Induced complete hereditary cotorsion pairs in D(R) with respect to Cartan-Eilenberg exact sequences

TL;DR

This paper constructs complete hereditary cotorsion pairs in the derived category of a ring, extending classical module-theoretic concepts to complexes using Cartan-Eilenberg sequences.

Contribution

It introduces a method to build cotorsion pairs in D(R) based on cohomologically ghost triangles, generalizing existing module-theoretic cotorsion pairs to complexes.

Findings

Constructed cotorsion pairs in D(R) using Cartan-Eilenberg sequences.

Identified classes of complexes forming hereditary cotorsion pairs.

Connected homological dimensions to the existence of such cotorsion pairs.

Abstract

Given a complete hereditary cotorsion pair (A,B) in ModR, we construct a complete hereditary cotorsion pair in the derived category D(R) of unbounded complexes with respect to the proper class {\xi} of cohomologically ghost triangles induced by the Cartan-Eilenberg exact sequences. More specifically, we prove that, each of the classes of projectively coresolved {\xi}-Gflat complexes PGF({\xi}), {\xi}-Gflat complexes GF({\xi}), {\xi}-Ginjective complexes GI({\xi}), {\xi}-Gprojective complexes GP({\xi}) (the last when R is virtually Gorenstein), forms one half of a complete hereditary cotorsion pair in D(R) with respect to {\xi}. Moreover, various homological dimensions offer additional way to obtain such cotorsion pairs in D(R) with respect to {\xi}.