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)

 
 

 

Finiteness of representation dimension


Author: Osamu Iyama
Journal: Proc. Amer. Math. Soc. 131 (2003), 1011-1014
MSC (2000): Primary 16G10; Secondary 16E10
DOI: https://doi.org/10.1090/S0002-9939-02-06616-9
Published electronically: July 17, 2002
MathSciNet review: 1948089
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We will show that any module over an artin algebra is a direct summand of some module whose endomorphism ring is quasi-hereditary. As a conclusion, any artin algebra has a finite representation dimension.


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

  • [A] M. Auslander: Representation dimension of Artin algebras, Lecture notes, Queen Mary College, London, 1971.
  • [ARS] M. Auslander, I. Reiten, S. O. Smale: Representation theory of Artin algebras. Cambridge Studies in Advanced Mathematics, 36, Cambridge University Press, Cambridge, 1995. MR 96c:16015, MR 98e:16011
  • [CPS] E. Cline, B. Parshall, L. Scott: Finite-dimensional algebras and highest weight categories. J. Reine Angew. Math. 391 (1988), 85-99. MR 90d:18005
  • [DR1] V. Dlab, C. M. Ringel: Quasi-hereditary algebras. Illinois J. Math. 33 (1989), no. 2, 280-291. MR 90e:16023
  • [DR2] V. Dlab, C. M. Ringel: Every semiprimary ring is the endomorphism ring of a projective module over a quasihereditary ring. Proc. Amer. Math. Soc. 107 (1989), no. 1, 1-5. MR 89m:16033
  • [FGR] R. M. Fossum, P. Griffith, I. Reiten: Trivial extensions of abelian categories. Lecture Notes in Mathematics, Vol. 456, Springer-Verlag, Berlin-New York, 1975. MR 52:10810
  • [HS] P. J. Hilton, U. Stammbach: A course in homological algebra. Graduate Texts in Mathematics, 4. Springer-Verlag, New York, 1997, xii+364 pp. MR 97k:18001
  • [I1] O. Iyama: $\tau$-categories II: Nakayama pairs and rejective subcategories, to appear in Algebras and Representation theory.
  • [I2] O. Iyama: $\tau$-categories III: Auslander orders and Auslander-Reiten quivers, to appear in Algebras and Representation theory.
  • [I3] O. Iyama: A proof of Solomon's second conjecture on local zeta functions of orders, to appear in J. Algebra.
  • [IT] K. Igusa, G. Todorov: On the finitistic global dimension conjecture, preprint.
  • [T] H. Tachikawa: Quasi-Frobenius rings and generalizations. Lecture Notes in Mathematics, Vol. 351, Springer-Verlag, Berlin-New York, 1973. MR 50:2233
  • [X1] C. C. Xi: On the representation dimension of finite dimensional algebras. J. Algebra 226 (2000), no. 1, 332-346. MR 2001d:16027
  • [X2] C. C. Xi: Representation dimension and quasi-hereditary algebras, to appear in Adv. Math.

Similar Articles

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

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


Additional Information

Osamu Iyama
Affiliation: Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan
Address at time of publication: Department of Mathematics, Himeji Institute of Technology, Himeji, 671-2201, Japan
Email: iyama@kusm.kyoto-u.ac.jp, iyama@sci.himeji-tech.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-02-06616-9
Received by editor(s): August 6, 2001
Received by editor(s) in revised form: October 29, 2001
Published electronically: July 17, 2002
Communicated by: Martin Lorenz
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society