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.
CITATION STYLE
Webb, P. (2018). Bilinear Forms on Grothendieck Groups of Triangulated Categories. In Springer Proceedings in Mathematics and Statistics (Vol. 242, pp. 465–480). Springer New York LLC. https://doi.org/10.1007/978-3-319-94033-5_19
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