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arXiv · 1709.03889

Bilinear forms on Grothendieck groups of triangulated categories

Peter Webb
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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.

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Peter Webb. 2017-09-11. Bilinear forms on Grothendieck groups of triangulated categories. https://arxiv.org/abs/1709.03889

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