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 RatliffRush 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 , (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 0 I (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.

j -Multiplicity and depth of associated graded modules

“…Let M be a finite module over a Noetherian local ring R and I an R-ideal. For every j ≥ 1, let I j M = ∪ t≥1 (I j+t M : M I t ) be the Ratliff-Rush filtration of I on M (see [16,15,20]). If depth I M > 0, by [20, Lemma 3.1], there exists an integer n 0 such that I j M = I j M for every j ≥ n 0 .…”

Equimultiple coefficient ideals

“…Because of Theorem 3.8, it suffices to show (I n ) u = I n for all n. The hypothesis grade I = s(I) gives the existence of a minimal reduction J of I generated by a regular sequence of length s. Since R satisfies S s+1 , the ideal J is unmixed. Further, by [HLS,(1.2)], we have (I n ) * = I n for all n. Hence, Ass(R/I n ) ⊆ Ass(R/I n+1 ) for all n, by [PZ,Lemma 6.6]. By hypothesis we then get Ass(R/I n ) = Min(R/I n ) for all n ≤ i.…”

“…Remark 2.17. The equation ((I n ) * ) u = I n for all n is equivalent to have (I n ) * = I n for all n and (I n ) u = I n for all n. Moreover if I is a regular ideal and only (I n ) u = I n for n ≫ 0, the condition (I n ) * = I n for all n implies (I n ) u = I n for all n, since Ass(R/(I n ) * ) ⊆ Ass(R/(I n+1 ) * ) for all n ≥ 1 (see [PZ,p.14]).…”

Hilbert Functions of Filtered Modules

“…Here we extend the construction to the general case of a filtered module by following the definition given by W. Heinzer et al in [43], Section 6. A further generalization was studied by T.J. Puthenpurakal and F. Zulfeqarr in [70].…”

“…It's worth recalling that T.J. Puthenpurakal and F. Zulfeqarr in [70] and in [69] further analyzed the case when q is not a regular ideal, i.e. it does not contain an M -regular element.…”