Ratliff-Rush Filtration, regularity and depth of Higher Associated graded modules: Part II

Abstract

Let $(A,\m)$ be a Noetherian local ring, let $M$ be a finitely generated \CM $A$-module of dimension $r\ge 2$ and let $I$ be an ideal of definition for $M$. Set $L^I(M)={⨁}_{n\ge 0}M/I^{n+1}M$. In part one of this paper we showed that $L^I(M)$ is a module over $\mathbb{R}$, the Rees algebra of $I$ and we gave many applications of $L^I(M)$ to study the associated graded module, $G_I(M)$. In this paper we give many further applications of our technique; most notable is a reformulation of a classical result due to Narita in terms of the Ratliff-Rush filtration. This reformulation can be extended to all dimensions $\ge 2$.

Part-1 of this paper is math.AC/0411324