Graded Version of Some Basic Theorems on Local Cohomology to a Pair of Ideals

TL;DR

This paper extends fundamental theorems on local cohomology to a graded setting for a pair of ideals, including vanishing, finiteness, and Artinian properties, providing generalized and new results.

Contribution

It introduces graded versions of key local cohomology theorems for pairs of ideals and generalizes Melkersson's theorem on Artinian modules.

Findings

Proved graded versions of the Independence and Vanishing Theorems.

Established a generalized Melkersson's theorem on Artinian modules.

Presented results on Artinianess of local cohomology modules.

Abstract

In this paper, we prove some well-known results on local cohomology with respect to a pair of ideals in graded version, such as, Independence Theorem, Lichtenbaum-Harshorne Vanishing Theorem, Basic Finiteness and Vanishing Theorem, among others. Besides, we present a generalized version of Melkersson Theorem about Artinianess of modules and a result concerning Artinianess of local cohomology modules.