Compositio Mathematica

On the core of ideals

Craig Huneke a1 and Ngô Viêt Trung a2
a1 Department of Mathematics, University of Kansas, Lawrence, KS 66045-7523, USA
a2 Institute of Mathematics, Box 631, Bò Hô, 10000 Hanoi, Vietnam

Article author query
huneke c   [Google Scholar] 
trung nv   [Google Scholar] 


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.

(Received October 22 2002)
(Accepted December 2 2003)
(Published Online December 1 2004)

Key Words: minimal reduction; core; generic element.

Maths Classification