Ratliff–Rush filtration, regularity and depth of higher associated graded modules: Part I

In this paper we introduce a new technique to study associated graded modules. Let (A,m)(A,m) be a Noetherian local ring with depthA≥2. Our techniques give a necessary and sufficient condition for depthGmn(A)≥2 for all n≫0n≫0. Other applications are also included; most notable is an upper bound regarding the Ratliff–Rush filtration.