Yoneda algebras and their singularity categories

Abstract

For a finite dimensional algebra Λ $\Lambda$ of finite representation type and an additive generator M $M$ for mod Λ $\operatorname{mod}\Lambda$ , we investigate the properties of the Yoneda algebra Γ = ⨁ i ⩾ 0 Ext Λ i ( M , M ) $\Gamma =\bigoplus _{i \geqslant 0}\operatorname{Ext}_\Lambda ^i(M,M)$ . We show that Γ $\Gamma$ is graded coherent and Gorenstein of self-injective dimension at most 1, and the graded singularity category D sg Z ( Γ ) $\mathrm{D_{sg}^\mathbb {Z}}(\Gamma )$ of Γ $\Gamma$ is triangle equivalent to the derived category of the stable Auslander algebra of Λ $\Lambda$ . These results remain valid for representation-infinite algebras. For this we introduce the Yoneda category Y $\mathcal {Y}$ of Λ $\Lambda$ as the additive closure of the shifts of the Λ $\Lambda$ -modules in the derived category D b ( mod Λ ) $\mathrm{D^b}(\operatorname{mod}\Lambda )$ . We show that Y $\mathcal {Y}$ is coherent and Gorenstein of self-injective dimension at most 1, and the singularity category of Y $\mathcal {Y}$ is triangle equivalent to the derived category D b ( mod ( mod ̲ Λ ) ) $\mathrm{D^b}(\operatorname{mod}(\operatorname{\underline{\operatorname{mod}}}\Lambda ))$ of the stable category mod ̲ Λ $\operatorname{\underline{\operatorname{mod}}}\Lambda$ . 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.