Bilinear Forms on Grothendieck Groups of Triangulated Categories
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.
KeywordsGreen ring Auslander–Reiten triangle Symmetric algebra Perfect complex
2010 Mathematics Subject ClassificationPrimary 16G70 Secondary 18E30 20C20
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