# On some applications of balanced pairs and their relation with cotorsion triplets

## TL;DR

This paper explores the relationship between balanced pairs and cotorsion triplets in abelian categories, providing new characterizations and extending known results using quiver representation techniques.

## Contribution

It introduces a new characterization of abelian categories with enough projectives and injectives via complete hereditary cotorsion triplets and offers a short proof regarding the imbalance of derived functors.

## Key findings

- Characterization of abelian categories with enough projectives and injectives
- Extension of Enochs' result on the imbalance of derived functors
- Application of quiver representation techniques to study balanced pairs

## Abstract

Balanced pairs appear naturally in the realm of Relative Homological Algebra associated to the balance of right derived functors of the $\mathsf{Hom}$ functor. A natural source to get such pairs is by means of cotorsion triplets. In this paper we study the connection between balanced pairs and cotorsion triplets by using recent quiver representation techniques. In doing so, we find a new characterization of abelian categories having enough projectives and injectives in terms of the existence of complete hereditary cotorsion triplets. We also give a short proof of the lack of balance for derived functors of $\mathsf{Hom}$ computed by using flat resolutions which extends the one showed by Enochs in the commutative case.