Bass numbers of local cohomology modules at the first and last non-vanishing levels

Abstract

Let $R $ be a commutative Noetherian ring, $\mathfrak{a}$ be an ideal of $R$ and $M$ be a finitely generated $R$-module. In this paper, we study the Bass numbers $\{\mu^i(\mathfrak{p}, H^j_{\mathfrak{a}}(M))\} $ of local cohomology modules with respect to an ideal $\mathfrak{p}\in Spec(R)$ in each of the following cases: $i\in \{ 0, 1, 2\}$ and $j= grade_{\mathfrak{a}}(M),$ $R$ is regular and $i\in\{ ht(\mathfrak{p}), ht(\mathfrak{p})- 1\}$ and $j= cd_{\mathfrak{a}}(M)$, the cohomological dimension of $M$ with respect to $\mathfrak{a}$.