$j$-multiplicity and depth of associated graded modules

TL;DR

This paper introduces the concepts of minimal and almost minimal j-multiplicity for ideals on modules over Noetherian local rings, and shows their implications for the Cohen-Macaulay property of associated graded modules, generalizing previous multiplicity results.

Contribution

It defines new multiplicity notions for ideals on modules and establishes their connection to the Cohen-Macaulayness of associated graded modules, extending prior multiplicity theories.

Findings

Minimal j-multiplicity implies Cohen-Macaulay associated graded modules.

Almost minimal j-multiplicity implies almost Cohen-Macaulay associated graded modules.

Generalizes classical results on minimal and almost minimal multiplicity.

Abstract

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 assumptions, the associated graded module ${\rm gr}_I(M)$ 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 VazPinto.