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

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 ${\mu }^0(\mathfrak{p},H_I^2(R))$ and ${\mu }^1(\mathfrak{p},H_I^2(R))$ are finite for all $\frak p\in \Spec R$, whenever $R$ is regular. As a consequence, it follows that the Goldie dimension of $H_I^2(R)$ is finite. Also, for a finitely generated $R$-module $M$ of dimension $d$, it is shown that the Bass numbers of $H_I^{d-1}(M)$ are finite if and only if $\Ext^i_R(R/I, H^{d-1}_{I}(M))$ be minimax for all $i\ge 0$. Finally, we prove that if $\dim R/I=2$, then the Bass numbers of $H_I^n(M)$ are finite if and only if $\Ext^i_R(R/I, H^{n}_{I}(M))$ be minimax, for all $i\ge 0$, where $n$ is a non-negative integer.

16 pages, to appear in Houston Journal of Mathematics

Cited by in corpus (2)

• Cominimaxness with respect to ideals of dimension one
• Cofiniteness of local cohomology modules for ideals of dimension one