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

TL;DR

This paper extends the study of the Ratliff-Rush filtration and its applications to the regularity and depth of higher associated graded modules of Cohen-Macaulay modules over Noetherian local rings, generalizing classical results.

Contribution

It provides a new reformulation of Narita's classical result using the Ratliff-Rush filtration, applicable in all dimensions greater than or equal to two.

Findings

Reformulation of Narita's result via Ratliff-Rush filtration

Applications to regularity and depth of associated graded modules

Extension of techniques to all dimensions ≥ 2

Abstract

Let $(A,\m)$ be a Noetherian local ring, let $M$ be a finitely generated \CM $A$-module of dimension $r \geq 2$ and let $I$ be an ideal of definition for $M$. Set $L^I(M) = \bigoplus_{n\geq 0}M/I^{n+1}M$. In part one of this paper we showed that $L^I(M)$ is a module over $\R$, 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 $\geq 2$.