Ratliff-Rush filtrations associated with ideals and modules over a Noetherian ring

Abstract

Let R be a commutative Noetherian ring, M a finitely generated R-module and I a proper ideal of R. In this paper we introduce and analyze some properties of r(I, M) = U-k >= 1 (Ik + 1 M : I-k M), the Ratliff-Rush ideal associated with I and M. When M = R (or more generally when M is projective) then r(I, M) = (I) over tilde, the usual Ratliff-Rush ideal associated with I. If I is a regular ideal and ann M = 0 we show that {r(I-n, M)}(n >= 0) is a stable I-filtration. If M-p is free for all p is an element of Spec R \ m-Spec R, then under mild condition on R we show that for a regular ideal I, l(r(I, M)/(I) over tilde) is finite. Further r(I, M) = (I) over tilde if A*(I) boolean AND m-Spec R = 0 (here A*(I) is the stable value of the sequence Ass(R / I-n)). Our generalization also helps to better understand the usual Ratliff-Rush filtration. When I is a regular m-primary ideal our techniques yield an easily computable bound for k such that (I) over tilde (n) = (In + k : I-k) for all n >= 1. For any ideal I we show that (I-n M) over tilde = I-n M + H-1(0)(M) for all n >> 0. This yields that (R) over tilde (I, M) = circle times(n >= 0) I-n M is Noetherian if and only if depth M > 0. Surprisingly if dim M = 1 then (G) over tilde (I)(M) = circle times(n >= 0) (I-n M) over tilde / In + 1 M is always a Noetherian and a Cohen-Macaulay G(I) (R)-module. Application to Hilbert coefficients is also discussed. (c) 2007