Skip to main content
SpringerLink
Log in

Tilting and cotilting modules over concealed canonical algebras

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We study infinite dimensional tilting modules over a concealed canonical algebra of domestic or tubular type. In the domestic case, such tilting modules are constructed by using the technique of universal localization, and they can be interpreted in terms of Gabriel localizations of the corresponding category of quasi-coherent sheaves over a noncommutative curve of genus zero. In the tubular case, we have to distinguish between tilting modules of rational and irrational slope. For rational slope the situation is analogous to the domestic case. In contrast, for any irrational slope, there is just one tilting module of that slope up to equivalence. We also provide a dual result for infinite dimensional cotilting modules and a description of the indecomposable pure-injective modules.

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. Angeleri Hügel, L., Archetti, M.: Tilting modules and universal localization. Forum Math. 24, 709–731 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Angeleri Hügel, L., Kussin, D.: Large tilting sheaves over weighted noncommutative regular projective curves (2016). Preprint arXiv:1508.03833

  3. Angeleri Hügel, L., Sánchez, J.: Tilting modules arising from ring epimorphisms. Algebra Represent. Theory 14, 217–246 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Angeleri Hügel, L., Sánchez, J.: Tilting modules over tame hereditary algebras. J. Reine Angew. Math. 682, 1–48 (2013)

    MathSciNet  MATH  Google Scholar 

  5. Angeleri Hügel, L., Tonolo, A., Trlifaj, J.: Tilting preenvelopes and cotilting precovers. Algebras Represent. Theory 4, 155–170 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Angeleri Hügel, L., Herbera, D., Trlifaj, J.: Baer and Mittag–Leffler modules over tame hereditary artin algebras. Math. Z. 265, 1–19 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Angeleri Hügel, L., Pospíšil, D., Šťovíček, J., Trlifaj, J.: Tilting, cotilting, and spectra of commutative noetherian rings. Trans. Am. Math. Soc. 366, 3487–3517 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bazzoni, S.: Cotilting modules are pure-injective. Proc. Am. Math. Soc. 131, 3665–3672 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bazzoni, S.: When are definable classes tilting and cotilting classes? J. Algebra 320, 4281–4299 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bazzoni, S., Herbera, D.: One dimensional tilting modules are of finite type. Algebras Represent. Theory 11, 43–61 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bazzoni, S., Herbera, D.: Cotorsion pairs generated by modules of bounded projective dimension. Isr. J. Math. 174, 119–160 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Beilinson, A.A., Bernstein, J., Deligne, P.: Faisceaux pervers. Astérisque 100, 5–171 (1982)

    Google Scholar 

  13. Braga, C., Coelho, F.U.: Limits of tilting modules. Coll. Math. 115, 207–217 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Buan, A.B., Krause, H.: Cotilting modules over tame hereditary algebras. Pac. J. Math. 211, 41–59 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cohn, P.M.: Free Rings and Their Relations, 2nd ed., London Math. Soc. Monographs 19. Academic Press Inc., London (1985)

  16. Colpi, R., Fuller, K.R.: Tilting objects in abelian categories and quasitilted rings. Trans. Am. Math. Soc. 359(2), 741–765 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Colpi, R., Gregorio, E., Mantese, F.: On the heart of a faithful torsion theory. J. Algebra 307, 841–863 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Colpi, R., Tonolo, A., Trlifaj, J.: Perpendicular categories of infinite dimensional partial tilting modules and transfers of tilting torsion classes. J. Pure Appl. Algebra 211, 223–234 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Colpi, R., Mantese, F., Tonolo, A.: Cotorsion pairs, torsion pairs, and \(\Sigma \)-pure-injective cotilting modules. J. Pure Appl. Algebra 214(5), 519–525 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Crawley-Boevey, W.W.: Locally finitely presented additive categories. Comm. Algebra 22, 1644–1674 (1994)

    MathSciNet  MATH  Google Scholar 

  21. Crawley-Boevey, W.W.: Infinite-dimensional modules in the representation theory of finite-dimensional algebras. In: Algebras and Modules, I (Trondheim, 1996), vol. 23 of CMS Conference Proceedings, pp. 29–54. Amer. Math. Soc., Providence (1998)

  22. Gabriel, P.: Des catégories abéliennes. Bull. Soc. Math. Fr. 90, 323–448 (1962)

    Article  MATH  Google Scholar 

  23. Geigle, W., Lenzing, H.: Perpendicular categories with applications to representations and sheaves. J. Algebra 144, 273–343 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gelfand, S.I., Manin, YuI: Methods of Homological Algebra. Springer, Berlin (1996)

