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LATTICE STRUCTURE OF TORSION CLASSES FOR HEREDITARY ARTIN ALGEBRAS

  • CLAUS MICHAEL RINGEL (a1)
Abstract

Let $\unicode[STIX]{x1D6EC}$ be a connected hereditary artin algebra. We show that the set of functorially finite torsion classes of $\unicode[STIX]{x1D6EC}$ -modules is a lattice if and only if $\unicode[STIX]{x1D6EC}$ is either representation-finite (thus a Dynkin algebra) or $\unicode[STIX]{x1D6EC}$ has only two simple modules. For the case of $\unicode[STIX]{x1D6EC}$ being the path algebra of a quiver, this result has recently been established by Iyama–Reiten–Thomas–Todorov and our proof follows closely some of their considerations.

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[AS] Auslander M. and Smalø S. O., Preprojective modules over artin algebras , J. Algebra 66 (1980), 61122.
[DR] Dlab V. and Ringel C. M., Indecomposable representations of graphs and algebras , Mem. Amer. Math. Soc. 173 (1976).
[IRTT] Iyama O., Reiten I., Thomas H. and Todorov G., Lattice structure of torsion classes for path algebras of quivers , Bull. Lond. Math. Soc. 47(4) (2015), 639650.
[ONFR] Obaid M. A. A., Nauman S. K., Fakieh W. M. and Ringel C. M., The Ingalls–Thomas bijections , Int. Electron. J. Algebra 20 (2016), 2844.
[R1] Ringel C. M., Representations of k-species and bimodules , J. Algebra 41 (1976), 269302.
[R2] Ringel C. M., Exceptional objects in hereditary categories, in Proceedings Constantza Conference, An. St. Univ. Ovidius Constantza, 4, f.2, Faculty of Mathematics and Computer Science, Ovidius University, Constanta, Romania, 1996, 150–158.
[R3] Ringel C. M., The Catalan combinatorics of the hereditary artin algebras, in Recent Developments in Representation Theory, Contemporary Mathematics, 673, American Mathematical Society, Providence, RI, 2016, 51–177.
[Ro] Roiter A. V., Unboundedness of the dimension of the indecomposable representations of an algebra which has infinitely many indecomposable representations , Izv. Akad. Nauk SSSR. Ser. Mat. 32 (1968), 12751282.
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Nagoya Mathematical Journal
  • ISSN: 0027-7630
  • EISSN: 2152-6842
  • URL: /core/journals/nagoya-mathematical-journal
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