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

M. Jahangiri, R. Ahangari Maleki

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 $\{μ^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 $\mathfrak{a}$.

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

Abstract

Let be a commutative Noetherian ring, be an ideal of and be a finitely generated -module. In this paper, we study the Bass numbers of local cohomology modules with respect to an ideal in each of the following cases: and is regular and and , the cohomological dimension of with respect to .
Paper Structure (2 sections, 9 theorems, 13 equations)

This paper contains 2 sections, 9 theorems, 13 equations.

Table of Contents

  1. introduction
  2. Results

Key Result

Theorem 1.2

Assume that $R$ is regular, $\mathfrak{p}\in \operatorname{Supp}_R(H^c_{\mathfrak{a}}(M))$ and set $d:= \operatorname{ht}(\mathfrak{p})$ and $c:= \operatorname{cd}_{\mathfrak{a}}(M)$. Then the following statements hold.

Theorems & Definitions (14)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • ...and 4 more