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 $\operatorname{mod}Λ$, we investigate the properties of the Yoneda algebra Γ = ⨁ i ⩾ 0 Ext Λ i ( M, M ) $Γ =\bigoplus _{i \geqslant 0}\operatorname{Ext}_Λ ^i(M, M)$ . 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 ) $\mathrm{D_{sg}^\mathbb {Z}}(Γ )$ 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 $\mathcal {Y}$ of Λ $Λ$ as the additive closure of the shifts of the Λ $Λ$ ‐modules in the derived category D b ( mod Λ ) $D^b(\operatorname{mod}Λ )$ . 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 ) ) $D^b(\operatorname{mod}(\operatorname{\underline{\operatorname{mod}}}Λ ))$ of the stable category mod ̲ Λ $\operatorname{\underline{\operatorname{mod}}}Λ$ . 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.