Yoneda algebras and their singularity categories

Abstract

For a finite dimensional algebra $\Lambda$ of finite representation type and an additive generator $M$ for $\mathrm{mod}\,\Lambda$, we investigate the properties of the Yoneda algebra $\Gamma=\bigoplus_{i \geq 0}\mathrm{Ext}_\Lambda^i(M,M)$. We show that $\Gamma$ is graded coherent and Gorenstein of self-injective dimension at most $1$, and the graded singularity category $\mathrm{D_{sg}^\mathbb{Z}}(\Gamma)$ of $\Gamma$ is triangle equivalent to the derived category of the stable Auslander algebra of $\Lambda$. These results remain valid for representation-infinite algebras. For this we introduce the Yoneda category $\mathcal{Y}$ of $\Lambda$ as the additive closure of the shifts of the $\Lambda$-modules in the derived category $\mathrm{D^b}(\mathrm{mod}\,\Lambda)$. We show that $\mathcal{Y}$ is coherent and Gorenstein of self-injective dimension at most $1$, and the singularity category of $\mathcal{Y}$ is triangle equivalent to the derived category $\mathrm{D^b}(\mathrm{mod}\,(\underline{\mathrm{mod}}\,\Lambda))$ of the stable category $\underline{\mathrm{mod}}\,\Lambda$. To give a triangle equivalence, we apply the theory of realization functors. We show that any algebraic triangulated category has an f-category over itself by formulating the filtered derived category of a DG category, which assures the existence of a realization functor.