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

TL;DR

This paper investigates the Bass numbers of local cohomology modules at key non-vanishing levels for finitely generated modules over Noetherian rings, focusing on specific cases involving regular rings and cohomological dimensions.

Contribution

It provides new insights into the structure of Bass numbers of local cohomology modules at the first and last non-vanishing levels, especially in regular rings and at specific cohomological degrees.

Findings

Bass numbers at the grade of M are characterized for i=0,1,2.

Bass numbers at the cohomological dimension are analyzed for i=ht(p) and ht(p)-1.

Results clarify the behavior of local cohomology modules in key non-vanishing degrees.

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 $\mathfrak{a}$.