Computing the core of ideals in arbitrary characteristic (Q926845)

Let \((R, \mathfrak m)\) denote a local ring with infinite residue field \(k = R/\mathfrak m.\) For an ideal \(I\) of \(R\) let \(\text{core } I\) denote the core of \(I,\) i.e. the intersection of all of the minimal reductions of \(I.\) Because there are infinitely many minimal reductions of \(I\) it is difficult to compute \(\text{core }I.\) Under certain additional conditions, \textit{C. Polini} and \textit{B. Ulrich} [Math. Ann. 331, No. 3, 487--503 (2005; Zbl 1089.13005)] have shown that \(\text{core} I = J^{n+1} : I^n\) for all \(n \gg 0,\) where \(J\) is a minimal reduction of \(I.\) One of the conditions is upon the characteristic of the residue field. Their result is a consequence of the following containement relations \(J^{n+1} : I^n \subseteq \text{core } I \subseteq J^{n+1} : \sum_{b \in I} (J, b)^n =: K\) for all \(n \gg 0.\) In the paper under review the author studied when \(\text{core } I = K\) for \(n \gg 0\) under the condition of the result of Polini and Ulrich [loc. cit.], but independently of the characteristic of \(k.\) To this end she is studying explicitly the ideal \(\sum_{b \in I} (J, b)^n\) as well as \(K.\) As a consequence of her considerations she gives a negative answer to this question for higher analytic spreads. Moreover she suggests a formula for the core of such ideals. Some of the explicit computations are done by \textsc{Macaulay II}.