Essential cohomology modules

TL;DR

This paper introduces $e$-injective modules as a generalization of injective modules, explores their properties, and applies them to define and analyze a new type of cohomology called essential cohomology modules, highlighting differences from classical cohomology.

Contribution

It proposes $e$-injective modules using $e$-exact sequences, reestablishes key homological theorems, and develops the concept of $e$-cohomology, expanding the framework of local cohomology.

Findings

$e$-injective modules generalize classical injective modules.

$e$-cohomology modules differ from classical cohomology in vanishing properties.

The torsion functor $ ext{Gamma}_a$ is $e$-exact on torsion-free modules.

Abstract

In this article, we give a generalization to injective modules by using $e$-exact sequences introduced by Akray in [1] and name it $e$-injective modules and investigate their properties. We reprove both Baer criterion and comparison theorem of homology using $e$-injective modules and $e$-injective resolutions. Furthermore, we apply the notion $e$-injective modules into local cohomology to construct a new form of the cohomology modules call it essential cohomology modules (briefly $e$-cohomology modules). We show that the torsion functor $\Gamma_a ( - )$ is an $e$-exact functor on torsion-free modules. We seek about the relationship of $e$-cohomology within the classical cohomology. Finally, we conclude that they are different on the vanishing of their $i_{th}$ cohomology modules.