For a finite dimensional algebra $\Lambda$ of finite representation type and an additive generator $M$ for $\mathrm{ mod}\,\Lambda$, we investigate the properties of the Yoneda algebra $\Gamma=\bigoplus_{ i \geq 0}\mathrm{ Ext}_\Lambda^i(M,M)$. We show that $\Gamma$ is graded coherent and Gorenstein of self-injective dimension at most $1$, and the graded singularity category $\mathrm{ D_{ sg}^\mathbb{ Z}}(\Gamma)$ of $\Gamma$ is triangle equivalent to the derived category of the stable Auslander algebra of $\Lambda$. These results remain valid for representation-infinite algebras. For this we introduce the Yoneda category $\mathcal{ Y}$ of $\Lambda$ as the additive closure of the shifts of the $\Lambda$-modules in the derived category $\mathrm{ D^b}(\mathrm{ mod}\,\Lambda)$. 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 $\mathrm{ D^b}(\mathrm{ mod}\,(\underline{ \mathrm{ mod}}\,\Lambda))$ of the stable category $\underline{ \mathrm{ mod}}\,\Lambda$. 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.
Categories: math.RT math.RA