Lower bounds of certain general local cohomology modules

TL;DR

This paper investigates the properties and bounds of general local cohomology modules over Noetherian rings, establishing conditions under which these modules belong to certain subcategories and analyzing their support and vanishing behavior.

Contribution

It provides new criteria relating the membership of local cohomology modules in Melkersson subcategories and explores their support and finiteness properties.

Findings

If certain local cohomology modules are in a subcategory, then others are as well.

Equivalence conditions for modules belonging to subcategories based on their support.

Support of specific local cohomology modules can be infinite in local rings.

Abstract

Let $R$ be a commutative Noetherian ring, $\Phi$ a system of ideals of $R$, $\fa \in \Phi$, $M$ an arbitrary $R$-module and $t$ a non-negative integer. Let $\mathcal{S}$ be a Melkersson subcategory of $R$-modules. Among other things, we prove that if $\lc^{i}_\Phi(M)$ is in $\mathcal{S}$ for all $i < t$ then $\lc^{i}_\fa(M)$ is in $\mathcal{S}$ for all $i < t$ and for all $\fa \in \Phi$. If $\mathcal{S}$ is the class of $R$-modules $N$ with $\dim N \leq k$ where $k \geq -1$, is an integer, then $\lc^{i}_\Phi(M)$ is in $\mathcal{S}$ for all $i < t$ (if and only if) $\lc^{i}_\fa(M)$ is in $\mathcal{S}$ for all $i < t$ and for all $\fa \in \Phi$. As consequences we study and compare vanishing, Artinianness and support of general local cohomology and ordinary local cohomology supported at ideals of its system of ideals at initial points $i <t$. We show that $\Supp_{R}(\lc^{\dim M-1}_{\Phi}(M))$ is not necessarily finite whenever $(R,\fm)$ is local and $M$ a finitely generated $R$-module.