Ratliff–Rush filtration, regularity and depth of higher associated graded modules. Part II

Let (A,m)(A,m) be a Noetherian local ring, let M be a finitely generated Cohen–Macaulay A  -module of dimension r≥2r≥2 and let I be an ideal of definition for M  . Set LI(M)=⨁n≥0M/In+1MLI(M)=⨁n≥0M/In+1M. In part one of this paper we showed that LI(M)LI(M) is a module over R(I)R(I), the Rees algebra of I   and we gave many applications of LI(M)LI(M) to study the associated graded module, GI(M)GI(M). In this paper we give many further applications of our technique; most notable is a complete characterization of good behavior of the Ratliff–Rush filtration modulo a superficial element.