On the vanishing, artinianness and finiteness of local cohomology modules

TL;DR

This paper investigates the properties of local cohomology modules over noetherian rings, establishing conditions under which they are artinian, finite, or minimax, and presents related vanishing and non-vanishing theorems.

Contribution

It proves that minimax local cohomology modules become artinian beyond a certain degree and establishes a local-global principle for minimax modules, extending understanding of their structure.

Findings

Minimax local cohomology modules are artinian for degrees above a threshold.

A local-global principle for minimax local cohomology modules is established.

Conditions are provided under which locally minimax modules are globally minimax.

Abstract

Let $R$ be a noetherian ring, $\fa$ an ideal of $R$, and $M$ an $R$--module. We prove that for a finite module $M$, if $\LC^{i}_{\fa}(M)$ is minimax for all $i\geq r\geq 1$, then $\LC^{i}_{\fa}(M)$ is artinian for $i\geq r$. A Local-global Principle for minimax local cohomology modules is shown. If $\LC^{i}_{\fa}(M)$ is coatomic for $i\leq r$ ($M$ finite) then $\LC^{i}_{\fa}(M)$ is finite for $i\leq r$. We give conditions for a module, which is locally minimax to be a minimax module. A non-vanishing theorem and some vanishing theorems are proved for local cohomology modules.