Advertisement

Bulletin of the Iranian Mathematical Society

, Volume 45, Issue 5, pp 1353–1366 | Cite as

Applications of Cotorsion Pairs on Triangulated Categories

  • Haixia ChengEmail author
  • Xiaosheng Zhu
Original Paper
  • 81 Downloads

Abstract

Giving a cotorsion pair in an abelian category \({\mathscr {C}}\), we have a sequence of exact functors between triangulated categories with respect to the pair, and construct right (left) adjoints of the exact functors such that the sequence is a (co)localization sequence. Further, for some especial cotorsion pairs, we gain a recollement and triangle-equivalences between corresponding triangulated categories. In particular, let (\({\mathcal {A}}, {\mathcal {Z}}, {\mathcal {B}}\)) and (\({\mathcal {A}}_{1}, {\mathcal {Z}}_{1}, {\mathcal {B}}_{1}\)) be two complete and hereditary cotorsion triples in \({\mathscr {C}}\) with \({\mathcal {A}}_{1}\subseteq {\mathcal {A}}\). We obtain a triangle-equivalence \(K({\mathcal {A}})\simeq K({\mathcal {B}})\), which restricts to an equivalence \(K({\mathcal {A}}_{1})\simeq K({\mathcal {B}}_{1})\).

Keywords

Cotorsion pair Localization sequence Relative derived category Gorenstein derive category 

Mathematics Subject Classification

18E30 18E35 55U35 

Notes

Acknowledgements

The authors would like to thank the referee for his/her helpful comments and suggestions. This work was partially supported by the National Natural Science Foundation of China (No. 11301007) and the Key University Science Research Project of Anhui Province (No. KJ2018A0302).

References

  1. 1.
    Asadollahi, J., Hafezi, R., Salarian, S.: Homotopy category of projective complexes and complexes of Gorenstein projective modules. J. Algebra 399, 423–444 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Avramov, L.L., Foxby, H.-B.: Homological dimensions of unbounded complexes. J. Pure Appl. Algebra 71, 129–155 (1991)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Beligiannis, A.: The homological theory of contravariantly finite subcategories: Auslander–Buchweitz contexts, Gorenstein categories and (co)stabilization. Commun. Algebra 28, 4547–4596 (2000)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bican, L., Bashir, R.E., Enochs, E.E.: All modules have flat covers. Bull. Lond. Math. Soc. 33(4), 385–390 (2001)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chen, X.W.: Homotopy equivalences induced by balanced pairs. J. Algebra 324, 2718–2731 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ding, N.Q., Chen, J.L.: The flat dimensions of injective modules. Manuscr. Math. 78(2), 165–177 (1993)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ding, N.Q., Chen, J.L.: Coherent rings with finite self-FP-injective dimension. Commun. Algebra 24, 2963–2980 (1996)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ding, N.Q., Li, Y.L., Mao, L.X.: Strongly Gorenstein flat modules. J. Aust. Math. Soc. 86(3), 323–338 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ding, N.Q., Mao, L.X.: Gorenstein FP-injective and Gorenstein flat modules. J. Algebra Appl. 7(4), 491–506 (2008)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Di, Z.X., Liu, Z.K., Yang, X.Y., Zhang, X.X.: Triangulated equivalence between a homotopy category and a triangulated quotient category. J. Algebra 506, 297–321 (2018)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Enochs, E.E., Jenda, O.M.G.: Relative Homological Algebra II. Walter de Gruyter, Berlin (2011)CrossRefGoogle Scholar
  12. 12.
    Gao, N., Zhang, P.: Gorenstein derived categories. J. Algebra 323, 2041–2057 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Rozas, J.R.G.: Covers and Envelopes in the Category of Complexes of Modules. CRC Press, Boca Raton (1999)zbMATHGoogle Scholar
  14. 14.
    Gillespie, J.: The flat model structure on \(Ch(R)\). Trans. Am. Math. Soc. 356(8), 3369–3390 (2004)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Gillespie, J.: Model structures on modules over Ding–Chen rings. Homol. Homotop. Appl. 12(1), 61–73 (2010)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Gillespie, J.: Gorenstein complexes and recollements from cotorsion pairs. Adv. Math. 291, 859–911 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Iyengar, S., Krause, H.: Acyclicity versus total acyclicity for complexes over notherian rings. Doc. Math. 11, 207–240 (2006)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Jørgensen, P.: The homotopy category of complexes of projective modules. Adv. Math. 193, 223–232 (2005)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Krause, H.: The stable derived category of a Noetherian scheme. Compos. Math. 141(5), 1128–1162 (2005)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Krause, H.: Localization theory for triangulated categories. LMS. Lect. Notes Ser. 275, 161–235 (2010)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Neeman, A.: Triangulated Categories. Annals of Mathematics Studies, vol. 148. Princeton University Press, Princeton (2001)CrossRefGoogle Scholar
  22. 22.
    Spaltenstein, N.: Resolutions of unbounded complexes. Compos. Math. 65, 121–154 (1988)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Verdier, J.L.: Catégories dérivées, etat 0. Spring. Lect. Note. Math. 569, 262–311 (1977)CrossRefGoogle Scholar
  24. 24.
    Verdier, JL.: Des cat\(\acute{e}\)gories d\(\acute{e}\)riv\(\acute{e}\)es des cat\(\acute{e}\)gories ab\(\acute{e}\)liennes. Aust\(\acute{e}\)risque 239 (1996)Google Scholar
  25. 25.
    Yang, G., Liu, Z.K.: Cotorsion pairs and model structures on Ch(R). Proc. Edinb. Math. Soc. 54(3), 783–797 (2011)MathSciNetCrossRefGoogle Scholar

Copyright information

© Iranian Mathematical Society 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsAnhui Normal UniversityWuhuChina
  2. 2.Department of MathematicsNanjing UniversityNanjingChina

Personalised recommendations