Yoneda algebras and their singularity categories

For a finite dimensional algebra $\mathrm{\Lambda }$ of finite representation type and an additive generator $M$ for $\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{\Lambda }$, we investigate the properties of the Yoneda algebra $\Gamma=\bigoplus_{i \geq 0}$\mathrm{E}\mathrm{x}\mathrm{t}$_\Lambda^i(M,M)$\mathrm{.}Weshowthat$\Gamma$ is graded coherent and Gorenstein of self-injective dimension at most $1$, and the graded singularity category ${\mathrm{D}}_{\mathrm{s}\mathrm{g}}^{\mathbb{Z}}(\mathrm{\Gamma })$ of $\mathrm{\Gamma }$ is triangle equivalent to the derived category of the stable Auslander algebra of $\mathrm{\Lambda }$. These results remain valid for representation-infinite algebras. For this we introduce the Yoneda category $\mathcal{Y}$ of $\mathrm{\Lambda }$ as the additive closure of the shifts of the $\mathrm{\Lambda }$-modules in the derived category ${\mathrm{D}}^{\mathrm{b}}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{\Lambda })$. We show that $\mathcal{Y}$ is coherent and Gorenstein of self-injective dimension at most $1$, and the singularity category of $\mathcal{Y}$ is triangle equivalent to the derived category ${\mathrm{D}}^{\mathrm{b}}(\mathrm{m}\mathrm{o}\mathrm{d}(\underset{‾}{\mathrm{m}\mathrm{o}\mathrm{d}}\mathrm{\Lambda }))$ of the stable category $\underset{‾}{\mathrm{m}\mathrm{o}\mathrm{d}}\mathrm{\Lambda }$. 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.