Version of A Generalization of Injective and Projective Complexes

TL;DR

This paper introduces a broad generalization of injective and projective complexes using a class of modules, extending known results to these new classes and their DG-variants.

Contribution

It defines and investigates $\\mathcal{X}$-(f.g.)injective and DG-$\mathcal{X}$-injective complexes, extending classical results to these generalized complexes.

Findings

Extended known results to generalized complexes.

Established properties of DG-$\mathcal{X}$-injective complexes.

Provided a unified framework for injective and projective complexes.

Abstract

Let $\mathcal{X}$ be a class of $R$-modules. In this paper, we investigate \;$\mathcal{X}$-(f.g.)injective ((f.g.)projective) and DG-$\mathcal{X}$-injective (projective) complexes which are generalizations of injective (projective) and DG-injective (projective) complexes. We prove that some known results can be extended to the class of \;$\mathcal{X}$-(f.g.)injective ((f.g.)projective) and DG-$\mathcal{X}$-injective (projective) complexes for this general settings.