On -good module categories without short cycles

Authors:
Bangming Deng and Bin Zhu

Journal:
Proc. Amer. Math. Soc. **129** (2001), 69-77

MSC (2000):
Primary 16G10, 16G60

DOI:
https://doi.org/10.1090/S0002-9939-00-05518-0

Published electronically:
June 14, 2000

MathSciNet review:
1694857

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a quasi-hereditary algebra, and the -good module category consisting of -modules which have a filtration by standard modules. An indecomposable module in is said to be on a short cycle in if there exist an indecomposable module in and a chain of two nonzero noninvertible maps . It is shown that two indecomposable modules in are isomorphic if they are not on short cycles in and have the same composition factors. Moreover, if there is no short cycle in , we show that is finite, that is, there are only finitely many isomorphism classes of indecomposables in . This is an analogue to a result in a complete module category proved by Happel and Liu.

**[1]**M.Auslander and I.Reiten, Modules determined by their composition factors, Illinois Journal of Mathematics, 29(2), 1985, 280-301. MR**86i:16032****[2]**B.M.Deng and C.C.Xi, On -good module categories of quasi-hereditary algebras, Chin. Ann. of Math., 18B(4), 1997, 467-480. MR**99b:16015****[3]**V.Dlab and C.M.Ringel, The module theoretical approach to quasi-hereditary algebras, LMS Lecture Notes Series 168, "Representations of algebras and related topics", ed. H. Tachikawa and S.Brenner, 1992, 200-224.**[4]**P.Gabriel and A.V.Roiter, Representations of finite-dimensional algebras, Encyclopaedia of Math. Sci. Vol.73, Algebra VIII (1992). MR**94h:16001b****[5]**D.Happel and S.Liu, Module categories without short cycles are of finite type, Proc. Amer. Math. Soc., 120(2), 1994, 371-375. MR**94d:16010****[6]**I.Reiten, A.Skowronski, and S.O.Smal, Short chains and short cycles of modules, Proc. Amer. Math. Soc., 117(2), 1993, 343-354. MR**93d:16013****[7]**C.M.Ringel, Report on the Brauer-Thrall conjectures, Springer Lecture Notes in Math. 831 (1980), 104-136. MR**82j:16055****[8]**C.M.Ringel, Tame algebras and integral quadratic forms, Springer Lecture Notes in Math. 1099, 1984. MR**87f:16027****[9]**C.M.Ringel, The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences, Math. Zeit., 208(1991), 209-223. MR**93c:16010****[10]**C.M.Ringel, The category of good modules over a quasi-hereditary algebra, Proc. of the sixth inter. conf. on representation of algebras, Carleton-Ottawa Math. Lecture Note Series, No. 14, 1992. MR**94b:16020****[11]**A.Skowronski, Cycles in module categories, Proc. CMS Annual Seminar/Nato Advanced Research Workshop (Ottawa, 1992), 309-345. MR**96a:16010****[12]**T.Wakamatsu, Stable equivalence of self-injective algebras and a generalization of tilting modules, J. Algebra, 134(1990), 298-325.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
16G10,
16G60

Retrieve articles in all journals with MSC (2000): 16G10, 16G60

Additional Information

**Bangming Deng**

Affiliation:
Department of Mathematics, Beijing Normal University, 100875 Beijing, People’s Republic of China

Email:
dengbm@bnu.edu.cn

**Bin Zhu**

Affiliation:
Department of Mathematics, Beijing Normal University, 100875 Beijing, People’s Republic of China

Address at time of publication:
Department of Mathematics, Tsinghua University, 100084 Beijing, People’s Republic of China

Email:
bzhu@math.tsinghua.edu.cn

DOI:
https://doi.org/10.1090/S0002-9939-00-05518-0

Received by editor(s):
September 21, 1998

Received by editor(s) in revised form:
March 31, 1999

Published electronically:
June 14, 2000

Dedicated:
To our teacher Shaoxue Liu on the occasion of his 70th birthday

Communicated by:
Ken Goodearl

Article copyright:
© Copyright 2000
American Mathematical Society