Skip to main content
SpringerLink
Log in

Triangulated categories of periodic complexes and orbit categories

  • Published:
Czechoslovak Mathematical Journal Aims and scope Submit manuscript

Abstract

We investigate the triangulated hull of orbit categories of the perfect derived category and the bounded derived category of a ring concerning the power of the suspension functor. It turns out that the triangulated hull corresponds to the full subcategory of compact objects of certain triangulated categories of periodic complexes. This specializes to Stai and Zhao’s result on the finite dimensional algebra of finite global dimension. As the first application, if A, B are flat algebras over a commutative ring and they are derived equivalent, then the corresponding derived categories of n-periodic complexes are triangle equivalent. As the second application, we get the periodic version of the Koszul duality.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. L. L. Avramov, R.-O. Buchweitz, S. B. Iyengar: Class and rank of differential modules. Invent. Math. 169 (2007), 1–35.

    Article  MathSciNet  MATH  Google Scholar 

  2. L. L. Avramov, R.-O. Buchweitz, S. B. Iyengar, C. Miller: Homology of perfect complexes. Adv. Math. 223 (2010), 1731–1781.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. A. Beilinson: Coherent sheaves on Pn and problems of linear algebra. Funkts. Anal. Prilozh. 12 (1978), 68–69. (In Russian.)

    Google Scholar 

  4. A. A. Beilinson, J. Bernstein, P. Deligne: Faisceaux pervers. Analysis and Topology on Singular Spaces. I. Astérisque 100. Société mathématique de France, Paris, 1982, pp. 5–171. (In French.)

    Google Scholar 

  5. A. A. Beilinson, V. Ginzburg, W. Soergel: Koszul duality patterns in representation theory J. Am. Math. Soc. 9 (1996), 473–527.

    Article  MathSciNet  MATH  Google Scholar 

  6. D. J. Benson, S. B. Iyengar, H. Krause: Stratifying modular representations of finite groups. Ann. Math. (2) 174 (2011), 1643–1684.

    Article  MathSciNet  MATH  Google Scholar 

  7. I. N. Bernstein, I. M. Gel’fand, S. I. Gel’fand: Algebraic vector bundles on Pn and problems of linear algebra. Funkts. Anal. Prilozh. 12 (1978), 66–67. (In Russian.)

    Google Scholar 

  8. M. Bökstedt, A. Neeman: Homotopy limits in triangulated categories. Compos. Math. 86 (1993), 209–234.

    MathSciNet  MATH  Google Scholar 

  9. W. Bruns, J. Herzog: Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics 39. Cambridge University Press, Cambridge, 1998.

    Book  MATH  Google Scholar 

  10. R.-O. Buchweitz: Maximal Cohen-Macaulay Modules and Tate Cohomology. Mathematical Surveys and Monographs 262. AMS, Providence, 2021.

    Book  MATH  Google Scholar 

  11. H. Cartan, S. Eilenberg: Homological Algebra. Princeton Mathematical Series 19. Princeton University Press, Princeton, 1956.

    MATH  Google Scholar 

  12. X.-W. Chen, J. Liu, R. Wang: Singular equivalences induced by bimodules and quadratic monomial algebras. To appear in Algebr. Represent. Theory.

  13. V. Drinfeld: DG quotients of DG categories. J. Algebra 272 (2004), 643–691.

    Article  MathSciNet  MATH  Google Scholar 

  14. D. Eisenbud, G. Fløystad, F.-O. Schreyer: Sheaf cohomology and free resolutions over exterior algebras. Trans. Am. Math. Soc. 355 (2003), 4397–4426.

    Article  MathSciNet  MATH  Google Scholar 

  15. E. E. Enochs, O. M. G. Jenda: Relative Homological Algebra. De Gruyter Expositions in Mathematics 30. Walter De Gruyter, Berlin, 2000.

    Book  MATH  Google Scholar 

  16. D. Happel: On the derived category of a finite-dimensional algebra. Comment. Math. Helv. 62 (1987), 339–389.

    Article  MathSciNet  MATH  Google Scholar 

  17. S. B. Iyengar, H. Krause: Acyclicity versus total acyclicity for complexes over Noetherian rings. Doc. Math. 11 (2006), 207–240.

    Article  MathSciNet  MATH  Google Scholar 

  18. S. B. Iyengar, J. C. Letz, J. Liu, J. Pollitz: Exceptional complete intersection maps of local rings. Pac. J. Math. 318 (2022), 275–293.

    Article  MathSciNet  MATH  Google Scholar 

  19. M. Kalck, D. Yang: Derived categories of graded gentle one-cycle algebras. J. Pure Appl. Algebra 222 (2018), 3005–3035.

    Article  MathSciNet  MATH  Google Scholar 

  20. B. Keller: Deriving DG categories. Ann. Sci. Éc. Norm. Supér. (4) 27 (1994), 63–102.

    Article  MathSciNet  MATH  Google Scholar 

  21. B. Keller: On the cyclic homology of exact categories. J. Pure Appl. Algebra 136 (1999), 1–56.

    Article  MathSciNet  MATH  Google Scholar 

  22. B. Keller: On triangulated orbit categories. Doc. Math. 10 (2005), 551–581.

    Article  MathSciNet  MATH  Google Scholar 

  23. B. Keller: Corrections to ‘On triangulated orbit categories’. Available at https://webusers.imj-prg.fr/∼bernhard.keller/publ/corrTriaOrbit.pdf (2009), 5 pages.

  24. H. Krause: The stable derived category of a Noetherian scheme. Compos. Math. 141 (2005), 1128–1162.

    Article  MathSciNet  MATH  Google Scholar 

  25. H. Krause: Localization theory for triangulated categories. Triangulated Categories. London Mathematical Society Lecture Note Series 375. Cambridge University Press, Cambridge, 2010, pp. 161–235.

    Book  MATH  Google Scholar 

  26. A. Neeman: The connection between the K-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel. Ann. Sci. Éc. Norm. Supér. (4) 25 (1992), 547–566.

    Article  MathSciNet  MATH  Google Scholar 

  27. A. Neeman: The Grothendieck duality theorem via Bousfield’s techniques and Brown representability. J. Am. Math. Soc. 9 (1996), 205–236.

    Article  MathSciNet  MATH  Google Scholar 

  28. A. Neeman: Triangulated Categories. Annals of Mathematics Studies 148. Princeton University Press, Princeton, 2001.

    Book  MATH  Google Scholar 

  29. D. Orlov: Derived categories of coherent sheaves and triangulated categories of singularities. Algebra, Arithmetic, and Geometry. Volume II. Progress in Mathematics 270. Birkhäuser, Boston, 2009, pp. 503–531.

    MATH  Google Scholar 

  30. L. Peng, J. Xiao: Root categories and simple Lie algebras. J. Algebra 198 (1997), 19–56.

    Article  MathSciNet  MATH  Google Scholar 

  31. L. Peng, J. Xiao: Triangulated categories and Kac-Moody algebras. Invent. Math. 140 (2000), 563–603.

    Article  MathSciNet  MATH  Google Scholar 

  32. J. Rickard: Morita theory for derived categories. J. Lond. Math. Soc., II. Ser. 39 (1989), 436–456.

    Article  MathSciNet  MATH  Google Scholar 

  33. J. Rickard: Derived equivalences as derived functors. J. Lond. Math. Soc., II. Ser. 43 (1991), 37–48.

    Article  MathSciNet  MATH  Google Scholar 

  34. C. M. Ringel, P. Zhang: Representations of quivers over the algebra of dual numbers. J. Algebra 475 (2017), 327–360.

    Article  MathSciNet  MATH  Google Scholar 

  35. T. Stai: The triangulated hull of periodic complexes. Math. Res. Lett. 25 (2018), 199–236.

    Article  MathSciNet  MATH  Google Scholar 

  36. X. Tang, Z. Huang: Higher differential objects in additive categories. J. Algebra 549 (2020), 128–164.

    Article  MathSciNet  MATH  Google Scholar 

  37. X. Zhao: A note on the equivalence of m-periodic derived categories. Sci. China, Math. 57 (2014), 2329–2334.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Part of the work was done during the author’s visit to the University of Utah. The author would like to thank his advisors Xiao-Wu Chen and Srikanth Iyengar for their encouragement and discussions, and the China Scholarship Council for their financial support. Special thanks to Benjamin Briggs for providing a similar project related to Section 6 which makes the article possible. The author thanks Janina Letz and Josh Pollitz for their discussions on this work. The author also thanks anonymous referees for their helpful comments and suggestions.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, China

    Jian Liu

Authors
  1. Jian Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jian Liu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, J. Triangulated categories of periodic complexes and orbit categories. Czech Math J 73, 765–792 (2023). https://doi.org/10.21136/CMJ.2023.0234-22

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.21136/CMJ.2023.0234-22

Keywords

MSC 2020

Search

Navigation