Tilting theory for hypersurface singularities of dimension one
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915575782965248 |
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| author | Iyama, Osamu Liu, Junyang |
| author_facet | Iyama, Osamu Liu, Junyang |
| contents | Any $\mathbb{N}$-graded commutative Gorenstein ring $R$ of Krull dimension one with $R_0$ a field admits a standard silting object $V$ in the stable category $\underline{\mathrm{CM}}_0^{\mathbb{Z}}R$, and the object $V$ is tilting if and only if the $a$-invariant $a$ is non-negative, as shown by Buchweitz, the first author, and Yamaura. In this article, under the additional assumption that $R$ is a hypersurface singularity, we prove that endomorphism algebra of $V$ is Iwanaga-Gorenstein of self-injective dimension at most $2$, and we give its explicit presentation in terms of a quiver with relation. In the case of where $a$ is negative, we prove that the dg endomorphism algebra of $V$ is Gorenstein, and we give its explicit presentation in terms of a dg path algebra. We explain our results by several examples including numerical semigroup algebras generated by two elements. Moreover, for each finite and countable Cohen-Macaulay representation type, we include the Auslander-Reiten quiver of the category $\mathrm{CM}_0^{\mathbb{Z}}R$ with the position of the standard silting object. As a step of the proof of our results, we give a characterization of Gorensteinness of homologically finite dg algebras in terms of Serre functors. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_12581 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tilting theory for hypersurface singularities of dimension one Iyama, Osamu Liu, Junyang Representation Theory Commutative Algebra Algebraic Geometry Rings and Algebras 18G80, 14F08, 13C14, 16E10, 13F70 Any $\mathbb{N}$-graded commutative Gorenstein ring $R$ of Krull dimension one with $R_0$ a field admits a standard silting object $V$ in the stable category $\underline{\mathrm{CM}}_0^{\mathbb{Z}}R$, and the object $V$ is tilting if and only if the $a$-invariant $a$ is non-negative, as shown by Buchweitz, the first author, and Yamaura. In this article, under the additional assumption that $R$ is a hypersurface singularity, we prove that endomorphism algebra of $V$ is Iwanaga-Gorenstein of self-injective dimension at most $2$, and we give its explicit presentation in terms of a quiver with relation. In the case of where $a$ is negative, we prove that the dg endomorphism algebra of $V$ is Gorenstein, and we give its explicit presentation in terms of a dg path algebra. We explain our results by several examples including numerical semigroup algebras generated by two elements. Moreover, for each finite and countable Cohen-Macaulay representation type, we include the Auslander-Reiten quiver of the category $\mathrm{CM}_0^{\mathbb{Z}}R$ with the position of the standard silting object. As a step of the proof of our results, we give a characterization of Gorensteinness of homologically finite dg algebras in terms of Serre functors. |
| title | Tilting theory for hypersurface singularities of dimension one |
| topic | Representation Theory Commutative Algebra Algebraic Geometry Rings and Algebras 18G80, 14F08, 13C14, 16E10, 13F70 |
| url | https://arxiv.org/abs/2508.12581 |