manuscripta mathematica

, Volume 132, Issue 3–4, pp 483–499 | Cite as

Large tilting modules and representation type

Article

Abstract

We study finiteness conditions on large tilting modules over arbitrary rings. We then turn to a hereditary artin algebra R and apply our results to the (infinite dimensional) tilting module L that generates all modules without preprojective direct summands. We show that the behaviour of L over its endomorphism ring determines the representation type of R. A similar result holds true for the (infinite dimensional) tilting module W that generates the divisible modules. Finally, we extend to the wild case some results on Baer modules and torsion-free modules proven in Angeleri Hügel, L., Herbera, D., Trlifaj, J.: Baer and Mittag-Leffler modules over tame hereditary algebras. Math. Z. 265, 1–19 (2010) for tame hereditary algebras.

Mathematics Subject Classification (2000)

16E30 16E40 16G10 16G60 16G70 18E40 

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References

  1. 1.
    Angeleri Hügel L.: A key module over pure-semisimple hereditary rings. J. Algebra 307, 361–376 (2007)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Angeleri Hügel L., Coelho F.: Infinitely generated tilting modules of finite projective dimension. Forum Math. 13, 239–250 (2001)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Angeleri Hügel L., Herbera D.: Mittag-Leffler conditions on modules. Indiana Univ. Math. J. 57, 2459–2517 (2008)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Angeleri Hügel L., Herbera D., Trlifaj J.: Tilting modules and Gorenstein rings. Forum Math. 18, 217–235 (2006)Google Scholar
  5. 5.
    AngeleriHügel L., Herbera D., Trlifaj J.: Baer and Mittag-Leffler modules over tame hereditary algebras. Math. Z. 265, 1–19 (2010)CrossRefGoogle Scholar
  6. 6.
    Angeleri Hügel L., Šaroch J., Trlifaj J.: On the telescope conjecture for module categories. J. Pure Appl. Algebra 212, 297–310 (2008)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    AngeleriHügel L., Trlifaj J.: Tilting theory and the finitistic dimension conjectures. Trans. Am. Math. Soc. 354, 4345–4358 (2002)CrossRefGoogle Scholar
  8. 8.
    Angeleri Hügel, L., Trlifaj, J.: Direct limits of modules of finite projective dimension. In: Facchini, A., Houston, E., Salce, L. (eds.) Rings, Modules, Algebras, and Abelian Groups, LNPAM 236, 27–44 (2004)Google Scholar
  9. 9.
    Angeleri Hügel L., Valenta H.: A duality result for almost split sequences. Colloq. Math. 80, 267–292 (1999)MATHMathSciNetGoogle Scholar
  10. 10.
    Auslander M., Reiten I.: Applications of contravariantly finite subcategories. Adv. Math. 86, 111–152 (1991)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Bazzoni S.: Cotilting modules are pure-injective. Proc. Am. Math. Soc. 131, 3665–3672 (2003)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Bazzoni S., Herbera D.: One dimensional tilting modules are of finite type. Algebras Represent. Theory 11, 43–61 (2008)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Bazzoni S., Šťovíček J.: All tilting modules are of finite type. Proc. Am. Math. Soc. 135, 3771–3781 (2007)MATHCrossRefGoogle Scholar
  14. 14.
    Crawley-Boevey W.: Locally finitely presented additive categories. Comm. Algebra 22, 1644–1674 (1994)Google Scholar
  15. 15.
    Crawley-Boevey, W.: Infinite dimensional modules in the representation theory of finite dimensional algebras. In: Reiten, I., Smalø, S., Solberg, O. (eds.) Algebras and Modules I, CMS Conf. Proc. 23 (1998), 29–54Google Scholar
  16. 16.
    Goebel R., Trlifaj J.: Approximations and Endomorphism Algebras of Modules. W. de Gruyter, Berlin (2006)MATHCrossRefGoogle Scholar
  17. 17.
    Kerner O., Trlifaj J.: Tilting classes over wild hereditary algebras. J. Algebra 290, 538–556 (2005)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Krause, H., Saorín, M.: On minimal approximations of modules. In: Green, E.L., Huisgen-Zimmermann, B. (eds) Trends in the representation theory of finite dimensional algebras, Contemp. Math. 229, 227–236 (1998)Google Scholar
  19. 19.
    Krause H., Solberg O.: Applications of cotorsion pairs. J. Lond. Math. Soc. 68, 631–650 (2003)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Lukas F.: Infinite-dimensional modules over wild hereditary algebras. J. Lond. Math. Soc. 44, 401–419 (1991)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Lukas F.: A class of infinite-rank modules over tame hereditary algebras. J. Algebra 158, 18–30 (1993)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Reiten I., Ringel C.M.: Infinite dimensional representations of canonical algebras. Can. J. Math. 58, 180–224 (2006)MATHMathSciNetGoogle Scholar
  23. 23.
    Ringel C.M.: Infinite dimensional representations of finite dimensional hereditary algebras. Symp. Math. 23, 321–412 (1979)MathSciNetGoogle Scholar
  24. 24.
    Šaroch J., Šťovíček J.: The countable telescope conjecture for module categories. Adv. Math. 219, 1002–1036 (2008)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Šťovíček J.: All n-cotilting modules are pure-injective. Proc. Am. Math. Soc. 134, 1891–1897 (2006)MATHCrossRefGoogle Scholar
  26. 26.
    Zimmermann-Huisgen B.: Strong preinjective partitions and representation type of artinian rings. Proc. Am. Math. Soc. 109, 309–322 (1990)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Lidia Angeleri Hügel
    • 1
  • Otto Kerner
    • 2
  • Jan Trlifaj
    • 3
  1. 1.Dipartimento di Informatica—Settore di MatematicaUniversità degli Studi di VeronaVeronaItaly
  2. 2.Mathematisches InstitutHeinrich-Heine-Universität DüsseldorfDüsseldorfGermany
  3. 3.Department of Algebra, Faculty of Mathematics and PhysicsCharles UniversityPrague 8Czech Republic

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