Bass numbers of local cohomology modules with respect a pair of ideals

TL;DR

This paper investigates the relationships between Bass numbers of modules and their local cohomology modules with respect to a pair of ideals, establishing inequalities and finiteness conditions under specific circumstances.

Contribution

It provides new inequalities linking Bass numbers of modules and their local cohomology, and proves finiteness of Bass numbers for minimax modules when one ideal is principal.

Findings

Established inequalities between Bass numbers of modules and local cohomology modules.

Proved finiteness of Bass numbers for local cohomology modules with minimax modules over principal ideals.

Extended understanding of the structure of local cohomology modules in Noetherian local rings.

Abstract

Let $R$ be a Noetherian local ring, $I$ and $J$ two ideals of $R$, $M$ an $R$-module and $s$ and $t$ two integers. We study the relationship between the Bass numbers of $M$ and $H^{i}_{I,J}(M)$. We show that $\mu^t(M)\leq\sum_{i=0}^{t}\mu^{t-i}(H^{i}_{I,J}(M))$ and $\mu^s(H^{t}_{I,J}(M))\leq \sum_{i=0}^{t-1}\mu^{s+t+1-i}(H^{i}_{I,J}(M))+\mu^{s+t}(M)+\sum_{i=t+1}^{s+t-1}\mu^{s+t-1-i}(H^{i}_{I,J}(M))$. As a consequence, it follows that if $I$ is a principal ideal of $R$ and $M$ is a minimax $R$-module, then $\mu^j(H^{i}_{I,J}(M))$ is finite for all $i\in\Bbb N_0$ and all $j\in\Bbb N_0$.