Homotopy Category of Cotorsion Flat Representations of Quivers

Document Type : Original Scientific Paper

Author

Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, I. R. Iran

Abstract

Recently in [10], it was proved that over any ring R, there exists a complete cotorsion pair (Kp(Flat-R); K(dg-CotF-R)) in K(Flat-R), the homotopy category of complexes of flat R-modules, where Kp(Flat-R) and K(dg-CotF-R) are the homotopy categories raised by flat (or pure) and dgcotorsion complexes of flat R-modules, respectively. This paper aims at recognition of a parallel cotorsion pair in K(Flat-Q), the homotopy category of flat representation of certain quivers Q, where Q may also be infinite. The importance of this result lies in the fact that this homotopy categories do not necessarily raise from the category of modules over some ring. In the other part of this paper, we give a classification of compact objects in K(dg-CotF-Q), the homotopy category of dg-cotorsion complexes of flat representations of certain Q, in terms of the corresponding vertex-complexes

Keywords

References

[1] S. T. Aldrich, E. E. Enochs, J. R. Garcia Rozas and L. Oyonarte, Covers and envelopes in Grothendieck categories: flat covers of complexes with applications, J. Algebra 243 (2) (2001) 615 - 630.
[2] J. Asadollahi, H. Eshrahgi, R. Hafezi and Sh. Salarian, On the homotopy categories of projective and injective representations of quivers, J. Algebra 346 (2011) 101 - 115.
[3] I. Assem, D. Simson and A. Skowronsky, Elements of the Reprensentation Theory of Associative Algebras, in: Techniques of Representation Theory, vol.1, Cambridge University Press, Cambridge, 2006.
[4] L. Bican, R. EL Bashir and E. E. Enochs, All modules have flat covers, Bull. London Math. Soc. 33 (4) (2001) 385 - 390.
[5] D. Bravo, E. E. Enochs, A. Iacob, O. Jenda and J.Rada, Cortorsion pairs in C(R-Mod), Rocky Mountain J. Math. 42 (6) (2012) 1787 - 1802.
[6] A. Beligiannis and I. Reiten, Homological and homotopical aspects of torsion theories, Mem. Amer. Math. Soc. 188 (883) (2007) viii+207 pp.
[7] E. Enochs and S. Estrada, Projective representations of quivers, Comm. Algebra 33 (2005) 3467 - 3478.
[8] E. Enochs, S. Estrada and J. R. García Rozas, Injective representations of infinite quivers, Applications. Canadian J. Math. 61 (2) (2009) 315 - 335.
[9] E. E. Enochs and J. R. García Rozas, Flat covers of complexes, J. Algebra 210 (1) (1998) 86 - 102.
[10] H. Eshraghi and A. Hajizamani, The homotopy category of cotorsion flat modules, (2020) preprint.
[11] H. Eshraghi, R. Hafezi, E. Hosseini and Sh. Salarian, Cotorsion pairs in the category of representations of quivers, J. Algebra Appl. 12 (6) (2013) 1350005.
[12] E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, Walter de Gruynter, Berlin-NewYork, 2000.
[13] E. Enochs, L. Oyonarte and B. Torrecillas, Flat covers and flat representations of quivers, Comm. Algebra 32 (4) (2004) 1319 - 1338.
[14] J. Gillespie, The flat model structure on Ch(R), Trans. Amer. Math. Soc. 356 (8) (2004) 3369 - 3390.
[15] E. Hosseini and Sh. Salarian, A cotorsion theory in the homotopy category of flat quasi-coherent sheaves, Proc. Amer. Math. Soc. 141 (3) (2013) 753-762.
[16] P. Jørgensen, The homotopy category of complexes of projective modules, Adv. Math. 193 (1) (2005) 223 - 232.
[17] H. Krause, The stable derived category of a Noetherian scheme, Compos. Math. 141 (5) (2005) 1128 - 1162.
[18] D. Murfet, The Mock Homotopy Category of Projectives and Grothendieck Duality, Ph.D thesis, Australian National University, 2007.
[19] A. Neeman, Triangulated Categories, Ann. of Math. Stud., vol. 148, Princeton University Press, Princeton, NJ, 2001.
[20] A. Neeman, The homotopy category of flat modules, and Grothendieck duality, Invent. Math. 174 (2008) 255 - 308.
[21] L. Oyonarte, Cotorsion representations of quivers, Comm. Algebra 29 (12) (2001) 5563 - 5574.
[22] C. A. Weible, An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994.



Mathematics Interdisciplinary Research
Volume 5, Issue 4
December 2020
Pages 279-294

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