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 I for a broader class of ideals, where I I is an ideal in a local Cohen-Macaulay ring. In this paper, we show that if I I is an ideal of analytic spread 1 1 in a Noetherian local ring with infinite residue field, then with some mild conditions on I I , we have core ⁡ ( I ) ⊇ J ( J n : I n ) = I ( J n : I n ) = ( J n + 1 : I n ) ∩ I \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 J of I I and for n ≫ 0 n\gg 0 .