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Reduction numbers of equimultiple ideals


Author: Cătălin Ciupercă
Journal: Proc. Amer. Math. Soc. 145 (2017), 2361-2371
MSC (2010): Primary 13A30, 13B22, 13H15; Secondary 13D40
DOI: https://doi.org/10.1090/proc/13402
Published electronically: November 30, 2016
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Abstract: Let $ (A, \mathfrak{M})$ be an unmixed local ring containing a field. If $ J$ is an $ \mathfrak{M}$-primary ideal with Hilbert-Samuel multiplicity $ e_A(J)$, a recent result of Hickel shows that every element in the integral closure $ \overline {J}$ satisfies an equation of integral dependence over $ J$ of degree at most $ \operatorname {e}_A(J)$. We extend this result to equimultiple ideals $ J$ by showing that the degree of such an equation of integral dependence is at most $ c_q(J)$, where $ c_q(J)$ is one of the elements of the so-called multiplicity sequence introduced by Achilles and Manaresi. As a consequence, if the characteristic of the field contained in $ A$ is zero, it follows that the reduction number of an equimultiple ideal $ J$ with respect to any minimal reduction is at most $ c_q(J)-1$.


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Cătălin Ciupercă
Affiliation: Department of Mathematics 2750, North Dakota State University, P.O. Box 6050, Fargo, North Dakota 58108-6050
Email: catalin.ciuperca@ndsu.edu

DOI: https://doi.org/10.1090/proc/13402
Received by editor(s): December 29, 2015
Received by editor(s) in revised form: July 26, 2016
Published electronically: November 30, 2016
Communicated by: Irena Peeva
Article copyright: © Copyright 2016 American Mathematical Society

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