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The dg Leavitt algebra, singular Yoneda category and singularity category

Xiao-Wu Chen, Bernhard Keller, Yu Wang, Zhengfang Wang

Abstract

For any finite dimensional algebra $Λ$ given by a quiver with relations, we prove that its dg singularity category is quasi-equivalent to the perfect dg derived category of a dg Leavitt path algebra. The result might be viewed as a deformed version of the known description of the dg singularity category of a radical-square-zero algebra in terms of a Leavitt path algebra with trivial differential. The above result is achieved in two steps. We first introduce the singular Yoneda dg category of $Λ$, which is quasi-equivalent to the dg singularity category of $Λ$. The construction of this new dg category follows from a general operation for dg categories, namely an explicit dg localization inverting a natural transformation from the identity functor to a dg endofunctor. This localization turns out to be quasi-equivalent to a dg quotient category. Secondly, we prove that the endomorphism algebra of the quotient of $Λ$ modulo its Jacobson radical in the singular Yoneda dg category is isomorphic to the dg Leavitt path algebra. The appendix is devoted to an alternative proof of the result using Koszul-Moore duality and derived localizations.

The dg Leavitt algebra, singular Yoneda category and singularity category

Abstract

For any finite dimensional algebra given by a quiver with relations, we prove that its dg singularity category is quasi-equivalent to the perfect dg derived category of a dg Leavitt path algebra. The result might be viewed as a deformed version of the known description of the dg singularity category of a radical-square-zero algebra in terms of a Leavitt path algebra with trivial differential. The above result is achieved in two steps. We first introduce the singular Yoneda dg category of , which is quasi-equivalent to the dg singularity category of . The construction of this new dg category follows from a general operation for dg categories, namely an explicit dg localization inverting a natural transformation from the identity functor to a dg endofunctor. This localization turns out to be quasi-equivalent to a dg quotient category. Secondly, we prove that the endomorphism algebra of the quotient of modulo its Jacobson radical in the singular Yoneda dg category is isomorphic to the dg Leavitt path algebra. The appendix is devoted to an alternative proof of the result using Koszul-Moore duality and derived localizations.

Paper Structure

This paper contains 31 sections, 38 theorems, 309 equations.

Table of Contents

  1. Introduction
  2. The background and main results
  3. Conventions and structure
  4. The Cohn and Leavitt algebras
  5. The dg Cohn and Leavitt algebras
  6. Quivers and path algebras
  7. Pretriangulated dg categories
  8. DG categories and functors
  9. Exact and pretriangulated dg categories
  10. The perfect dg derived categories
  11. An explicit dg localization
  12. The Yoneda dg category
  13. The bar and Yoneda dg categories
  14. The dg derived categories
  15. The dg tensor algebra as an endomorphism algebra
  16. ...and 16 more sections

Key Result

Theorem A

Let $\Lambda=\mathbb{K}Q/I$ be a finite dimensional algebra, and $\widetilde{Q}$ be its radical quiver. Then there is a quasi-equivalence Consequently, there are triangle equivalences

Theorems & Definitions (74)

  • Theorem A
  • Theorem B
  • Example 2.1
  • Proposition 2.2
  • proof
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • proof
  • ...and 64 more