Integral closure of ideals and equisingularity
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 Lojasiewicz inequality. The second point is that \({\overline \nu }_I(x)\) can be seen as the slope of some Newton polyhedra. The developments on the Newton Polyhedra are very instructive. The third point is about Izumi's work. The fourth point is a generalization due to \textit{C. Ciuperca, F. Enescu} and \textit{S. Spiroff} [Ill. J. Math. 51, No. 1, 29--39 (2007; Zbl 1137.13002)] of the rationality of \({\overline \nu }\) to the case of several ideals. The fifth point presents the connection of \({\overline \nu }\) with the notion of \textit{type} of ideals introduced by \textit{J. P. d'Angelo} [Ann. Math., II. Ser. 115, 615--637 (1982; Zbl 0488.32008)] and used recently by \textit{G. Heier} [Commun. Algebra 36, No. 8, 2947--2957 (2008; Zbl 1149.14047)] for the proof of an effective Nullstellensatz. The sixth point concern results of the reviewer and others [e.g. \textit{M. Morales, Ngô Viêt Trung} and \textit{O. Villamayor}, J. Algebra 129, No.~1, 96--102 (1990; Zbl 0701.13004)] about Hilbert function associated to the integral closed powers of a primary ideal in an excellent ring. Finally it is pointed the importance of the graded algebra \(\overline{ gr}_{ I}{A}_X\) in the equisingularity theory of a plane analytic branch. The reviewer should say that the abstracts of this paper is more complete than this review, and that this paper is very interesting and useful to everybody interested to the integral closure of ideals. The references are appropriated and limited to the subject of the Seminar. The reading of this paper should be completed with the books of \textit{W. Vasconcelos} [Integral Closure (Springer Monographs in Mathematics, Springer-Verlag, Berlin) (2005; Zbl 1082.13006)], and \textit{I. Swanson} and \textit{C. Huneke} [London Mathematical Society Lecture Note Series 336. Cambridge: Cambridge University Press (2006; Zbl 1117.13001)].
scientific article; zbMATH DE number 5543510
scientific article