Bilinear Forms on Grothendieck Groups of Triangulated Categories

  • Peter WebbEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 242)


We extend the theory of bilinear forms on the Green ring of a finite group developed by Benson and Parker to the context of the Grothendieck group of a triangulated category with Auslander–Reiten triangles, taking only relations given by direct sum decompositions. We examine the non-degeneracy of the bilinear form given by dimensions of homomorphisms, and show that the form may be modified to give a Hermitian form for which the standard basis given by indecomposable objects has a dual basis given by Auslander–Reiten triangles. An application is given to the homotopy category of perfect complexes over a symmetric algebra, with a consequence analogous to a result of Erdmann and Kerner.


Green ring Auslander–Reiten triangle Symmetric algebra Perfect complex 

2010 Mathematics Subject Classification

Primary 16G70 Secondary 18E30 20C20 


  1. 1.
    A. Adem and J.H. Smith, Periodic complexes and group actions, Ann. of Math. 154 (2001), 407–435.MathSciNetCrossRefGoogle Scholar
  2. 2.
    M. Auslander, Relations for Grothendieck groups of Artin algebras, Proc. Amer. Math. Soc. 91 (1984), 336–340.MathSciNetCrossRefGoogle Scholar
  3. 3.
    D.J. Benson and R.A. Parker, The Green ring of a finite group, J. Algebra 87 (1984), 290–331.MathSciNetCrossRefGoogle Scholar
  4. 4.
    K. Bongartz, A generalization of a result of M. Auslander, Bull. London Math. Soc. 21 (1989), 255–256.Google Scholar
  5. 5.
    M. Broué, Equivalences of blocks of group algebras, Finite-dimensional algebras and related topics (Ottawa, ON, 1992), 126, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 424, Kluwer Acad. Publ., Dordrecht, 1994.Google Scholar
  6. 6.
    K. Diveris, M. Purin and P.J. Webb, Combinatorial restrictions on the tree class of the Auslander-Reiten quiver of a triangulated category, Math. Z. 282 (2016), 405–410.MathSciNetCrossRefGoogle Scholar
  7. 7.
    K. Erdmann and O. Kerner, On the stable module category of a self-injective algebra, Trans. Amer. Math. Soc. 352 (2000), 2389–2405.MathSciNetCrossRefGoogle Scholar
  8. 8.
    K. Erdmann and A. Skowroński, On Auslander-Reiten components of blocks and self-injective biserial algebras, Trans. Amer. Math. Soc. 330 (1992), 165–189.MathSciNetCrossRefGoogle Scholar
  9. 9.
    D. Happel, Triangulated Categories in the Representation Theory of Finite-Dimensional Algebras, London Math. Soc. Lecture Note Ser. 119, Cambridge University Press, Cambridge, 1988.Google Scholar
  10. 10.
    D. Happel, B. Keller and I. Reiten, Bounded derived categories and repetitive algebras, Journal of Algebra 319 (2008), 1611–1635.MathSciNetCrossRefGoogle Scholar
  11. 11.
    B.T. Jensen, X. Su and A. Zimmermann, Degeneration-like orders in triangulated categories, J. Algebra Appl. 4 (2005), 587–597.MathSciNetCrossRefGoogle Scholar
  12. 12.
    J. Rickard, Morita theory for derived categories, J. London Math. Soc. 39 (1989), 436–456.MathSciNetCrossRefGoogle Scholar
  13. 13.
    P.J. Webb, On the orthogonality coefficients for character tables of the Green ring of a finite group, J. Algebra 89 (1984), 247–263.MathSciNetCrossRefGoogle Scholar
  14. 14.
    P.J. Webb, A split exact sequence of Mackey functors, Commentarii Math. Helv. 66 (1991), 34–69.MathSciNetCrossRefGoogle Scholar
  15. 15.
    W. Wheeler, The triangulated structure of the stable derived category, J. Algebra 165 (1994), 23–40.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA

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