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The Hom-Ext quiver and applications to exceptional collections

Kiyoshi Igusa, Ray Maresca

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.

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 , 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 , 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.

Paper Structure

This paper contains 10 sections, 33 theorems, 18 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quivers of type $\tilde{\mathbb{A}}$
  4. A geometric model for the category of representations of $\tilde{\mathbb{A}}$ quivers
  5. A geometric model for the derived category of quivers of type $\tilde{\mathbb{A}}$
  6. The Hom-Ext quiver
  7. Applications to exceptional sets
  8. A geometric realization of the Hom-Ext quiver in type $\tilde{\mathbb{A}}$
  9. Hom-Ext quiver and derived autoequivalences
  10. Superquivers

Key Result

Theorem A

For a finite acyclic quiver, the number of exceptional sequences associated to an exceptional collection of representations is equal to the number of linear extensions of the poset defined by the Hom-Ext quiver.

Figures (2)

  • Figure 1: The six possible local configurations of an arc $a_1$ being clockwise from another arc $a_2$.
  • Figure 2: The four local possibilities for three arcs to intersect at an endpoint $p$. The arc corresponding to $X_i$ is depicted in blue, $X_k$ in red, and $X_j$ in black.

Theorems & Definitions (96)

  • Theorem A: Theorem \ref{['thm: linear extensions and exceptional sequences']}
  • Theorem B: Theorem \ref{['thm: iso H-E quiver iff Dehn twist']}
  • Theorem C: Corollary \ref{['cor: iso HE quivers iff derived equiv']}
  • Theorem D: Theorem \ref{['thm: superquiver twists']}
  • Remark 2.1
  • Lemma 2.1: Lemma 4.1 in happel1982tilted
  • Lemma 2.2: crawley1992exceptional, ringel1994braid
  • Definition 2.1
  • Example 2.1
  • Lemma 2.3
  • ...and 86 more