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Degenerations in graded quiver varieties and in derived categories of Dynkin quivers

Alessandro Contu, Fang Yang

Abstract

For any acyclic quiver, Keller-Scherotzke provided a stratifying functor from the category of finite-dimensional modules of the singular Nakajima category to the derived category of the quiver. Under this functor, a degeneration of strata of a graded quiver variety corresponds to a degeneration, in the sense of Jensen-Su-Zimmermann, in the derived category. In this article, for any Dynkin quiver, we further investigate Jensen-Su-Zimmermann's partial order and show that any degeneration of objects in the derived category can be obtained in this way.

Degenerations in graded quiver varieties and in derived categories of Dynkin quivers

Abstract

For any acyclic quiver, Keller-Scherotzke provided a stratifying functor from the category of finite-dimensional modules of the singular Nakajima category to the derived category of the quiver. Under this functor, a degeneration of strata of a graded quiver variety corresponds to a degeneration, in the sense of Jensen-Su-Zimmermann, in the derived category. In this article, for any Dynkin quiver, we further investigate Jensen-Su-Zimmermann's partial order and show that any degeneration of objects in the derived category can be obtained in this way.
Paper Structure (12 sections, 22 theorems, 116 equations)

This paper contains 12 sections, 22 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Happel's correspondence
  4. Degeneration order in $D^b(Q)$
  5. Quantum Grothendieck rings
  6. Derived Hall algebras
  7. Graded quiver varieties
  8. Correspondence between partial orders
  9. Chains of ext-degenerations
  10. Minimal degenerations
  11. Main theorem
  12. Degenerations in type $A_n$

Key Result

Theorem 1.1

For any $n,m\in {\mathcal{M}}^+$,

Theorems & Definitions (36)

  • Theorem 1.1: Theorem \ref{['Thm:Correspondence']}
  • Theorem 2.1: Happel_articolo_1987
  • Definition 2.2: Jensen_Su_Zimmerman_2005II_tring_catJense_Su_Zimmermann_2005_der_cat
  • Definition 2.3
  • Proposition 2.4: Qin_tqchar_2014
  • Lemma 2.5: Keller_Scherotzke_2016
  • Proposition 2.6: Keller_Scherotzke_2016
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 26 more