Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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



Tilting theory and the finitistic dimension conjectures

Authors: Lidia Angeleri-Hügel and Jan Trlifaj
Journal: Trans. Amer. Math. Soc. 354 (2002), 4345-4358
MSC (2000): Primary 16E10, 16E30, 16G10
Published electronically: June 24, 2002
MathSciNet review: 1926879
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $R$ be a right noetherian ring and let $\mathcal{P}^{<\infty}$ be the class of all finitely presented modules of finite projective dimension. We prove that findim $R = n < \infty$ iff there is an (infinitely generated) tilting module $T$ such that pd$T = n$ and $T ^\perp = (\mathcal P^{<\infty})^\perp$. If $R$ is an artin algebra, then $T$ can be taken to be finitely generated iff $\mathcal P^{<\infty}$ is contravariantly finite. We also obtain a sufficient condition for validity of the First Finitistic Dimension Conjecture that extends the well-known result of Huisgen-Zimmermann and Smalø.

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

  • 1. L. Angeleri Hügel, F. U. Coelho, Infinitely generated tilting modules of finite projective dimension, Forum Math. 13 (2001), 239-250. MR 2002b:16009
  • 2. L. Angeleri Hügel, F. U. Coelho, Infinitely generated complements to partial tilting modules, Math. Proc. Cambridge Philos. Soc. 132 (2002), 88-95.
  • 3. L. Angeleri Hügel, A. Tonolo, J. Trlifaj, Tilting preenvelopes and cotilting precovers, Algebras and Representation Theory 4 (2001), 155-170.MR 2002e:16010
  • 4. M. Auslander, Functors and morphisms determined by objects, in Representation Theory of Algebras, Lecture Notes in Pure Appl. Math.37, M.Dekker, New York, 1978, pp. 1-244.MR 58:844
  • 5. M. Auslander, I. Reiten, Applications of contravariantly finite subcategories, Adv. Math. 86 (1991), 111-152. MR 92e:16009
  • 6. R. Colpi, J. Trlifaj, Tilting modules and tilting torsion classes, J. Algebra 178 (1995), 614-634. MR 97e:16003
  • 7. W. Crawley-Boevey, Infinite-dimensional modules in the representation theory of finite-dimensional algebras, Canad. Math. Soc. Conf. Proc.23(1998), 29-55. MR 99m:16016
  • 8. E. E. Enochs, O. M. G. Jenda, Relative Homological Algebra, de Gruyter, Berlin, 2000. MR 2001h:16013
  • 9. P. Eklof, J. Trlifaj, How to make Ext vanish, Bull. London Math. Soc. 33 (2001), 41-51. MR 2001i:16015
  • 10. R. Göbel, S. Shelah, Cotorsion theories and splitters, Trans. Amer. Math. Soc. 352 (2000), 5357-5379. MR 2001b:20098
  • 11. D. Happel, C. M. Ringel, Tilted algebras, Trans. Amer Math. Soc. 274 (1982), 399-443. MR 84d:16027
  • 12. B. Huisgen-Zimmermann, Homological assets of positively graded representations of finite dimensional algebras, Canad. Math. Soc. Conf. Proc.14(1993), 463-475.
  • 13. B. Huisgen-Zimmermann, S. O. Smalø, A homological bridge between finite and infinite dimensional representations, Algebras and Representation Theory 1 (1998), 169-188. MR 2000b:16014
  • 14. H. Krause, M. Saorín, On minimal approximations of modules, In: Trends in the representation theory of finite dimensional algebras (ed. by E. L. Green and B. Huisgen-Zimmermann), Contemp. Math. 229 (1998) 227-236. MR 99m:16002
  • 15. H. Lenzing, Homological transfer from finitely presented to infinite modules, Lecture Notes in Math. 1006 (1983), 734-761. MR 85f:16034
  • 16. Y. Miyashita, Tilting modules of finite projective dimension, Math. Z. 193 (1986), 113-146. MR 87m:16055
  • 17. L. Salce, Cotorsion theories for abelian groups, Symposia Math. XXIII (1979), 11-32. MR 81j:20078
  • 18. S. O. Smalø, Homological differences between finite and infinite dimensional representations of algebras, Trends in Math., Birkhäuser, Basel, 2000, 425-439. MR 2001i:16013
  • 19. J. Trlifaj, Cotorsion theories induced by tilting and cotilting modules, Abelian Groups, Rings and Modules, Contemp. Math. 273 (2001), 285-300.MR 2001m:16012
  • 20. J. Trlifaj, Approximations and the little finitistic dimension of artinian rings, J. Algebra 246 (2001), 343-355.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 16E10, 16E30, 16G10

Retrieve articles in all journals with MSC (2000): 16E10, 16E30, 16G10

Additional Information

Lidia Angeleri-Hügel
Affiliation: Mathematisches Institut der Universität, Theresienstrasse 39, D-80333 München, Germany
Address at time of publication: Universitat Autònoma de Barcelona, Departament de Matemàtiques, E-08193 Bellaterra (Barcelona), Spain

Jan Trlifaj
Affiliation: Katedra algebry MFF UK, Sokolovsk\a’a 83, 186 75 Prague 8, Czech Republic

Received by editor(s): July 13, 2001
Received by editor(s) in revised form: February 7, 2002
Published electronically: June 24, 2002
Additional Notes: Research supported by an HWP-grant of LMU Munich and by grants GAČR 201/00/0766 and MSM 113200007
Dedicated: Dedicated to Idun Reiten on the occasion of her sixtieth birthday
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society