Integral closure of ideals and equisingularity
type: Item
instance of: scholarly article
title: Integral closure of ideals and equisingularity
zbMATH Open document ID: 1171.13005
author: Monique Lejeune-Jalabert
author: Bernard Teissier
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 2009-04-14T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/10106
review text: This text has two parts. The first one is the integral version of the 1973-1974 Seminar Lejeune-Teissier on integral dependence in complex analytic geometry. The second part is a short survey of recent results directly related to the first part. In the first chapter, the notions of valuations and semi valuations on a reduced analytic algebra \(A\) and their associated filtrations are introduced. As an example, the authors define the Rees function \({\overline \nu }_I(x)=\lim_{k\rightarrow \infty } \nu_I(x^k)/k\), where \(\nu_I\) is the \(I-\)adic valuation. The integral closure appears as one ideal in the filtration associated to \({\overline \nu }_I\), namely \({\overline I}=\{x\in A \mid {\overline \nu }_I(x)\geq 1\}\). In the third and the fourth chapters they establish the link between the normalized blow up, and the function \({\overline \nu }_I\), and they prove that \({\overline \nu }_I(x)\in \mathbb Z/q\), where \(q\) is an integer number. (This last fact was proved before by Nagata but unknown to the authors). Another important point is that given a coherent sheaf \({\mathcal I}\) of \({\mathcal O}_X\)-ideals on a reduced analytic space \(X\) they define for each positive real number \(\nu \) the sheaves \(\overline{{\mathcal I}^\nu }\) and \(\overline{{\mathcal I}^{\nu_+} }\), and the graded \({\mathcal O}_X/{\mathcal I}-\)algebra \[ \overline{ gr}_{\mathcal I}{\mathcal O}_X=\bigoplus_{\nu} \overline{{\mathcal I}^\nu }/\overline{{\mathcal I}^{\nu_+} }. \] This algebra is locally finitely generated. In chapter five, the authors compute \({\overline \nu }\) by using analytic arcs, and in chapter six, it is shown that Lojasiewicz exponents are the inverses of \({\overline \nu }\), and so they are rational. The seminar is completed by the Risler's appendix on the Lojasiewicz exponents in the real case. The second part of this paper contains several developments of the first part. The first part is the proof in the spirit of the seminar of the classical Lojasie…
reviewed by: Marcel Morales
Mathematics Subject Classification ID: 13B22
Mathematics Subject Classification ID: 13A18
Mathematics Subject Classification ID: 13A30
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 14B12
zbMATH DE Number: 5543510
zbMATH Keywords: Rees valuation
zbMATH Keywords: Lojasiewicz's number
zbMATH Keywords: integral closure
zbMATH Keywords: graded associated ring
zbMATH Keywords: toric ring
zbMATH Keywords: blow-up
zbMATH Keywords: normalized blow-up
MaRDI profile type: Publication
OpenAlex ID: W1677443536
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Integral closure of ideals and equisingularity (DOI)
DOI: 10.5802/AFST.1203

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