Local cohomology based on a nonclosed support defined by a pair of ideals

TL;DR

This paper generalizes local cohomology modules using a pair of ideals, exploring their properties, vanishing theorems, and connections to traditional local cohomology.

Contribution

It introduces a new concept of local cohomology with respect to a pair of ideals and investigates its fundamental properties and relationships.

Findings

Established vanishing and nonvanishing theorems for the generalized modules.

Analyzed the connection between the new and classical local cohomology.

Provided foundational properties and potential applications of the generalized theory.

Abstract

We introduce an idea for generalization of a local cohomology module, which we call a local cohomology module with respect to a pair of ideals (I,J), and study their various properties. Some vanishing and nonvanishing theorems are given for this generalized version of local cohomology. We also discuss its connection with the ordinary local cohomology.