Reduction numbers of equimultiple ideals
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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 $c_q(J)-1$.References
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Additional Information
- Cătălin Ciupercă
- Affiliation: Department of Mathematics 2750, North Dakota State University, P.O. Box 6050, Fargo, North Dakota 58108-6050
- MR Author ID: 676535
- ORCID: 0000-0003-2716-4905
- Email: catalin.ciuperca@ndsu.edu
- 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
- © Copyright 2016 American Mathematical Society
- 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
- MathSciNet review: 3626495