The Hom-Ext quiver and applications to exceptional collections

Abstract

We study what we call the Hom-Ext quiver and characterize it as a type of `superquiver'. In type $\tilde{\mathbb{A}}$, the Hom-Ext quiver of an exceptional set is the tiling algebra of the corresponding geometric model. And, in that case, Hom-Ext quivers classify exceptional sets up to Dehn twist of the corresponding geometric model. We show that these Dehn twists are realized by twist functors and give autoequivalences of the derived category. We provide a generating set for the group of autoequivalences of the derived category in type $\tilde{\mathbb{A}}$, and show that the Hom-Ext quiver classifies exceptional sets up to the action of the subgroup of the automorphism group of the derived category generated by twist functors associated to exceptional cycles. We introduce superquivers, which are a generalization of Hom-Ext quivers. Exceptional sets over finite acyclic quivers are realized as representations of superquivers. Throughout, we list several questions and conjectures that make for, what we believe, exciting new research.