Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Equivalences induced by infinitely generated tilting modules


Author: Silvana Bazzoni
Journal: Proc. Amer. Math. Soc. 138 (2010), 533-544
MSC (2000): Primary 16D90, 16E30, 18E30; Secondary 16S90, 16G10
DOI: https://doi.org/10.1090/S0002-9939-09-10120-X
Published electronically: October 7, 2009
MathSciNet review: 2557170
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We generalize Brenner and Butler's Theorem as well as Happel's Theorem on the equivalences induced by a finitely generated tilting module over Artin algebras, to the case of an infinitely generated tilting module over an arbitrary associative ring establishing the equivalences induced between subcategories of module categories and also at the level of derived categories.


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

  • 1. L. Angeleri Hügel, A. Tonolo, J. Trlifaj, Tilting preenvelopes and cotilting precovers, Algebr. Represent. Theory 4 (2001), 155-170. MR 1834843 (2002e:16010)
  • 2. S. Bazzoni, Cotilting modules are pure-injective, Proc. Amer. Math. Soc. 131 (2003), 3665-3672. MR 1998172 (2004f:16049)
  • 3. S. Bazzoni, D. Herbera, One dimensional tilting modules are of finite type, Algebr. Represent. Theory 11 (2008), no. 1, 43-61. MR 2369100 (2009a:16010)
  • 4. M. Bökstedt and A. Neeman, Homotopy limits in triangulated categories, Compositio Math. 86 (1993), 209-234. MR 1214458 (94f:18008)
  • 5. S. Brenner, M. Butler, Generalizations of the Bernstein-Gelfand-Ponomarev reflection functors, Lecture Notes in Math., 832, Springer, 1980, 103-169. MR 607151 (83e:16031)
  • 6. R. R. Colby, K. R. Fuller, Tilting, cotilting and serially tilted rings, Comm. Algebra 18(5) (1990), 1585-1615. MR 1059750 (91h:16011)
  • 7. R. Colpi, Tilting in Grothendieck categories, Forum Math. 11 (1999), 735-759. MR 1725595 (2000h:18018)
  • 8. R. Colpi and J. Trlifaj, Tilting modules and tilting torsion theories, J. Alg. 178 (1995), 614-634. MR 1359905 (97e:16003)
  • 9. R. Colpi, G. D'Este, A. Tonolo, Quasi-tilting modules and counter equivalences, J. Alg. 191 (1997), 461-494. MR 1448804 (98g:16003)
  • 10. R. Colpi, A. Tonolo, J. Trlifaj, Partial cotilting modules and the lattices induced by them, Comm. Algebra 25 (10) (1997), 3225-3237. MR 1465112 (98i:16003)
  • 11. A. Facchini, Divisible modules over integral domains, Ark. Mat. 26 (1988), no. 1, 67-85. MR 948281 (90a:13018)
  • 12. A. Facchini, A tilting module over commutative integral domains, Comm. Alg. 15(11) (1987), 2235-2250. MR 912770 (89c:13028)
  • 13. L. Fuchs, On divisible modules over domains, Abelian groups and modules, Proc. of the Udine Conference, CISM Courses and Lectures, 287, Springer-Verlag, Wien-New York, 1984, 341-356. MR 789830 (86j:13012)
  • 14. P. Gabriel, M. Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 35, Springer-Verlag, New York, 1967. MR 0210125 (35:1019)
  • 15. D. Happel, On the derived category of a finite-dimensional algebra, Comment. Math. Helv. 62 (1987), no. 3, 339-389. MR 910167 (89c:16029)
  • 16. D. Happel, C. Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), 399-443. MR 675063 (84d:16027)
  • 17. D. Happel, I. Reiten, S. Smalø, Tilting in abelian categories and quasitilted algebras, Memoirs Amer. Math. Soc. 120, no. 575 (1996). MR 1327209 (97j:16009)
  • 18. B. Keller, Derived categories and their uses, Handbook of Algebra, Vol. 1, edited by M. Hazewinkel, North-Holland, Amsterdam, 1996. MR 1421815 (98h:18013)
  • 19. B. Keller, Derived categories and tilting, Handbook of Tilting Theory, LMS Lecture Note Series, 332, Cambridge University Press, Cambridge, 2007, 49-97. MR 2384608 (2009b:16029)
  • 20. N. Spaltenstein, Resolutions of unbounded complexes, Compositio Mathematica 65 (1988), 121-154. MR 932640 (89m:18013)
  • 21. B. Stenström, Rings of quotients, Die Grundleheren der Math. Wissenschaften, 217, Springer-Verlag, New York-Heidelberg, 1975. MR 0389953 (52:10782)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 16D90, 16E30, 18E30, 16S90, 16G10

Retrieve articles in all journals with MSC (2000): 16D90, 16E30, 18E30, 16S90, 16G10


Additional Information

Silvana Bazzoni
Affiliation: Dipartimento di Matematica Pura e Applicata, Università di Padova, Via Trieste 63, 35121 Padova, Italy
Email: bazzoni@math.unipd.it

DOI: https://doi.org/10.1090/S0002-9939-09-10120-X
Keywords: Tilting modules, equivalences and derived equivalences.
Received by editor(s): October 8, 2008
Received by editor(s) in revised form: July 2, 2009
Published electronically: October 7, 2009
Additional Notes: Supported by MIUR, PRIN 2005, project “Perspectives in the theory of rings, Hopf algebras and categories of modules” and by Università di Padova (Progetto di Ateneo CPDA071244/07 “Algebras and cluster categories”).
Communicated by: Birge Huisgen-Zimmermann
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society