On the graded singularity category of Abelian quotient singularities, I. Smooth categorical compactification

Given a singularity which is the quotient of an affine space $V$ by a finite Abelian group $G\subseteq \mathrm{S}\mathrm{L}(V)$, we study the DG enhancement ${\mathcal{D}}^b(\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{l}\mathrm{s}(k[V{]}^G))$ of the bounded derived category of the non-commutative projective space $\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{l}\mathrm{s}(k[V{]}^G)$ and the DG enhancement ${\mathcal{D}}_{sg}^{\mathbb{Z}}(k[V{]}^G)$ of its graded singularity category. In this paper, we construct smooth categorical compactifications, in the sense of Efimov, of ${\mathcal{D}}^b(\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{l}\mathrm{s}(k[V{]}^G))$ and ${\mathcal{D}}_{sg}^{\mathbb{Z}}(k[V{]}^G)$ respectively, via the non-commutative crepant resolution of $k[V{]}^G$. We give explicit constructions of canonical classical generators in the kernels of such compactifications.