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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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References

1.Bass, H., Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488.CrossRefGoogle Scholar
2.Beligiannis, A. and Reiten, I., Homological and homotopical aspects of torsion theories, Mem. Am. Math. Soc. 883 (2007), 207.Google Scholar
3.Chen, X.-W., Homotopy equivalences induced by balanced pairs, J. Algebra. 324(10) (2010), 2718–2731.10.1016/j.jalgebra.2010.09.002CrossRefGoogle Scholar
4.Ding, N. and Chen, J., The flat dimensions of injective modules, Manuscripta Math. 78(2) (1993), 165–177.10.1007/BF02599307CrossRefGoogle Scholar
5.Enochs, E. E., Balance with flat objects, J. Pure Appl. Algebra. 219(3) (2015), 488–493.10.1016/j.jpaa.2014.05.007CrossRefGoogle Scholar
6.Enochs, E. E. and Herzog, I., A homotopy of quiver morphisms with applications to representations, Can. J. Math. 51(2) (1999), 294–308.10.4153/CJM-1999-015-0CrossRefGoogle Scholar
7.Enochs, E. E. and Jenda, O. M. G., Relative homological algebra. 2nd revised and extended edn, Volume 1 (Walter de Gruyter, Berlin, 2011).Google Scholar
8.Enochs, E. E. and Jenda, O. M. G., Relative homological algebra. 2nd revised edn, Volume 2 (Walter de Gruyter, Berlin, 2011).Google Scholar
9.Enochs, E. E., Jenda, O. M. G., Torrecillas, B. and Xu, J., Torsion theory relative to Ext. Research Report 98–11. Department of Mathematics, University of Kentucky (1998).Google Scholar
10.Enochs, E. E., Estrada, S., García Rozas, J. R. and Iacob, A., Gorenstein quivers, Arch. Math. 88(3) (2007), 199–206.10.1007/s00013-006-1921-5CrossRefGoogle Scholar
11.Eshraghi, H., Hafezi, R., Hosseini, E. and Salarian, S., Cotorsion theory in the category of quiver representations, J. Algebra Appl. 12(6) (2013), 1350005.CrossRefGoogle Scholar
12.Estrada, S., Iacob, A. and Odabaşı, S., Gorenstein flat and projective and precovers, Pub. Math. Debrecen 91(1-2(7)) (2017), 111–121.10.5486/PMD.2017.7689CrossRefGoogle Scholar
13.Gillespie, J., Kaplansky classes and derived categories, Math. Z. 257(4) (2007), 811–843.CrossRefGoogle Scholar
14.Gillespie, J., Cotorsion pairs and degreewise homological model structures, Homology Homotopy Appl. 10(1) (2008), 283–304.10.4310/HHA.2008.v10.n1.a12CrossRefGoogle Scholar
15.Gillespie, J., Model structures on modules over Ding–Chen rings, Homology Homotopy Appl. 12(1) (2010), 61–73.10.4310/HHA.2010.v12.n1.a6CrossRefGoogle Scholar
16.Gillespie, J., On Ding injective, Ding projective and Ding flat modules and complexes, Rocky Mountain J. Math. 47(8) (2017), 2641–2673.10.1216/RMJ-2017-47-8-2641CrossRefGoogle Scholar
17.Grothendieck, A. and Dieudonné, J. A., Eléments de géométrie algébrique. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Volume 166 (Springer-Verlag, Berlin, 1971).Google Scholar
18.Guil Asensio, P. A. and Herzog, I., Sigma-cotorsion rings, Adv. Math. 191(1) (2005), 11–28.10.1016/j.aim.2004.01.006CrossRefGoogle Scholar
19.Hartshorne, R., Residues and duality, Lecture Notes in Mathematics, Volume 20 (Springer-Verlag, Berlin–New York, 1966).10.1007/BFb0080482CrossRefGoogle Scholar
20.Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, Volume 52 (Springer-Verlag, New York–Heidelberg, 1977).10.1007/978-1-4757-3849-0CrossRefGoogle Scholar
21.Holm, H. and Jørgensen, P., Cotorsion pairs in categories of quiver representations, Kyoto J. Math., in press.Google Scholar
22.Hovey, M., Cotorsion pairs, model category structures, and representation theory, Math. Z. 241(3) (2002), 553–592.10.1007/s00209-002-0431-9CrossRefGoogle Scholar
23.Krause, H., The stable derived category of a Noetherian scheme, Compos. Math. 141(5) (2005), 1128–1162.10.1112/S0010437X05001375CrossRefGoogle Scholar
24.Li, Z.-W. and Zhang, P., A construction of Gorenstein-projective modules, J. Algebra 323(6) (2010), 1802–1812.10.1016/j.jalgebra.2009.12.030CrossRefGoogle Scholar
25.Luo, X.-H. and Zhang, P., Monic representations and Gorenstein-projective modules, Pac. J. Math. 264(1) (2013), 163–194.10.2140/pjm.2013.264.163CrossRefGoogle Scholar
26.Mitchell, B., Rings with several objects, Adv. Math. 8 (1972), 1–161.CrossRefGoogle Scholar
27.Odabaşı, S., Completeness of the induced cotorsion pairs in categories of quiver representations, J. Pure Appl. Algebra 233(10) (2019), 4536–4559.10.1016/j.jpaa.2019.02.003CrossRefGoogle Scholar
28.Šaroch, J. and Šťovíček, J., Singular compactness and definability for Σ-cotorsion and Gorenstein modules (arXiv:1804.09080, 2018).Google Scholar
29.Stenström, B., Rings of quotients, Die Grundlehren der Mathematischen Wissenschaften, Band 217: An Introduction to Methods of Ring Theory (Springer-Verlag, New York–Heidelberg, 1975).10.1007/978-3-642-66066-5CrossRefGoogle Scholar
30.Zareh-Khoshchehreh, F., Asgharzadeh, M. and Divaani-Aazar, K., Gorenstein homology, relative pure homology and virtually Gorenstein rings, J. Pure Appl. Algebra 218(12) (2014), 2356–2366.10.1016/j.jpaa.2014.04.005CrossRefGoogle Scholar