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 $r(I,M)={\bigcup }_{k⩾1}(I^{k+1}M:I^{k}M)$, {\it the Ratliff-Rush ideal associated with $I$ and $M$}. When $M=R$ (or more generally when $M$ is projective) then $r(I,M)=\tilde{I}$, 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 ${\p}\in \spec R\setminus \mspec R,$ then under mild condition on $R$ we show that for a regular ideal $I$, $\ell (r(I,M)/\tilde{I})$ is finite. Further $r(I,M)=\tilde{I}$ if $A^*(I)\cap \mspec R =\emptyset $ (here $A^{\ast }(I)$ 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 $\widetilde{I^n}=(I^{n+k}:I^k)$ for all $n⩾1$. For any ideal $I$ we show that $\widetilde{I^nM}=I^nM+H^0_I(M)\quad\mbox{for all} n\gg 0.$ This yields that $\tilde{\mathcal{R}}(I,M)={⨁}_{n⩾0}\widetilde{I^{n}M}$ is Noetherian if and only if $\depth M>0$. Surprisingly if $\dim M=1$ then ${\tilde{G}}_I(M)={⨁}_{n⩾0}\widetilde{I^{n}M}/\widetilde{I^{n+1}M}$ is always a Noetherian and a Cohen-Macaulay $G_I(R)$-module. Application to Hilbert coefficients is also discussed.

27 pages. Many minor revisions made, including little changes in title and abstract. Five additional refernces added. To appear in Journal of algebra