Algebras and Representation Theory

, Volume 11, Issue 1, pp 43–61 | Cite as

One Dimensional Tilting Modules are of Finite Type

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Abstract

We prove that every tilting module of projective dimension at most one is of finite type, namely that its associated tilting class is the Ext-orthogonal of a family of finitely presented modules of projective dimension at most one.

Keywords

Tilting modules Finite type Ext-orthogonal 

Mathematics Subject Classifications (2000)

Primary 16D90 16E30 Secondary 16G99 
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References

  1. 1.
    Angeleri Hügel, L., Coelho, F.U.: Infinitely generated tilting modules of finite projective dimension. Forum Math. 13, 239–250 (2001)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Angeleri Hügel, L., Saorin, M.: Modules with perfect decompositions. Math. Scand. 98(1), 19–43 (2006)MathSciNetGoogle Scholar
  3. 3.
    Angeleri Hügel, L., Herbera, D., Trlifaj, J.: Tilting modules and Gorenstein rings. Forum Math. 18, 217–235 (2006)CrossRefGoogle Scholar
  4. 4.
    Angeleri Hügel, L., Tonolo, A., Trlifaj, J.: Tilting preenvelopes and cotilting precovers. Algebr. Represent. Theory 4, 155–170 (2001)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Azumaya, G.: Finite splitness and projectivity. J. Algebra 106, 114–134 (1987)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bass, H.: Finitistic dimension and a homological generalization of semi-primary rings. Trans. Amer. Math. Soc. 95, 466–488 (1960)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bazzoni, S.: Cotilting modules are pure-injective. Proc. Amer. Math. Soc. 131, 3665–3672 (2003)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Bazzoni, S.: A characterization of n-cotilting and n-tilting modules. J. Algebra 273, 359–372 (2004)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Bazzoni, S., Eklof, P., Trlifaj, J.: Tilting cotorsion pairs. Bull. London Math. Soc. 37, 683–696 (2005)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Brenner, S., Butler, M.: Generalizations of the Bernstein-Gelfand-Ponomarev reflection functors. In: Proc. ICRA III. LNM, vol. 832, pp. 103–169. Springer, Berlin-Heidelberg-New York (1980)Google Scholar
  11. 11.
    Colby, R.R., Fuller, K.R.: Tilting, cotilting and serially tilted rings. Comm. Algebra 25(10), 3225–3237 (1997)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Colpi, R., Trlifaj, J.: Tilting modules and tilting torsion theories. J. Algebra 178, 614–634 (1995)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Crawley-Boevey, W.W.: Infinite-dimensional modules in the representation theory of finite-dimensional algebras. In: Algebras and modules, I (Trondheim, 1996), pp. 29–54. CMS Conf. Proc., vol. 23. American Mathematical Society Providence, RI (1998)Google Scholar
  14. 14.
    Eklof, P.C., Trlifaj, J.: How to make Ext vanish. Bull. London Math. Soc. 33, 41–51 (2001)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Grothendieck, A.: Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I, Publ. Math. Inst. Hautes Études Sci. 11, 5–167 (1961)CrossRefGoogle Scholar
  16. 16.
    Happel, D., Ringel, C.: Tilted algebras. Trans. Amer. Math. Soc. 215, 81–98 (1976)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Keisler, H.J.: Ultraproducts and elementary classes. Indag. Math. 23, 477–495 (1961)MathSciNetGoogle Scholar
  18. 18.
    Puninski, G.: When every projective module is a direct sum of finitely generated modules. Preprint available at www.maths.man.ac.uk/~gpuninski/prog-fg.pdf (2004)
  19. 19.
    Raynaud, M., Gruson, L.: Critères de platitude et de projectivité. Invent. Math 13, 1–89 (1971)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Shelah, S.: Every two elementarily equivalent models have isomorphic ultrapowers. Israel J. Math. 10, 224–233 (1971)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Weibel, C.A.: An introduction to homological algebra. In: Cambridge Studies in Advanced Mathematics, vol. 38. Cambridge University Press, Cambridge (1994)Google Scholar
  22. 22.
    Whitehead, J.M.: Projective modules and their trace ideals. Comm. Algebra 8(19), 1873–1901 (1980)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Ziegler, M.: Model theory of modules. Ann. Pure Appl. Logic 26, 149–213 (1984)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Zimmermann, W.: Π-projektive Moduln. J. Reine Angew. Math. 292, 117–124 (1977)MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.Dipartimento di Matematica Pura e ApplicataUniversità di PadovaPadovaItaly
  2. 2.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterra (Barcelona)Spain

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