Abstract
In this paper, we study the category \(\mathbb {C} (\mathrm{{Rep}}(\mathcal {Q}, \mathfrak {A}))\) of complexes of representations of quiver \(\mathcal {Q}\) with values in an abelian category \(\mathfrak {A}\). We develop a method for constructing some model structures on \(\mathbb {C} (\mathrm{{Rep}}(\mathcal {Q}, \mathfrak {A}))\) based on componentwise notion. Moreover, we also show that these model structures are monoidal. As an application of these model structures, we introduce some descriptions of the derived category of complexes of representations of \(\mathcal {Q}\) in \(\mathrm{Mod\hbox {-}}R\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Asadollahi, J., Bahiraei, P., Hafezi, R., Vahed, R.: Sheaves over infinite posets. J. Algebra Appl. (2017). https://doi.org/10.1142/S0219498817500268
Beke, T.: Sheafifiable homotopy model categories. Math. Proc. Camb. Philos. Soc. 129, 447–475 (2000)
Bican, L., El Bashir, R., Enochs, E.: All modules have flat covers. Bull. Lond. Math. Soc. 33(4), 385–390 (2001)
Birkhoff, G.: Subgroups of abelian groups. Proc. Lond. Math. Soc. II Ser. 38, 385–401 (1934)
Cortes Izurdiaga, M., Estrada, S., Guil, Asensio P.: A Quillen model structure approach to the finistic dimension conjectures. Math. Nachr 285(7), 821–833 (2012)
Dalezios, G.: Abelian model structures on categories of quiver representations. J. Algebra Appl. (2019). https://doi.org/10.1142/S0219498820501959
Dwyer, W.G., Spalinski, J.: Homotopy Theories and Model Categories Handbook of Algebraic Topology, 73126. North-Holland, Amsterdam (1995)
Enochs, E., Estrada, S.: Projective representations of quivers. Commun. Algebra. 33, 3467–3478 (2005)
Enochs, E., Herzog, I.: A homotopy of quiver morphisms with applications to representations. Can. J. Math 51(2), 294–308 (1999)
Enochs, E., Ozdemir, L., Torrecillas, B.: Flat covers and flat representations of quivers. Commun. Algebra 32(4), 1319–1338 (2004)
Enochs, E., Estrada, S., Garcia-Rozas, J.R.: Locally projective monoidal model structure for complexes of quasi-coherent shaves on \(\mathbb{P}^1(k)\). J. Lond. Math. Soc 77, 253–269 (2008)
Enochs, E., Estrada, S., Garcia-Rozas, J.R.: Injective representations of infinite quivers. Appl. Can. J. Math 61(2), 284–295 (2009)
Enochs, E., Estrada, S., Iacob, I.: Cotorsion pairs, model structures and adjoints in homotopy categories. Houston J. Math. 40(1), 43–61 (2014)
Eshraghi, H., Hafezi, R., Hosseini, E., Salarian, Sh: Cotorsion theory in the category of quiver representations. J. Algebra Appl. (2013). https://doi.org/10.1142/S0219498813500059
Estrada, S., Guil Asensio, P., Prest, M., Trlifaj, J.: Model category structures arising from Drinfeld vector bundles. Adv. Math. 231(3–4), 1417–1438 (2012)
Gelfand, S.I., Manin, Y.I.: Methods of Homological Algebra. Springer, Berlin (2003)
Gillespie, J.: The flat model structure on \(Ch(R)\). Trans. Am. Math. Soc. 356(8), 3369–3390 (2004)
Gillespie, J.: The flat model structure on complexes of sheaves. Trans. Am. Math. Soc. 358(7), 2855–2874 (2006)
Gillespie, J.: Cotorsion pairs and degreewise homological model structures. Homol Homotopy Appl. 10(1), 283–304 (2008)
Gillespie, J.: Model structure on exact category. J. Pure Appl. Algebra 215, 2892–2902 (2011)
Happel, D.: Triangulated Categories in the Representation Theory of Finite-Dimensional Algebras. Cambridge University Press, Cambridge (1988)
Holm, H., Jørgensen, P.: Cotorsion pairs in categories of quiver representations. Kyoto J. Math. 59(3), 575–606 (2019)
Holm, H., Jørgensen, P.: Model categories of quiver representations. Adv. Math. 357, 46 (2019)
Hovey, M.: Model Categories. Amer. Math. Soc., Providence (1999)
Hovey, M.: Cotorsion pair, model category structures, and representation theory. Math. Zeit. 241, 553–592 (2002)
Joyal, Letter to A. Grothendieck (1984)
Krause, H.: Localization theory for triangulated categories, Triangulated categories, 161-235, London Math. Soc. Lecture Note Ser., 375, Cambridge Univ. Press, Cambridge, (2010)
Mac Lane, S.: Categories for the Working Mathematician, Graduate Texts in Mathematics, vol. 5. Springer, Berlin (1971)
Quillen, D.G.: Homotopical Algebra. Lecture Notes inMathematics, vol. 43. Springer, Berlin (1967)
Salce, L.: Cotorsion Theory for Abelian Groups. Symposia Math, vol. 23, pp. 11–32. Academic Press, New York (1979)
Acknowledgements
I would like to thank Rasool Hafezi for many useful hints and comments that improved the exposition.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Javad Asadollahi.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Bahiraei, P. Model Structures on the Category of Complexes of Quiver Representations. Bull. Iran. Math. Soc. 47 (Suppl 1), 103–117 (2021). https://doi.org/10.1007/s41980-020-00515-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s41980-020-00515-1