Yoneda algebras and their singularity categories. Issue 6 (11th May 2022) - Hanihara, Norihiro
Abstract: For a finite dimensional algebra Λ Λ of finite representation type and an additive generator M M for mod Λ modΛ, we investigate the properties of the Yoneda algebra Γ = ⨁ i ⩾ 0 Ext Λ i ( M, M ) Γ =_i 0Ext_Λ ^i(M, M) . We show that Γ Γ is graded coherent and Gorenstein of self‐injective dimension at most 1, and the graded singularity category D sg Z ( Γ ) D_sg^Z(Γ ) of Γ Γ is triangle equivalent to the derived category of the stable Auslander algebra of Λ Λ . These results remain valid for representation‐infinite algebras. For this we introduce the Yoneda category Y Y of Λ Λ as the additive closure of the shifts of the Λ Λ ‐modules in the derived category D b ( mod Λ ) D^b(modΛ ) . We show that Y Y is coherent and Gorenstein of self‐injective dimension at most 1, and the singularity category of Y Y is triangle equivalent to the derived category D b ( mod ( mod ̲ Λ ) ) D^b(mod(modΛ )) of the stable category mod ̲ Λ modΛ . 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.