Finiteness of representation dimension

Author:
Osamu Iyama

Journal:
Proc. Amer. Math. Soc. **131** (2003), 1011-1014

MSC (2000):
Primary 16G10; Secondary 16E10

Published electronically:
July 17, 2002

MathSciNet review:
1948089

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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]**Maurice Auslander, Idun Reiten, and Sverre O. Smalø,*Representation theory of Artin algebras*, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1995. MR**1314422**

Maurice Auslander, Idun Reiten, and Sverre O. Smalø,*Representation theory of Artin algebras*, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1997. Corrected reprint of the 1995 original. MR**1476671****[CPS]**E. Cline, B. Parshall, and L. Scott,*Finite-dimensional algebras and highest weight categories*, J. Reine Angew. Math.**391**(1988), 85–99. MR**961165****[DR1]**Vlastimil Dlab and Claus Michael Ringel,*Quasi-hereditary algebras*, Illinois J. Math.**33**(1989), no. 2, 280–291. MR**987824****[DR2]**Vlastimil Dlab and Claus Michael 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**943793**, 10.1090/S0002-9939-1989-0943793-2**[FGR]**Robert M. Fossum, Phillip A. Griffith, and Idun Reiten,*Trivial extensions of abelian categories*, Lecture Notes in Mathematics, Vol. 456, Springer-Verlag, Berlin-New York, 1975. Homological algebra of trivial extensions of abelian categories with applications to ring theory. MR**0389981****[HS]**P. J. Hilton and U. Stammbach,*A course in homological algebra*, 2nd ed., Graduate Texts in Mathematics, vol. 4, Springer-Verlag, New York, 1997. MR**1438546****[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]**Hiroyuki Tachikawa,*Quasi-Frobenius rings and generalizations. 𝑄𝐹-3 and 𝑄𝐹-1 rings*, Lecture Notes in Mathematics, Vol. 351, Springer-Verlag, Berlin-New York, 1973. Notes by Claus Michael Ringel. MR**0349740****[X1]**Changchang Xi,*On the representation dimension of finite dimensional algebras*, J. Algebra**226**(2000), no. 1, 332–346. MR**1749892**, 10.1006/jabr.1999.8177**[X2]**C. C. Xi: Representation dimension and quasi-hereditary algebras, to appear in Adv. Math.

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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