On graded local cohomology modules defined by a pair of ideals

TL;DR

This paper investigates the graded structure, finiteness, vanishing, Artinian property, and tameness of local cohomology modules defined by a pair of ideals in standard graded rings, providing new insights into their algebraic properties.

Contribution

It introduces a detailed study of the graded structure and properties of local cohomology modules defined by a pair of ideals, extending existing theories.

Findings

Finiteness and vanishing conditions for graded components.

Artinian property of certain submodules and quotients.

Tameness of specific modules related to local cohomology.

Abstract

Let $R = \bigoplus_{n \in \mathbb{N}_{0}} R_{n}$ be a standard graded ring, $M$ be a finite graded $R$-module and $J$ be a homogenous ideal of $R$. In this paper we study the graded structure of the $i$-th local cohomology module of $M$ defined by a pair of ideals $(R_{+},J)$, i.e. $H^{i}_{R_{+},J}(M)$. More precisely, we discuss finiteness property and vanishing of the graded components $H^{i}_{R_{+},J}(M)_{n}$. Also, we study the Artinian property and tameness of certain submodules and quotient modules of $H^{i}_{R_{+},J}(M)$.