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The Inductive Step of the Second Brauer-Thrall Conjecture

Published online by Cambridge University Press:  20 November 2018

Sverre O. Smalø
Sverre O. Smalø*
Affiliation:
Universitet i Trondheim NLHT, Trondheim, Norway
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In this paper we are going to use a result of H. Harada and Y. Sai concerning composition of nonisomorphisms between indecomposable modules and the theory of almost split sequences introduced in the representation theory of Artin algebras by M. Auslander and I. Reiten to obtain the inductive step in the second Brauer-Thrall conjecture.

Section 1 is devoted to giving the necessary background in the theory of almost split sequences.

As an application we get the first Brauer-Thrall conjecture for Artin algebras. This conjecture says that there is no bound on the length of the finitely generated indecomposable modules over an Artin algebra of infinite type, i.e., an Artin algebra such that there are infinitely many nonisomorphic indecomposable finitely generated modules. This result was first proved by A. V. Roiter [8] and later in general for Artin rings by M. Auslander [2] using categorical methods.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 2 , February 1980 , pp. 342 - 349
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Auslander, M., Representation theory of Artin algebras II, Comm. in Algebra (1974), 269–310.Google Scholar
2. Auslander, M., Applications of morphisms determined by modules. Representation theory of algebras (Proc. Conf. Temple University, 1976) Lecture Notes in Pure and Applied Math. Vol. 37, (M. Dekker, N.Y., 1978).Google Scholar
3. Auslander, M. and Reiten, I., Representation theory of Artin algebras III, Comm. in Algebra (1975), 239–294.Google Scholar
4. Auslander, M. and Reiten, I., Representation theory of Artin algebras IV, Comm. in Algebra (1977), 443–518.Google Scholar
5. Auslander, M. and Reiten, I., Representation theory of Artin algebras VI, Comm. in Algebra (1978), 257–300.Google Scholar
6. Harada, H. and Sai, Y., On categories of indecomposable modules I, Osaka J. Math. 7 (1970), 323–344.Google Scholar
7. Nazarova, L. A. and Roiter, A. V., Categorical matrix problems and the Brauer-Thrall conjecture, Mitt. Math. Sem. Gissen (1975), 115–153.Google Scholar
8. Roiter, A. V., Unboundness of the dimension of the indecomposable representations of an algebra which has infinitely many indecomposable representations, Tzv. Akad. SSR, Sev. Math. 32 (1968), 1275–1282.Google Scholar
9. Yamagata, K., On Artin rings of finite representation type, J. of Algebra 50, No. 2, 1978.Google Scholar