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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)

Abstract

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.

Keywords

Green ring Auslander–Reiten triangle Symmetric algebra Perfect complex 

2010 Mathematics Subject Classification

Primary 16G70 Secondary 18E30 20C20 

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Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA

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