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Minimal representation-infinite artin algebras

Published online by Cambridge University Press:  24 October 2008

Andrzej Skowroński
Andrzej Skowroński
Affiliation:
Institute of Mathematics, Nicholas Copernicus University, Chopina 12/18, 87-100 Toruń, Poland
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Extract

Let A be an artin algebra over a commutative artin ring R, mod A be the category of finitely generated right A-modules, and rad∞ (modA) be the infinite power of the Jacobson radical rad(modA) of modA. Recall that A is said to be representation-finite if mod A admits only finitely many non-isomorphic indecomposable modules. It is known that A is representation-finite if and only if rad∞ (mod A) = 0. Moreover, from the validity of the First Brauer–Thrall Conjecture [26, 2] we know that A is representation-finite if and only if there is a common bound on the length of indecomposable modules in mod A.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 116 , Issue 2 , September 1994 , pp. 229 - 243
Copyright
Copyright © Cambridge Philosophical Society 1994

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References

REFERENCES

[1]Assem, I. and Coelho, F.. Gluings of tilted algebras (to appear).Google Scholar
[2]Auslander, M.. Representation theory of artin algebras II. Comm. Algebra 1 (1974), 269–310.CrossRefGoogle Scholar
[3]Bautista, R.. On algebras of strongly unbounded representation type. Comment. Math. Helv. 60 (1985), 392–399.CrossRefGoogle Scholar
[4]Bretscher, O. and Todorov, G.. On a theorem of Nazarova and Roiter. In Representation Theory I. Finite Dimensional Algebras. Lecture Notes in Math. 1177 (Springer-Verlag, 1980), pp. 50–54.CrossRefGoogle Scholar
[5]Crawley-Boevey, W. W.. On tame algebras and bocses. Proc. London Math. Soc. 56 (1988), 451–483.CrossRefGoogle Scholar
[6]Crawley-Boevey, W. W.. Regular modules for tame hereditary algebras. Proc. London Math. Soc. 62 (1991), 490–508.CrossRefGoogle Scholar
[7]Crawley-Boevey, W. W.. Tame algebras and generic modules. Proc. London Math. Soc. 63 (1991), 241–265.CrossRefGoogle Scholar
[8]Crawley-Boevey, W. W.. Modules of finite length over their endomorphism rings. In Representations of Algebras and Related Topics. London Math. Soc. Lecture Note Series 168 (Cambridge Univ. Press, 1992), pp. 127–184.CrossRefGoogle Scholar
[9]Dlab, V. and Ringel, C. M.. On algebras of finite representation type. J. Algebra 33 (1975) 306–394.CrossRefGoogle Scholar
[10]Dlab, V. and Ringel, C. M.. Indecomposable representations of graphs and algebras. Memoirs Amer. Math. Soc. 173 (1976).Google Scholar
[11]Dlab, V. and Ringel, C. M.. The representations of tame hereditary algebras. In Representation Theory of Algebras. Pure and Applied Mathematics 37 (Marcel Dekker, 1978), pp. 329–353.Google Scholar
[12]Fischbacher, V.. A new proof of a theorem of Nazarova and Roiter. C.R. Acad. Sc. Paris, t. 300, Serie I, no. 9 (1985), 259–262.Google Scholar
[13]Happel, D. and Liu, S.. Module categories without short cycles are of finite type. Proc. Amer. Math. Soc. (1994) (to appear).CrossRefGoogle Scholar
[14]Happel, D., Preiser, V. and Ringel, C. M.. Vinberg's characterization of Dynkin diagrams using subadditive functions with application to D Tr-periodic modules. In Representation Theory II, Lecture Notes in Math. 832 (Springer-Verlag, 1980), pp. 280–294.CrossRefGoogle Scholar
[15]Happel, D. and Unger, L.. Factors of wild concealed algebras. Math. Z. 201 (1989), 474–483.CrossRefGoogle Scholar
[16]Harada, M. and Sai, Y.. On categories of indecomposable modules I. Osaka J. Math. 7 (1970), 323–344.Google Scholar
[17]Igusa, K. and Todorov, G.. A characterization of finite Auslander–Reiten quivers. J. Algebra 89 (1984), 148–177.CrossRefGoogle Scholar
[18]Liu, S.. Degrees of irreducible maps and the shapes of the Auslander–Reiten quivers. J. London Math. Soc. 45 (1992), 32–54.CrossRefGoogle Scholar
[19]Liu, S.. Semi-stable components of an Auslander–Reiten quiver. J. London Math. Soc. 47 (1983), 405–416.Google Scholar
[20]Nazarova, L. A. and Roiter, A. V.. Categorical matrix problems and the Brauer–Thrall conjecture. Preprint. Inst. Math. Acad. Sci. Kiev 1973, German transl. Mitt. Math. Sem. Giessen 115 (1975).Google Scholar
[21]Ringel, C. M.. Representation of K-species and bimodules. J. Algebra 41 (1976), 269–302.CrossRefGoogle Scholar
[22]Ringel, C. M.. Finite dimensional hereditary algebras of wild representation type. Math. Z. 161 (1978), 236–255.CrossRefGoogle Scholar
[23]Ringel, C. M.. Report on the Brauer–Thrall conjectures. In Representation Theory I, Lecture Notes in Math. 831 (Springer-Verlag, 1980), pp. 104–136.CrossRefGoogle Scholar
[24]Ringel, C. M.. Tame algebras and integral quadratic forms. Lecture Notes in Math. 1099 (Springer, 1984).CrossRefGoogle Scholar
[25]Ringel, C. M.. The canonical algebras. In Topics in Algebra, Banach Center Publications, vol. 26, part 1 (Warsaw, 1990), pp. 407–432.Google Scholar
[26]Roiter, A. V.. The unboundness of the dimension of the indecomposable representations of algebras that have infinite number of indecomposable representations. Izv. Acad. Nauk SSR 32 (1968), 1275–1282.Google Scholar
[27]Skowroński, A.. Regular Auslander–Reiten components containing directing modules. Proc. Amer. Math. Soc. (1994) (to appear).CrossRefGoogle Scholar
[28]Skowroński, A.. Generalized standard Auslander–Reiten components. J. Math. Soc. Japan (1994) (to appear).CrossRefGoogle Scholar
[29]Skowroński, A.. Generalized standard Auslander–Reiten components without oriented cycles. Osaka J. Math. 30 (1993), 515–527.Google Scholar
[30]Skowroński, A.. Cycles in module categories. In Representations of Algebras and Related Topics, Proc. CMS Annual Seminar/NATO Advanced Research Workshop (Ottawa, 1992) Kluwer Acad. Publ. (1994) (to appear).Google Scholar
[31]Strauss, H.. On the perpendicular category of a partial tilting module. J. Algebra 144 (1991), 43–66.CrossRefGoogle Scholar
[32]Zhang, Y.. The structure of stable components. Canad. J. Math. 43 (1991), 652–672.CrossRefGoogle Scholar