On the graded singularity category of Abelian quotient singularities, I. Smooth categorical compactification
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912502625861632 |
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| author | Chen, Xiaojun Zeng, Jieheng |
| author_facet | Chen, Xiaojun Zeng, Jieheng |
| contents | Given a singularity which is the quotient of an affine space $V$ by a finite Abelian group $G \subseteq \mathrm{SL}(V)$, we study the DG enhancement $\mathcal{D}^{b}(\mathrm{tails}(k[V]^G))$ of the bounded derived category of the non-commutative projective space $\mathrm{tails}(k[V]^G)$ 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 $\mathcal{D}^b(\mathrm{tails}(k[V]^G))$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_19815 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the graded singularity category of Abelian quotient singularities, I. Smooth categorical compactification Chen, Xiaojun Zeng, Jieheng Algebraic Geometry Rings and Algebras Representation Theory Given a singularity which is the quotient of an affine space $V$ by a finite Abelian group $G \subseteq \mathrm{SL}(V)$, we study the DG enhancement $\mathcal{D}^{b}(\mathrm{tails}(k[V]^G))$ of the bounded derived category of the non-commutative projective space $\mathrm{tails}(k[V]^G)$ 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 $\mathcal{D}^b(\mathrm{tails}(k[V]^G))$ 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. |
| title | On the graded singularity category of Abelian quotient singularities, I. Smooth categorical compactification |
| topic | Algebraic Geometry Rings and Algebras Representation Theory |
| url | https://arxiv.org/abs/2507.19815 |