• Vanishing of the Top Local Cohomology Modules over Noetherian Rings

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/119/01/0023-0035

• # Keywords

Artinian modules; attached prime ideals; cohomological dimension; formally isolated; local cohomology; secondary representations.

• # Abstract

Let 𝑅 be a (not necessarily local) Noetherian ring and 𝑀 a finitely generated 𝑅-module of finite dimension 𝑑. Let $\mathfrak{a}$ be an ideal of 𝑅 and $\mathfrak{M}$ denote the intersection of all prime ideals $\mathfrak{p}\in\mathrm{Supp}_R H^d_a(M)$. It is shown that

$$H^d_a(M)\simeq H^d_{\mathfrak{M}}(M)/\sum\limits_{n\in\mathbb{N}}\langle \mathfrak{M}\rangle(0:_{H^d_{\mathfrak{M}}(M)}a^n),$$

where for an Artinian 𝑅-module 𝐴 we put $\langle\mathfrak{M}\rangle A=\cap_{n\in\mathbb{N}}\mathfrak{M}^n A$. As a consequence, it is proved that for all ideals $\mathfrak{a}$ of 𝑅, there are only finitely many non-isomorphic top local cohomology modules $H^d_a(M)$ having the same support. In addition, we establish an analogue of the Lichtenbaum–Hartshorne vanishing theorem over rings that need not be local.

• # Author Affiliations

1. Department of Mathematics, Az-Zahra University, Vanak, Post Code 19834, Tehran, Iran

• # Proceedings – Mathematical Sciences

Current Issue
Volume 129 | Issue 5
November 2019

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019