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Combinatorial restrictions on the tree class of the Auslander–Reiten quiver of a triangulated category

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Abstract

We show that if a connected, Hom-finite, Krull–Schmidt triangulated category has an Auslander–Reiten quiver component with Dynkin tree class then the category has Auslander–Reiten triangles and that component is the entire quiver. This is an analogue for triangulated categories of a theorem of Auslander, and extends a previous result of Scherotzke. We also show that if there is a quiver component with extended Dynkin tree class, then other components must also have extended Dynkin class or one of a small set of infinite trees, provided there is a non-zero homomorphism between the components. The proofs use the theory of additive functions.

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References

  1. Amiot, C.: On the structure of triangulated categories with finitely many indecomposables. Bull. Soc. Math. Fr. 135, 435–474 (2007)

    MathSciNet  MATH  Google Scholar 

  2. Dlab, V., Ringel, C.M.: Indecomposable Representations of Graphs and Algebras. In: Memoirs of the American Mathematical Society, vol. 6, no. 173 (1976)

  3. Erdmann, K., Skowroński, A.: On Auslander–Reiten components of blocks and self-injective biserial algebras. Trans. Am. Math. Soc. 330, 165–189 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. Happel, D.: Triangulated Categories in the Representation Theory of Finite-Dimensional Algebras. London Mathematical Society Lecture Note Series, vol. 119, Cambridge University Press, Cambridge (1988)

  5. Happel, D., Preiser, U., Ringel, C.M.: Vinberg’s characterization of Dynkin diagrams using subadditive functions with application to \(DTr\)-periodic modules. In: Representation Theory II. Lecture Notes in Math, vol. 832, pp. 280-294. Springer, Berlin (1980)

  6. Riedtmann, C.: Algebren, Darstellungsköcher, Überlagerungen und zurück. Comment. Math. Helv. 55, 199–224 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  7. Reiten, I., Van den Bergh, M.: Noetherian hereditary abelian categories satisfying Serre duality. J. Am. Math. Soc 15, 295–366 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Scherotzke, S.: Finite and bounded Auslander–Reiten components in the derived category. J. Pure Appl. Algebra 215, 232–241 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Webb, P.J.: The Auslander–Reiten quiver of a finite group. Math. Z. 179, 97–121 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  10. Webb, P.J.: Consequences of the existence of Auslander-Reiten triangles with applications to perfect complexes for self-injective algebras. Preprint arXiv:1301.4701

  11. Xiao, J., Zhu, B.: Relations for the Grothendieck groups of triangulated categories. J. Algebra 257, 37–50 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Xiao, J., Zhu, B.: Locally finite triangulated categories. J. Algebra 290, 473–490 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, St. Olaf College, Northfield, MN, 55057, USA

    Kosmas Diveris & Marju Purin

  2. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Peter Webb

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  1. Kosmas Diveris
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  2. Marju Purin
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  3. Peter Webb
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Correspondence to Peter Webb.

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Diveris, K., Purin, M. & Webb, P. Combinatorial restrictions on the tree class of the Auslander–Reiten quiver of a triangulated category. Math. Z. 282, 405–410 (2016). https://doi.org/10.1007/s00209-015-1545-1

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  • DOI: https://doi.org/10.1007/s00209-015-1545-1

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