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Balanced pairs, cotorsion triplets and quiver representations

Part of: Homological algebra Abelian categories Representation theory of rings and algebras Categories and theories

Published online by Cambridge University Press:  13 August 2019

Sergio Estrada ,
Marco A. Pérez  and
Haiyan Zhu
Sergio Estrada
Affiliation:
Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, Espinardo Murcia 30100, Spain ([email protected])
Marco A. Pérez
Affiliation:
Instituto de Matemática y Estadística ‘Prof. Ing. Rafael Laguardia’, Universidad de la República, Montevideo11300, Uruguay ([email protected])
Haiyan Zhu*
Affiliation:
College of Science, Zhejiang University of Technology, Hangzhou310023, China ([email protected])
*
*Corresponding author:
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Abstract

Balanced pairs appear naturally in the realm of relative homological algebra associated with the balance of right-derived functors of the Hom functor. Cotorsion triplets are a natural source of such pairs. 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 that have enough projectives and injectives in terms of the existence of complete hereditary cotorsion triplets. We also provide a short proof of the lack of balance for derived functors of Hom computed using flat resolutions, which extends the one given by Enochs in the commutative case.

Keywords

balanced paircotorsion tripletquiver representationflat balance

MSC classification

Primary: 18G25: Relative homological algebra, projective classes
Secondary: 16G20: Representations of quivers and partially ordered sets 18E10: Exact categories, abelian categories 18C15: Triples (= standard construction, monad or triad), algebras for a triple, homology and derived functors for triples
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 1 , February 2020 , pp. 67 - 90
Copyright
Copyright © Edinburgh Mathematical Society 2019

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