    Book  MATH  Google Scholar 

  25. Goebel, R., Trlifaj, J.: Approximations and Endomorphism Algebras of Modules. W. de Gruyter, Berlin (2006)

    Book  MATH  Google Scholar 

  26. Happel, D., Reiten, I., Smalø, S.O.: Tilting in abelian categories and quasitilted algebras. Mem. Am. Math. Soc. 575, viii\(+\)88 pp (1996)

  27. Harland, R.: Pure-injective modules over tubular algebras and string algebras. Ph.D. Thesis, University of Manchester (2011)

  28. Harland, R., Prest, M.: Modules with irrational slope over tubular algebras. Proc. Lond. Math. Soc. 110, 695–720 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. Herzog, I.: The Ziegler spectrum of a locally coherent Grothendieck category. Proc. Lond. Math. Soc. 74, 503–558 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kerner, O., Trlifaj, J.: Tilting classes over wild hereditary algebras. J. Algebra 290, 538–556 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Krause, H.: The spectrum of a locally coherent category. J. Pure Appl. Algebra 114, 259–271 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kussin, D.: Graduierte Faktorialität und die Parameterkurven tubularer Familien. Dissertation Doctoral Thesis, Paderborn (1997)

  33. Kussin, D.: Noncommutative curves of genus zero: related to finite dimensional algebras. Mem. Am. Math. Soc. 201 (2009), 942, x+128 pp

  34. Lenzing, H.: Representations of finite-dimensional algebras and singularity theory, Trends in ring theory (Miskolc, 1996), pp. 71–97, CMS Conference Proceedings, vol. 22, Amer. Math. Soc., Providence (1998)

  35. Lenzing, H.: Hereditary Categories, Handbook of Tilting Theory. Cambridge University Press, Cambridge (2007)

    Google Scholar 

  36. Lenzing, H., de la Peña, J.A.: Concealed-canonical algebras and separating tubular families. Proc. Lond. Math. Soc. 78, 513–540 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  37. Lukas, F.: Infinite-dimensional modules over wild hereditary algebras. J. Lond. Math. Soc. 44, 401–419 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  38. Prest, M.: Purity, Spectra and Localisation. Cambridge University Press, Cambridge (2009)

    Book  MATH  Google Scholar 

  39. Prest, M.: Definable additive categories: purity and model theory. Mem. Am. Math. Soc. 210(987), vi+109 (2011)

    MathSciNet  MATH  Google Scholar 

  40. Reiten, I., Ringel, C.M.: Infinite dimensional representations of canonical algebras. Can. J. Math. 58, 180–224 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  41. Ringel, C.M.: Infinite-dimensional representations of finite-dimensional hereditary algebras. In: Symposia Mathematica, Vol. XXIII, Conference Abelian Groups and their Relationship to the Theory of Modules, INDAM, Rome, 1977, pp. 321–412. Academic Press, London (1979)

  42. Ringel, C.M.: Algebraically compact modules arising from tubular families. A survey. Algebra Colloq. 11, 155–172 (2004)

    MathSciNet  MATH  Google Scholar 

  43. Schofield, A.H.: Representations of Rings over Skew Fields, London Math. Soc. Lecture Note Series 2. Cambridge University Press (1985)

  44. Schofield, A.H.: Severe right Ore sets and universal localization (2007). Preprint arXiv:0708.0246

  45. Schofield, A.H.: Universal localization of hereditary rings (2007). Preprint arXiv:0708.0257

  46. S̆tóvíc̆ek, J.: Diploma Thesis, Charles University Prague (2004)

  47. Šťovíček, J., Kerner, O., Trlifaj, J.: Tilting via torsion pairs and almost hereditary noetherian rings. J. Pure Appl. Algebra 215, 2072–2085 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  48. Trlifaj, J.: Infinite Dimensional Tilting Modules and Cotorsion Pairs, Handbook of Tilting Theory. Cambridge University Press, Cambridge (2007)

    Google Scholar 

Download references

Acknowledgments

The authors would like to thank Helmut Lenzing and Mike Prest for valuable discussions. This research started while the second named author was visiting the University of Verona with a research grant of the Department of Computer Science. The first named author is partially supported by Fondazione Cariparo, Progetto di Eccellenza ASATA.

Author information

Authors and Affiliations

  1. Università degli Studi di Verona, Strada Le Grazie 15 - Ca’ Vignal 2, 37134, Verona, Italy

    Lidia Angeleri Hügel

  2. Institut für Mathematik, Universität Paderborn, 33095, Paderborn, Germany

    Dirk Kussin

Authors
  1. Lidia Angeleri Hügel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dirk Kussin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lidia Angeleri Hügel.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Angeleri Hügel, L., Kussin, D. Tilting and cotilting modules over concealed canonical algebras. Math. Z. 285, 821–850 (2017). https://doi.org/10.1007/s00209-016-1729-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-016-1729-3

Keywords

Mathematics Subject Classification

Search

Navigation