Bilinear forms on Grothendieck groups of triangulated categories
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
Cite
@article{arxiv.1709.03889,
title = {Bilinear forms on Grothendieck groups of triangulated categories},
author = {Peter Webb},
journal= {arXiv preprint arXiv:1709.03889},
year = {2017}
}
Comments
arXiv admin note: substantial text overlap with arXiv:1301.4701