On the finiteness of Bass numbers of local cohomology modules and Cominimaxness

TL;DR

This paper investigates the finiteness of Bass numbers of local cohomology modules in Noetherian rings, establishing conditions under which these invariants are finite and exploring their relation to minimax modules.

Contribution

It proves the finiteness of certain Bass numbers in regular rings and characterizes finiteness conditions via minimax properties of Ext modules.

Findings

Bass numbers of H^2_I(R) are finite in regular rings.

Finiteness of Bass numbers of H^{d-1}_I(M) is characterized by minimax Ext modules.

Bass numbers of H^n_I(M) are finite when dim R/I=2 if and only if related Ext modules are minimax.

Abstract

In this paper, we continue the study of cominimaxness modules with respect to an ideal of a commutative Noetherian ring (cf. \cite{ANV}), and Bass numbers of local cohomology modules. Let $R$ denote a commutative Noetherian local ring and $I$ an ideal of $R$. We first show that the Bass numbers $\mu^0(\frak p, H^2_I(R))$ and $\mu^1(\frak p, H^2_I(R))$ are finite for all $\frak p\in \Spec R$, whenever $R$ is regular. As a consequence, it follows that the Goldie dimension of $H^2_I(R)$ is finite. Also, for a finitely generated $R$-module $M$ of dimension $d$, it is shown that the Bass numbers of $H^{d-1}_{I}(M)$ are finite if and only if $\Ext^i_R(R/I, H^{d-1}_{I}(M))$ be minimax for all $i\geq0$. Finally, we prove that if $\dim R/I=2$, then the Bass numbers of $H^{n}_{I}(M)$ are finite if and only if $\Ext^i_R(R/I, H^{n}_{I}(M))$ be minimax, for all $i\geq0$, where $n$ is a non-negative integer.