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 $\{{\mu }^i(\mathfrak{p},H_{\mathfrak{a}}^j(M))\}$ 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 $j=grad{e}_{\mathfrak{a}}(M),$ $R$ is regular and $i\in \{ht(\mathfrak{p}),ht(\mathfrak{p})-1\}$ and $j=c{d}_{\mathfrak{a}}(M)$, the cohomological dimension of $M$ with respect to $\mathfrak{a}$.

9 pages, comments are well come!