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  of Λ as the additive closure of the shifts of the Λ-modules in the derived category D b (mod Λ). We show that  is coherent and Gorenstein of self-injective dimension at most 1, and the singularity category of  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 f-category over itself by formulating the filtered derived category of a DG category, which assures the existence of a realization functor.

Enhanced Auslander-Reiten duality and tilting theory for singularity categories

“…Tilting theory of Z-graded singularity categories is an active subject in various branches of mathematics including representation theory, commutative algebra, algebraic geometry and mathematical physics, see e.g. [AIR,BIY,DL1,DL2,FU,Han1,Han2,Hap,HI,HIMO,HO,IKU,IO,IT,KST1,KST2,Ki1,Ki2,KLM,LP,LZ,MY,MU,N,SV,U2,Ya] and a survey article [Iy4].…”