Local Homology with Respect to a Pair of Ideals

TL;DR

This paper generalizes local homology modules to pairs of ideals, exploring their properties and relationships with traditional local homology, advancing the theoretical framework in algebraic topology and commutative algebra.

Contribution

It introduces the concept of local homology with respect to a pair of ideals and investigates its fundamental properties and connections to existing theories.

Findings

Characterization of vanishing conditions

Analysis of co-support and co-associated primes

Relationship with classical local homology

Abstract

We introduce a generalization of the notion of local homology module, which we call a local homology module with respect to a pair of ideals $\left(I,J\right)$, and study its various properties such as vanishing, co-support and co-associated. We also discuss its connection with ordinary local homology.