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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
MR Author ID: 348795
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
MR Author ID: 340709
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
MR Author ID: 175910
Email: ulrich@math.purdue.edu

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