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 ) = ⋃ k ⩾ 1 ( I k + 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 ˜ , 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 ∈ Spec R ∖ m - Spec R , then under mild condition on R we show that for a regular ideal I, ℓ ( r ( I , M ) / I ˜ ) is finite. Further r ( I , M ) = I ˜ if A ∗ ( I ) ∩ m - Spec R = ∅ (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 n ˜ = ( I n + k : I k ) for all n ⩾ 1 . For any ideal I we show that I n M ˜ = I n M + H I 0 ( M ) for all n ≫ 0 . This yields that R ˜ ( I , M ) = ⊕ n ⩾ 0 I n M ˜ is Noetherian if and only if depth M > 0 . Surprisingly if dim M = 1 then G ˜ I ( M ) = ⊕ n ⩾ 0 I n M ˜ / I n + 1 M ˜ is always a Noetherian and a Cohen–Macaulay G I ( R ) -module. Application to Hilbert coefficients is also discussed.