SOME COVERS AND ENVELOPES IN THE CHAIN COMPLEX CATEGORY OF R-MODULES

Abstract

AbstractWe study the existence of some covers and envelopes in the chain complex category of R-modules. Let (𝒜,ℬ) be a cotorsion pair in R-Mod and let ℰ𝒜 stand for the class of all exact complexes with each term in 𝒜. We prove that (ℰ𝒜,ℰ𝒜⊥) is a perfect cotorsion pair whenever 𝒜 is closed under pure submodules, cokernels of pure monomorphisms and direct limits and so every complex has an ℰ𝒜-cover. As an application we show that every complex of R-modules over a right coherent ring R has an exact Gorenstein flat cover. In addition, the existence of $\overline {\mathcal {A}}$-covers and $\overline {\mathcal {B}}$-envelopes of special complexes is considered where $\overline {\mathcal {A}}$ and $\overline {\mathcal {B}}$ denote the classes of all complexes with each term in 𝒜 and ℬ, respectively.