Core of ideals of Noetherian local rings

Abstract:

The core of an ideal is the intersection of all its reductions. In 2005, Polini and Ulrich explicitly described the core as a colon ideal of a power of a single reduction and a power of $I$ for a broader class of ideals, where $I$ is an ideal in a local Cohen-Macaulay ring. In this paper, we show that if $I$ is an ideal of analytic spread $1$ in a Noetherian local ring with infinite residue field, then with some mild conditions on $I$, we have $\operatorname {core} (I)\supseteq J(J^n: I^n)=I(J^n: I^n)=(J^{n+1}: I^n)\cap I$ for any minimal reduction $J$ of $I$ and for $n\gg 0$.