Some properties of generalized local cohomology modules with respect to a pair of ideals

TL;DR

This paper introduces a generalized notion of local cohomology modules with respect to a pair of ideals, extending existing concepts, and explores their computation via Čech cohomology and their artinianness properties.

Contribution

It generalizes local cohomology modules to pairs of ideals and establishes their computation method and properties, advancing the theoretical framework.

Findings

Generalized local cohomology modules can be computed using Čech cohomology.

The paper studies conditions under which these modules are artinian.

Provides new insights into the structure of cohomology modules with respect to ideal pairs.

Abstract

We introduce a notion of generalized local cohomology modules with respect to a pair of ideals $(I,J)$ which is a generalization of the concept of local cohomology modules with respect to $(I,J).$ We show that generalized local cohomology modules $H^i_{I,J}(M,N)$ can be computed by the \v{C}ech cohomology modules. We also study the artinianness of generalized local cohomology modules $H^i_{I,J}(M,N).$