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Transactions of the American Mathematical Society

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Core and residual intersections of ideals


Authors: Alberto Corso, Claudia Polini and Bernd Ulrich
Journal: Trans. Amer. Math. Soc. 354 (2002), 2579-2594
MSC (2000): Primary 13H10; Secondary 13A30, 13B22, 13C40, 13D45
DOI: https://doi.org/10.1090/S0002-9947-02-02908-2
Published electronically: February 1, 2002
MathSciNet review: 1895194
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Abstract: D. Rees and J. Sally defined the core of an $R$-ideal $I$ as the intersection of all (minimal) reductions of $I$. However, it is not easy to give an explicit characterization of it in terms of data attached to the ideal. Until recently, the only case in which a closed formula was known is the one of integrally closed ideals in a two-dimensional regular local ring, due to C. Huneke and I. Swanson. The main result of this paper explicitly describes the core of a broad class of ideals with good residual properties in an arbitrary local Cohen-Macaulay ring. We also find sharp bounds on the number of minimal reductions that one needs to intersect to get the core.


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Additional Information

Alberto Corso
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
Email: corso@ms.uky.edu

Claudia Polini
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
Address at time of publication: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: cpolini@nd.edu

Bernd Ulrich
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Address at time of publication: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: ulrich@math.purdue.edu

DOI: https://doi.org/10.1090/S0002-9947-02-02908-2
Keywords: Integral closure, reductions, residual intersections of ideals
Received by editor(s): April 10, 2001
Published electronically: February 1, 2002
Additional Notes: The first author was partially supported by the NATO/CNR Advanced Fellowships Programme during an earlier stage of this work. The second and third authors were partially supported by the NSF
Dedicated: To Professor Craig Huneke on the occasion of his fiftieth birthday
Article copyright: © Copyright 2002 American Mathematical Society

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