Singularity categories of derived categories of hereditary algebras are derived categories

TL;DR

This paper establishes a triangle equivalence between the singularity category of the derived category of a hereditary algebra's module category and the derived category of the functor category of its stable module category, extending known results.

Contribution

It extends Iyama-Oppermann's result to hereditary algebras of any acyclic quiver by establishing a functor category analog of Happel's triangle equivalence.

Findings

Singularity category of derived category is equivalent to derived category of functor category

Extension of Iyama-Oppermann's result to all acyclic quivers

Establishment of a functor category analog of Happel's equivalence

Abstract

We show that for the path algebra $A$ of an acyclic quiver, the singularity category of the derived category $\mathsf{D}^{\rm b}(\mathsf{mod}\,A)$ is triangle equivalent to the derived category of the functor category of $\underline{\mathsf{mod}}\,A$, that is, $\mathsf{D}_{\rm sg}(\mathsf{D}^{\rm b}(\mathsf{mod}\,A))\simeq \mathsf{D}^{\rm b}(\mathsf{mod}(\underline{\mathsf{mod}}\,A))$. This extends a result of Iyama-Oppermann for the path algebra $A$ of a Dynkin quiver. An important step is to establish a functor category analog of Happel's triangle equivalence for repetitive algebras.