Yoneda algebras and their singularity categories

Abstract

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