Equivalence and finite determinacy of mappings
type: Item
instance of: scholarly article
title: Equivalence and finite determinacy of mappings
zbMATH Open document ID: 0955.13010
author: Steven Dale Cutkosky
author: Hema Srinivasan
published in: Journal of Algebra
publication date: 2001-02-16T00:00:00Z
review text: The aim of this article is to consider the following problem: Given two ideals \(I\) and \(J\) in \(R\) a power series ring over a field \(k\), or a ring of convergent power series, when is there an isomorphism \(R/I \simeq R/J\)? This article is the continuation of a previous article of the same authors where they give an answer in the \(j(I)\)-adic completion \(\widehat R\) of \(R\), where \(j(I)\) is the Jacobian ideal of \(I\). In the first part the authors give ``the extension to the analytic case'' of this previous result. Theorem: Let \(k\) be a field of characteristic zero with a non-trivial valuation, let \(I\) be a reduced and equidimensional ideal of the ring of convergent power series \(R = k \{ x_1 , \ldots , x_r \}\). Then there is an integer \(a\) depending only on \(I\), such that if \(J\) is any another reduced equidimensional ideal of the same height as \(I\), and if \(I + j(I) ^{ac} = J + j(I) ^{ac}\) for some \(c \geq 1\), then there exists an automorphism \(s\) of \(R\) such that \(s(I) = J\) and \(s \equiv\) id mod \(j(I) ^c\). This theorem is an extension of Hironaka's theorem for isolated singularities. To prove this result, the authors use their previous theorem for \(\widehat R\) and the approximation theorem of Artin for solutions of analytic equations. If \(F\) is an homomorphism, \(F \in \Hom(R^n,R)\), and \(H\) an ideal of \(R\), we say that \(F\) is finitely determined by \(H\) if there is a positive integer \(a\), such that \(F \cong G\bmod H\) implies that there exist an automorphism \(\phi\) of \(R\) and an \(R\)-isomorphism \(A\) of \(R^n\) such that \(\phi (F) = A G\). If \(I\) is the ideal of \(R\), image of \(F\), and if \(F\) is determined by an ideal \(H\) which depends only on \(I\), we get a criterion for determining when \(R/I\) would be isomorphic to \(R/J\). (The idea of finite determination was introduced by Mather for analytic and \(C^{\infty}\) mappings.) Then the authors give a condition for the existence of a non-tri…
Mathematics Subject Classification ID: 13F25
Mathematics Subject Classification ID: 13J05
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 14E15
zbMATH DE Number: 1013912
zbMATH Keywords: equivalence of analytic mapping
zbMATH Keywords: Artin approximation
zbMATH Keywords: finitely determined ideal
reviewed by: Michel Vaquié
MaRDI profile type: Publication
full work available at URL: jabr.1996.6822
OpenAlex ID: W1977501606
cites work: On the solutions of analytic equations
cites work: Some structure theorems for finite free resolutions
cites work: An Intrinsic Criterion for Isomorphism of Singularities
cites work: Q5602063
cites work: Solutions d'équations à coefficients dans un anneau hensélien
cites work: Q5525145
cites work: Q5605086
cites work: Uniform bounds in noetherian rings
cites work: Stability of \(C^ \infty\) mappings. III: Finitely determined map germs
cites work: Q3739243
cites work: On the approximation property of excellent rings
cites work: Singularités des variétés algébriques
cites work: Idéaux de fonctions différentiables. I
DOI: 10.1006/JABR.1996.6822
description: scientific article; zbMATH DE number 1013912