Algebras and Representation Theory

, Volume 17, Issue 2, pp 565–591 | Cite as

Torsion Pairs and Rigid Objects in Tubes

  • Karin Baur
  • Aslak Bakke Buan
  • Robert J. MarshEmail author


We classify the torsion pairs in a tube category and show that they are in bijection with maximal rigid objects in the extension of the tube category containing the Prüfer and adic modules. We show that the annulus geometric model for the tube category can be extended to the larger category and interpret torsion pairs, maximal rigid objects and the bijection between them geometrically. We also give a similar geometric description in the case of the linear orientation of a Dynkin quiver of type A.


Tube category Torsion pair Torsion theory Pruefer module Adic module Annulus Maximal rigid object Arc model Direct limit Inverse limit 

Mathematics Subject Classifications (2010)

Primary 16G10 16G70 55N45; Secondary 13F60 16G20 


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  1. 1.
    Assem, I., Simson, D., Skowronski, A.: Elements of the Representation Theory of Associative Algebras. 1: Techniques of Representation Theory. LMS Student Texts, vol. 65 (2006)Google Scholar
  2. 2.
    Auslander, M.: Functors and morphisms determined by objects. In: Representation Theory of Algebras (Proc. Conf., Temple Univ., Philadelphia, Pa., 1976), p. 1244. Lecture Notes in Pure Appl. Math., vol. 37. Dekker, New York (1978)Google Scholar
  3. 3.
    Baur, K., Marsh, R.J.: A geometric model of tube categories. J. Algebra 362, 178–191 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Beligiannis, A., Reiten, I.: Homological and homotopical aspects of torsion theories. Mem. Amer. Math. Soc. 188(883), 207 (2007)MathSciNetGoogle Scholar
  5. 5.
    Bongartz, K.: Tilted algebras. In: Auslander, M., Lluis, E. (eds.) Representations of Algebras (Puebla, 1980). Lecture Notes in Math., vol. 903, pp. 26–38. Springer, Berlin (1981)CrossRefGoogle Scholar
  6. 6.
    Brüstle, T., Zhang, J.: On the cluster category of a marked surface without punctures. Algebra Number Theory 5(4), 529–566 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Buan, A.B., Krause, H.: Cotilting modules over tame hereditary algebras. Pac. J. Math. 211(1), 41–59 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Buan, A.B., Krause, H.: Tilting and cotilting for quivers of type \(\widetilde{A}_n\). J. Pure Appl. Algebra 190(1–3), 1–21 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Colpi, R.: Tilting in Grothendieck categories. Forum Math. 11(6), 735–759 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Crawley-Boevey, W.: Infinite-dimensional modules in the representation theory of finite-dimensional algebras. Canadian Math. Soc. Conf. Proc. 23, 29–54 (1998)MathSciNetGoogle Scholar
  11. 11.
    Dickson, E.S.: A torsion theory for Abelian categories. Trans. Am. Math. Soc. 121, 223–235 (1966)CrossRefzbMATHGoogle Scholar
  12. 12.
    Gehrig, B.: Geometric realizations of cluster categories. Master’s thesis, Winter Semester 2009/2010.
  13. 13.
    Holm, T., Jørgensen, P., Rubey, M.: Ptolemy diagrams and torsion pairs in the cluster category of Dynkin type An. J. Algebr. Comb. 34, 507–523 (2011)CrossRefzbMATHGoogle Scholar
  14. 14.
    Holm, T., Jørgensen, P., Rubey, M.: Torsion pairs in cluster tubes. arXiv:1207.3206v1 [math.RT] (2012, preprint)
  15. 15.
    Iyama, O., Yoshino, Y.: Mutation in triangulated categories and rigid Cohen–Macaulay modules. Invent. Math. 172(1), 117–168 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Jensen, C.U., Lenzing, H.: Model-theoretic algebra with particular emphasis on fields, rings, modules. In: Algebra, Logic and Applications, vol. 2. Gordon and Breach Science Publishers, New York (1989)Google Scholar
  17. 17.
    Koenig, S., Zhu, B.: From triangulated categories to abelian categories: cluster tilting in a general framework. Math. Z. 258(1), 143–160 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Krause, H., Solberg, Ø.: Applications of torsion pairs. J. Lond. Math. Soc. (2) 68, 631–650 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Nakaoka, H.: General heart construction on a triangulated category (I): unifying t-structures and cluster tilting subcategories. Appl. Categ. Struct. 19(6), 879–899 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Ng, P.: A characterization of torsion theories in the cluster category of Dynkin type A. arXiv:1005.4364v1 [math.RT] (2010, preprint)
  21. 21.
    Ringel, C.M.: Infinite Length Modules. Some Examples as Introduction. In: Infinite Length Modules (Bielefeld, 1998), pp. 1–73. Trends Math., Birkhäuser, Basel (2000)Google Scholar
  22. 22.
    Prest, M.: Purity, spectra and localisation. In: Encyclopedia of Mathematics and its Applications, vol. 121. Cambridge University Press, Cambridge (2009)Google Scholar
  23. 23.
    Trlifaj, J.: Ext and inverse limits. Ill. J. Math. 47(1–2), 529–538 (2003)zbMATHMathSciNetGoogle Scholar
  24. 24.
    Warkentin, M.: Fadenmoduln über \(\widetilde{A_n}\) und cluster-kombinatorik (String modules over \(\widetilde{A_n}\) and cluster combinatorics). Diploma Thesis, University of Bonn. Available from (2008)
  25. 25.
    Zhou, Y., Zhu, B.: Mutation of torsion pairs in triangulated categories and its geometric realization. arXiv:1105.3521v1 [math.RT] (2011, preprint)

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© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Karin Baur
    • 1
  • Aslak Bakke Buan
    • 2
  • Robert J. Marsh
    • 3
    Email author
  1. 1.Institut für Mathematik und wissenschaftliches RechnenUniversität GrazGrazAustria
  2. 2.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway
  3. 3.School of MathematicsUniversity of LeedsLeedsEngland

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