Briançon-skoda for Noetherian Filtrations

In this note the Briançon-Skoda theorem is extended to Noetherian filtrations of ideals in a regular ring. The method of proof couples the Lipman-Sathaye approach with results due to Rees. Let A be a regular ring of dimension d and I an ideal of A. Let I denote the integral closure of I. The Briançon-Skoda theorem asserts that if I is generated by l elements then In+l ⊆ I, for all nonnegative integers n. Moreover, In+d ⊆ I for all nonnegative integers n. Both statements were proved by Lipman-Sathaye [4]. These results have generated a considerable number of papers in commutative algebra and algebraic geometry, for a general discussion see for example Chapter 13 in [7] and Chapter 9 in [3]. In this paper, our aim is to prove a Briançon-Skoda type theorem for Noetherian filtrations. Our treatment will follow classical arguments by Lipman and Sathaye, and, respectively, Rees. Let F = {In}n be a filtration of ideals of A: that is, I0 = A, In+1 ⊆ In, and InIm ⊆ In+m, for all nonnegative n, m. For any nonnegative integer k, let F(k) = {In+k}n (technically this is not a filtration according to the above definition since on the zeroth spot we have Ik and not A, but this will not affect what follows). Also, given two filtrations F = {In}n and G = {Jn}n, we write F ⊆ G if In ⊆ Jn for all n. The filtration F is called Noetherian if its Rees algebra R = ⊕n≥0Intn is Noetherian over A. This holds if and only if its extended Rees algebra S = R[t−1] ⊂ A[t−1, t] is Noetherian. There are various definitions of Noetherian filtrations in the literature. We follow the terminology used by Rees in [6], although the reader should be