Algebras and Representation Theory

, Volume 21, Issue 2, pp 447–470 | Cite as

Injective Presentations of Induced Modules over Cluster-Tilted Algebras

  • Ralf Schiffler
  • Khrystyna Serhiyenko


Every cluster-tilted algebra B is the relation extension \(C\ltimes \textup {Ext}^{2}_{C}(DC,C)\) of a tilted algebra C. A B-module is called induced if it is of the form M C B for some C-module M. We study the relation between the injective presentations of a C-module and the injective presentations of the induced B-module. Our main result is an explicit construction of the modules and morphisms in an injective presentation of any induced B-module. In the case where the C-module, and hence the B-module, is projective, our construction yields an injective resolution. In particular, it gives a module theoretic proof of the well-known 1-Gorenstein property of cluster-tilted algebras.


Cluster-tilted algebra Induction Coinduction Relation extension 

Mathematics Subject Classification (2010)

16G20 16G70 13F60 


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  1. 1.
    Amiot, C.: Cluster categories for algebras of global dimension 2 and quivers with potential. Ann. Inst. Fourier 59(6), 2525–2590 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Assem, I., Brüstle, T., Schiffler, R.: Cluster-tilted algebras as trivial extensions. Bull. Lond. Math. Soc. 40, 151–162 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Assem, I., Brüstle, T., Schiffler, R.: Cluster-tilted algebras and slices. J. Algebra 319, 3464–3479 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Assem, I., Brüstle, T., Schiffler, R.: On the Galois covering of a cluster-tilted algebra. J. Pure Appl. Alg. 213(7), 1450–1463 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Assem, I., Brüstle, T., Schiffler, R.: Cluster-tilted algebras without clusters. J. Algebra 324, 2475–2502 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Assem, I., Bustamante, J.C., Igusa, K., Schiffler, R.: The first Hochschild cohomology group of a cluster-tilted algebra revisited. Int. J. Alg. Comput. 23(4), 729–744 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Assem, I., Gatica, M. A., Schiffler, R., Taillefer, R.: Hochschild cohomology of relation extension algebras. J. Pure Appl. Alg. 220(7), 2471–2499 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Assem, I., Redondo, M. J.: The first Hochschild cohomology group of a schurian cluster-tilted algebra. Manuscripta Math. 128(3), 373–388 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Assem, I., Redondo, M. J., Schiffler, R.: On the first Hochschild cohomology group of a cluster-tilted algebra. Algebr. Represent. Theory 18(6), 1547–1576 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Assem, I., Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras, 1: Techniques of Representation Theory, London Mathematical Society Student Texts 65. Cambridge University Press, Cambridge (2006)CrossRefzbMATHGoogle Scholar
  11. 11.
    Barot, M., Fernandez, E., Pratti, I., Platzeck, M. I., Trepode, S.: From iterated tilted to cluster-tilted algebras. Adv. Math. 223(4), 1468–1494 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Beaudet, L., Brüstle, T., Todorov, G.: Projective dimension of modules over cluster-tilted algebras. Algebr. Represent. Theory 17(6), 1797–1807 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bertani-Økland, M.A., Oppermann, S., Wrȧlsen, A.: Constructing tilted algebras from cluster-tilted algebras. J. Algebra 323(9), 2408–2428 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Buan, A. B., Marsh, R., Reineke, M., Reiten, I., Todorov, G.: Tilting theory and cluster combinatorics. Adv. Math. 204(2), 572–618 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Buan, A.B., Marsh, R., Reiten, I.: Cluster-tilted algebras. Trans. Amer. Math. Soc. 359(1), 323–332 (2007). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Buan, A. B., Marsh, R., Reiten, I.: Cluster-tilted algebras of finite representation type. J. Algebra 306(2), 412–431 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Buan, A. B., Marsh, R., Reiten, I.: Cluster mutation via quiver representations. Comment. Math. Helv. 83(1), 143–177 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Caldero, P., Chapoton, F.: Cluster algebras as Hall algebras of quiver representations. Comment Math. Helv. 81(3), 595–616 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations arising from clusters (A n case). Trans. Amer. Math. Soc. 358(3), 1347–1364 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations and cluster tilted algebras. Algebr. Represent Theory 9(4), 359–376 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Caldero, P., Keller, B.: From triangulated categories to cluster algebras. Invent Math. 172, 169–211 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    David-Roesler, L., Schiffler, R.: Algebras from surfaces without punctures. J. Algebra 350, 218–244 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Derksen, H., Weyman, J., Zelevinsky, A.: Quivers with potentials and their representations. I. Mutations. Selecta Math.(N.S.) 14(1), 59–119 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Fomin, S., Zelevinsky, A.: Cluster algebras I: Foundations. J. Amer. Math. Soc. 15, 497–529 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Keller, B.: On triangulated orbit categories. Documenta Math. 10, 551–581 (2005)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Keller, B., Reiten, I.: Cluster-tilted algebras are Gorenstein and stably Calabi-Yau. Adv. Math. 211(1), 123–151 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Ladkani, S.: Hochschild cohomology of the cluster-tilted algebras of finite representation type, arXiv:1205.0799
  28. 28.
    Plamondon, P.G.: Cluster algebras via cluster categories with infinite-dimensional morphism spaces. Compos. Math. 147(6), 1921–1954 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Schiffler, R.: Quiver Representations, CMS Books in Mathematics, Springer International Publishing (2014)Google Scholar
  30. 30.
    Schiffler, R., Serhiyenko, K.: Induced and coinduced modules over cluster-tilted algebras. J. Algebra 472, 226–258 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ConnecticutStorrsUSA
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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