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.

**[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: -categories II: Nakayama pairs and rejective subcategories, to appear in Algebras and Representation theory.**[I2]**O. Iyama: -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.

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