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On the core of ideals

Published online by Cambridge University Press:  01 December 2004

Craig Huneke
Department of Mathematics, University of Kansas, Lawrence, KS 66045-7523,
Ngô Viêt Trung
Institute of Mathematics, Box 631, Bò Hô, 10000 Hanoi,
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This paper studies the core of an ideal in a Noetherian local or graded ring. By definition, the core of an ideal is the intersection of all reductions of the ideal. We provide computational formulae for the determination of the core of a graded ring, meaning the core of the unique homogeneous maximal ideal. We then apply the formulae to give answers to several questions raised by Corso, Polini and Ulrich. We are also able to answer in the positive a conjecture raised by these three authors concerning a closed formula for the core. We give a positive answer to their question in the case in which the ring is Cohen–Macaulay with a residue field of characteristic 0, and in the case the ideal is equimultiple.

Research Article
Foundation Compositio Mathematica 2005