Essential cohomology modules

In this article, we present a generalization of injective modules using e-exact sequences introduced by Akray (Commun Korean Math Soc 35(2), 469–480, 2020), naming it e-injective modules. We then explore their properties, reproving both the Baer criterion and the comparison theorem of homology using e-injective modules and e-injective resolutions. Additionally, we incorporate the concept of e-injective modules into local cohomology, resulting in the construction of a new form of cohomology modules referred to as essential cohomology modules (briefly, e-cohomology modules). We show that the torsion functor \(\Gamma _a ( - )\) ${\mathrm{\Gamma }}_a(-)$ is an e-exact functor on torsion-free modules. We seek about the relationship of e-cohomology with in the classical cohomology. Finally, we conclude that they are different on the vanishing of their \(i_{th}\) $i_{\mathit{th}}$ cohomology modules.