The core of ideals in arbitrary characteristic

Abstract

In this paper we provide explicit formulas for the core of an ideal. Recall that for an ideal I in a Noetherian ring R, the core of I, core(I ), is the intersection of all reductions of I. For a subideal J ⊂ I we say that J is a reduction of I, or that I is integral over J, if I r+1 = JI r for some r ≥ 0; the smallest such r is called the reduction number of I with respect to J and is denoted by rJ(I ). If (R,m) is local with infinite residue field k then every ideal has a minimal reduction, which is a reduction minimal with respect to inclusion. Minimal reductions of a given ideal I are far from unique, but they all share the same minimal number of generators, called the analytic spread of I and written (I ). Minimal reductions arise from Noether normalizations of the special fiber ring F(I ) = grI(R)⊗k of I, and therefore (I ) = dimF(I ). From this one readily sees that ht I ≤ (I ) ≤ dimR; these inequalities are equalities for any m-primary ideal, and if the first inequality is an equality then I is called equimultiple. Obviously, the core can be obtained as an intersection of minimal reductions of a given ideal. Through the study of the core one hopes to better understand properties shared by all reductions. The notion was introduced by Rees and Sally for the purpose of generalizing the Briancon–Skoda Theorem [17]. As an a priori infinite intersection of reductions, the core is difficult to compute, and there have been considerable efforts to find explicit formulas; see [3; 4; 9; 10; 11; 12; 15]. We quote the following result from [15].