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Multiplicity sequence and integral dependence

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We prove that two arbitrary ideals \(I \subset J\) 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.

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Acknowledgements

The main results of this paper were obtained at the American Institute of Mathematics (AIM) in San Jose, California, while the authors participated in a SQuaRE. We are very appreciative of the hospitality offered by AIM and by the support of the National Science Foundation.

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Correspondence to Ngo Viet Trung.

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Communicated by Vasudevan Srinivas.

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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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Polini, C., Trung, N.V., Ulrich, B. et al. Multiplicity sequence and integral dependence. Math. Ann. 378, 951–969 (2020). https://doi.org/10.1007/s00208-020-02059-5

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