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Endomorphism Algebras of Equivariant Exceptional Collections

Andreas Krug, Erik Nikolov

Abstract

Given an action of a finite group on a triangulated category with a suitable strong exceptional collection, a construction of Elagin produces an associated strong exceptional collection on the equivariant category. We prove that the endomorphism algebra of the induced exceptional collection is the basic reduction of the skew group algebra of the endomorphism algebra of the original exceptional collection.

Endomorphism Algebras of Equivariant Exceptional Collections

Abstract

Given an action of a finite group on a triangulated category with a suitable strong exceptional collection, a construction of Elagin produces an associated strong exceptional collection on the equivariant category. We prove that the endomorphism algebra of the induced exceptional collection is the basic reduction of the skew group algebra of the endomorphism algebra of the original exceptional collection.
Paper Structure (17 sections, 15 theorems, 49 equations)

This paper contains 17 sections, 15 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Categorical set-up
  4. Generators, Tilting Objects and Exceptional Collections
  5. Basic algebras
  6. Group actions on categories, equivariant categories
  7. Skew groups algebras
  8. Proof of the Main Theorem
  9. Induced generators and tilting objects in the equivariant category
  10. The set-up and the induced equivariant exceptional sequence
  11. Comparison of the direct summands
  12. Proof of the main results
  13. Further Remarks and Examples
  14. Morita equivalence vs. derived equivalence
  15. Non-full exceptional sequences
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $G$ act on a smooth variety $X$ over an algebraically closed field of characteristic zero or coprime to $|G|$. Assume that $D^b(X)$ possesses a full exceptional collection of the form such that for every $i=1,\dots,k$, there is a transitive $G$-action on the index set $\{1,\dots,\ell_i\}$ with the property that $g_*E_{i,\ell_i}\cong E_{i,g(\ell_i)}$. Let $H_i=\mathop{\mathrm{\mathsf{Stab}}}\n

Theorems & Definitions (38)

  • Theorem 1.1: Ela
  • Theorem 1.2
  • Definition 1
  • Proposition 1
  • proof
  • Remark 1
  • Definition 2
  • Remark 2
  • Definition 3
  • Remark 3
  • ...and 28 more