Multiplicity sequence and integral dependence
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Mathematische Annalen
Multiplicity sequence and integral dependence Claudia Polini1 · Ngo Viet Trung2
· Bernd Ulrich3 · Javid Validashti4
Received: 29 January 2020 / Revised: 30 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We prove that two arbitrary ideals I ⊂ J in an equidimensional and universally catenary Noetherian local ring have the same integral closure if and only if they have the same multiplicity sequence. We also obtain a Principle of Specialization of Integral Dependence, which gives a condition for integral dependence in terms of the constancy of the multiplicity sequence in families. Mathematics Subject Classification Primary 13B22 · 13D40 · 13A30; Secondary 14B05 · 14D99
Communicated by Vasudevan Srinivas. Claudia Polini and Bernd Ulrich were partially supported by NSF grants DMS-1601865 and DMS-1802383, respectively. Ngo Viet Trung was partially supported by grant 101.04-2019.313 of the Vietnam National Foundation for Science and Technology Development.
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Ngo Viet Trung [email protected] Claudia Polini [email protected] Bernd Ulrich [email protected] Javid Validashti [email protected]
1
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA
2
International Centre for Research and Postgraduate Training, Institute of Mathematics, Vietnam Academy of Science and Technology, 10307 Hanoi, Vietnam
3
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
4
Department of Mathematics, DePaul University, Chicago, IL 60604, USA
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C. Polini et al.
1 Introduction The aim of this paper is to prove a numerical criterion for integral dependence of arbitrary ideals, which is an important topic in commutative algebra and singularity theory. The first numerical criterion for integral dependence was proved by Rees in 1961 [23]: let I ⊂ J be two m-primary ideals in an equidimensional and universally catenary Noetherian local ring (R, m). Then I and J have the same integral closure if and only if they have the same Hilbert–Samuel multiplicity. This multiplicity theorem plays an important role in Teissier’s work on the equisingularity of families of hypersurfaces with isolated singularities, as it is used in the proof of his principle of specialization of integral dependence (PSID) [27,28]. For hypersurfaces with non-isolated singularities, one needs a similar numerical criterion for integral dependence of non-m-primary ideals. In 1969, Böger [6] extended Rees’ multiplicity theorem to the case of equimultiple ideals. We refer to a survey of Lipman for the geometric significance of equimultiplicity [14]. Subsequently, there were further generalizations that still maintain remnants of the m-primary assumption, see for instance [11,18,22,24]. For a long time, it was not clear how to extend Rees’ multiplicity theorem to arbitrary ideals. Since the Hilbert– Samuel multiplicity is no longer defined for non-m-primary ideals, the need arose to use other notions of multiplicities that can be used to check for integral dependence.
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