Singularity categories of derived categories of hereditary algebras are derived categories

Abstract

We show that for the path algebra $A$ of an acyclic quiver, the singularity category of the derived category ${\mathsf{D}}^{\mathrm{b}}(\mathsf{mod}A)$ is triangle equivalent to the derived category of the functor category of $\underline{\mathsf{mod}}A$, that is, ${\mathsf{D}}_{\mathrm{sg}}({\mathsf{D}}^{\mathrm{b}}(\mathsf{mod}A))\simeq {\mathsf{D}}^{\mathrm{b}}(\mathsf{mod}(\underline{\mathsf{mod}}A))$. 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.

21 pages

References in corpus (1)

• Tilting and cluster tilting for preprojective algebras and Coxeter groups