j-Multiplicity and depth of associated graded modules

Let R be a Noetherian local ring. We define the minimal j-multiplicity and almost minimal j-multiplicity of an arbitrary R-ideal on any finite R-module. For any ideal I with minimal j-multiplicity or almost minimal j-multiplicity on a Cohen–Macaulay module M, we prove that under some residual conditions, the associated graded module grI(M) is Cohen–Macaulay or almost Cohen–Macaulay, respectively. Our work generalizes the results for minimal multiplicity and almost minimal multiplicity obtained by Sally, Rossi, Valla, Wang, Huckaba, Elias, Corso, Polini, and Vaz Pinto.