Integral Closure of Ideals in Excellent Local Rings

Abstract

In [1] Briancon and Skoda proved, using analytic methods, that if I is an ideal in the convergent power series ring C{x1, . . . , xn} then In, the integral closure of I, is contained in I. Extensive work has been done in the direction of proving “Briancon-Skoda type theorems”, that is, statements about I t being contained in (I t−k)#, where k is a constant independent of t, and # is a closure operation on ideals (cf. [5], [6], [8], [11]). In this paper we study the following related problem: given an ideal I of a Noetherian local ring (R, m), find a “linear” integer-valued function f(n) such that I + mn ⊆ I +m for all n or for all sufficiently large n. An element x of R is said to be in the integral closure J of an ideal J if it satisfies a relation of the form x+α1x n−1+α2xn−2+. . .+αn = 0, with αt ∈ J t for all t. We first observe that if (R, m) is a noetherian local ring which is complete in the m-adic topology, then there exists an integer-valued function f(n), with limn→∞ f(n) = ∞, such that I + mn ⊆ I + m. To see this, we use the fact that I = ∩ V φ V −1(IV ), where the intersection is over all discrete valuation domains V which are R algebras via φ V and whose maximal ideal contracts to m. With this it is easily shown that ∩ n I + mn = I. By Chevalley’s theorem ([12, p. 270]) (for R/I and the descending sequence of ideals {I + mn / I}n) then I + mn ⊆ I + m for some function f(n) such that limn→∞ f(n) = ∞. Chevalley’s theorem does not help in determining the order of growth of f(n), and, in fact, it cannot because it takes into account only the topology determined by a given descending sequence of ideals. By also considering the algebraic properties of the sequence we prove a stronger statement: