Skip to main content
SpringerLink
Log in

Differential Modules over Quadratic Monomial Algebras

  • Published:
Algebras and Representation Theory Aims and scope Submit manuscript

Abstract

We compare the so-called clock condition to the gradability of certain differential modules over quadratic monomial algebras. These considerations show that a stably hereditary or gentle one-cycle algebra is piecewise hereditary if and only if the orbit category of its bounded derived category with respect to a positive power of the shift functor is triangulated.

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

Similar content being viewed by others

References

  1. Amiot, C., Oppermann, S.: The image of the derived category in the cluster category. Int. Math. Res. Not. IMRN 4, 733–760 (2013). MR 3024264

    Article  MathSciNet  MATH  Google Scholar 

  2. Assem, I., Skowroński, A.: Iterated tilted algebras of type à n . Math. Z. 195(2), 269–290 (1987). MR 892057 (88m:16033)

    Article  MathSciNet  MATH  Google Scholar 

  3. Auslander, M., Reiten, I.: Stable equivalence of Artin algebras. In: Proceedings of the Conference on Orders, Group Rings and Related Topics (Ohio State Univ., Columbus, Ohio, 1972). Lecture Notes in Math., Vol. 353 MR 0335575, pp 8–71. Springer, Berlin (1973)

    Chapter  Google Scholar 

  4. Avramov, L.L., Buchweitz, R., Iyengar, S.: Class and rank of differential modules. Invent. Math. 169(1), 1–35 (2007). MR 2308849 (2008h:13032)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bautista, R., Liu, S.: The bounded derived category of an algebra with radical square zero, Preprint

  6. Bekkert, V., Drozd, Y.: Derived categories for algebras with radical square zero, Algebras, representations and applications. Contemp. Mat. 483, 55–62 (2009). MR 2497950 (2010b:16013)

    Article  MATH  Google Scholar 

  7. Bongartz, K., Riedtmann, C.: Algèbres stablement héréditaires. C. R. Acad. Sci. Paris A-B 288(15), 703–706 (1979). MR 532393 (81i:16055)

    MathSciNet  MATH  Google Scholar 

  8. Farnsteiner, R.: Support varieties, AR-components and good filtrations. Lecture notes

  9. Gordon, R., Green, E.L.: Graded Artin algebras. J. Algebra 76(1), 111–137 (1982). MR 659212 (83m:16028a)

    Article  MathSciNet  MATH  Google Scholar 

  10. Happel, D., Zacharia, D.: A homological characterization of piecewise hereditary algebras. Math. Z. 260(1), 177–185 (2008). MR 2413349 (2009g:16011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Keller, B.: On triangulated orbit categories. Doc. Math. 10, 551–581 (2005). MR 2184464 (2007c:18006)

    MathSciNet  MATH  Google Scholar 

  12. Kerner, O., Skowroński, A., Yamagata, K., Zacharia, D.: Finiteness of the strong global dimension of radical square zero algebras. Cent. Eur. J. Math. 2(1), 103–111 (2004). (electronic). MR 2041672 (2005g:16012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lévesque, J.: Nakayama oriented pullbacks and stably hereditary algebras. J. Pure Appl. Algebra 212(5), 1149–1161 (2008). MR 2387592 (2009b:16038)

    Article  MathSciNet  MATH  Google Scholar 

  14. Pogorzały, Z., Skowroński, A.: Self-injective biserial standard algebras. J. Algebra 138(2), 491–504 (1991). MR 1102821 (92f:16012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Stai, T.: The triangulated hull of periodic complexes, Mathematical Research Letters, to appear

  16. Xi, C.: Representation dimension and quasi-hereditary algebras. Adv. Math. 168 (2), 193–212 (2002)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institutt for Matematiske Fag, NTNU, 7491, Trondheim, Norway

    Torkil Stai

Authors
  1. Torkil Stai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Torkil Stai.

Additional information

Presented by Yuri Drozd.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Stai, T. Differential Modules over Quadratic Monomial Algebras. Algebr Represent Theor 20, 1239–1247 (2017). https://doi.org/10.1007/s10468-017-9684-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10468-017-9684-1

Keywords

Mathematics Subject Classification (2010)

Search

Navigation