Core and residual intersections of ideals (Q2782654)

The core of an ideal was defined by Rees and Sally as the intersection of all the reductions of the ideal [\textit{D. Rees} and \textit{J. Sally}, Mich. Math. J. 35, No. 2, 241-254 (1988; Zbl 0666.13004)]. Thus a priori the core is an infinite intersection and thus hard to compute. In a previous paper the authors showed that, under some quite general restrictions on the ring and ideals, the core is actually a finite intersection and behaves well [\textit{A. Corso, C. Polini} and \textit{B. Ulrich}, Math. Ann. 321, No. 1, 89-105 (2001; Zbl 0992.13003)]. In this paper the authors concentrate on the issue of an explicit formulation of the core. An explicit formulation was previously given for integrally closed ideals in two-dimensional regular rings by \textit{C. Huneke} and \textit{I. Swanson} [Mich. Math. J. 42, No. 1, 193-208 (1995; Zbl 0829.13014)]. NEWLINENEWLINENEWLINEIn the paper under review, the authors greatly generalize that result: They weaken the dimension, the regularity, and the integral closure conditions. They introduce the notion of balanced ideals: An ideal \(I\) is ``balanced'' if \(I\):\(J\) is independent of the minimal reduction \(J\) of \(I\). For these ideals they then prove (under some conditions on the ring) that the core of \(I\) is \((J\):\(I) I\). The authors prove further properties of cores, such as when they are integrally closed, and they give other explicit characterizations of the cores of several special classes of ideals.