  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.
