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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Tilting objects in abelian categories and quasitilted rings

Authors: Riccardo Colpi and Kent R. Fuller
Journal: Trans. Amer. Math. Soc. 359 (2007), 741-765
MSC (2000): Primary 16E10, 16G99, 16S50, 18E40, 18E25, 18G20; Secondary 16B50, 16D90
Published electronically: August 24, 2006
MathSciNet review: 2255195
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Abstract: D. Happel, I. Reiten and S. Smalø initiated an investigation of quasitilted artin $ K$-algebras that are the endomorphism rings of tilting objects in hereditary abelian categories whose Hom and Ext groups are all finitely generated over a commutative artinian ring $ K$. Here, employing a notion of $ \ast $-objects, tilting objects in arbitrary abelian categories are defined and are shown to yield a version of the classical tilting theorem between the category and the category of modules over their endomorphism rings. This leads to a module theoretic notion of quasitilted rings and their characterization as endomorphism rings of tilting objects in hereditary cocomplete abelian categories.

References [Enhancements On Off] (What's this?)

  • 1. F. W. Anderson and K. R. Fuller.
    Rings and Categories of Modules.
    Springer-Verlag, Inc., New York, Heidelberg, Berlin, second edition, 1992. MR 1245487 (94i:16001)
  • 2. K. Bongartz.
    Tilted algebras.
    ``Proc. ICRA III (Puebla, 1980)'', LNM 903, Springer, 26-38, 1981. MR 0654701 (83g:16053)
  • 3. S. Brenner and M. Butler.
    Generalizations of the Bernstein-Gelfand-Ponomarev reflection functors.
    ``Proc. ICRA II (Ottawa, 1979)'', LNM 832, Springer, 103-169, 1980. MR 0607151 (83e:16031)
  • 4. R. R. Colby and K. R. Fuller.
    Tilting, cotilting and serially tilted rings.
    Comm. Algebra, 18, 1585-1615, 1990. MR 1059750 (91h:16011)
  • 5. R. R. Colby and K. R. Fuller.
    Tilting and torsion theory counter equivalences.
    Comm. Algebra, 23, 4833-4849, 1995. MR 1356105 (96k:16015)
  • 6. R. R. Colby and K. R. Fuller.
    Equivalence and Duality for Module Categories.
    Cambridge University Press, 2004. MR 2048277 (2005d:16001)
  • 7. R. Colpi.
    Tilting in Grothendieck Categories.
    Forum Math., 11, 735-759, 1999. MR 1725595 (2000h:18018)
  • 8. R. Colpi and J. Trlifaj.
    Tilting modules and tilting torsion theories.
    J. Algebra, 178, 614-634, 1995. MR 1359905 (97e:16003)
  • 9. S. E. Dickson.
    A torsion theory for abelian categories.
    Trans. Amer. Math. Soc., 121, 223-235, 1966. MR 0191935 (33:162)
  • 10. E. E. Enochs and O. M. Jenda.
    Relative homological algebra.
    Walter de Gruyter & Co., Berlin, 2000. MR 1753146 (2001h:16013)
  • 11. C. Faith.
    Rings with ascending condition on annihilators.
    Nagoya Math. J., 27, 179-191, 1966. MR 0193107 (33:1328)
  • 12. R. Fossum, P. Griffith, I. Reiten.
    Trivial Extensions of Abelian Categories.
    Springer-Verlag Lect. Notes in Math. 456, 1975. MR 0389981 (52:10810)
  • 13. D. Happel and I. Reiten.
    An introduction to quasitilted algebras.
    An. St. Univ. Ovidius Constanta, 4, 137-149, 1996. MR 1428462 (98g:16009)
  • 14. D. Happel, I. Reiten, S. O. Smalø.
    Tilting in Abelian Categories and Quasitilted Algebras.
    Memoirs of the A.M.S., vol. 575, 1996. MR 1327209 (97j:16009)
  • 15. D. Happel and C. M. Ringel.
    Tilted algebras.
    Trans. Amer. Math. Soc., 274, 399-443, 1982. MR 0675063 (84d:16027)
  • 16. B. Keller.
    Derived Categories and Tilting
    (to appear in Handbook of Tilting Theory).
  • 17. C. Menini and A. Orsatti.
    Representable equivalences between categories of modules and applications.
    Rend. Sem. Mat. Univ. Padova, 82, 203-231, 1989. MR 1049594 (91h:16026)
  • 18. B. Mitchell.
    Theory of Categories.
    Academic Press, London and New York, 1965. MR 0202787 (34:2647)
  • 19. Y. Miyashita.
    Tilting modules of finite projective dimension.
    Math. Z., 193, 113-146, 1986. MR 0852914 (87m:16055)
  • 20. N. Popescu.
    Abelian Categories with applications to Rings and Modules.
    Academic Press, London and New York, 1973. MR 0340375 (49:5130)
  • 21. L. Small.
    An example in noetherian rings.
    Proc. Natl. Acad. Sci. USA, 54, 1035-1036, 1965. MR 0188252 (32:5691)
  • 22. Bo Stentröm.
    Rings of Quotients.
    Springer-Verlag, Berlin, Heidelberg, New York, 1975. MR 0389953 (52:10782)

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Additional Information

Riccardo Colpi
Affiliation: Department of Pure and Applied Mathematics, University of Padova, via Belzoni 7, I 35100 Padova, Italy

Kent R. Fuller
Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242-1419

Received by editor(s): September 21, 2004
Received by editor(s) in revised form: December 3, 2004
Published electronically: August 24, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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