Regular $\mathbb{Z}$-graded local rings and Graded Isolated Singularities
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866908481610579968 |
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| author | Li, Haonan Wu, Quanshui |
| author_facet | Li, Haonan Wu, Quanshui |
| contents | In this note we first study regular $\mathbb{Z}$-graded local rings. We characterize commutative noetherian regular $\mathbb{Z}$-graded local rings in similar ways as in the usual local case. Then, we characterize graded isolated singularity for a commutative $\mathbb{Z}$-graded semilocal algebra in terms of the global dimension of its associated noncommutative projective scheme. As a corollary, we obtain that a commutative affine $\mathbb{N}$-graded algebra generated in degree $1$ is a graded isolated singularity if and only if its associated noncommutative projective scheme is smooth; if and only if the category of coherent sheaves on its projective scheme has finite global dimension, which are known in literature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_05667 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regular $\mathbb{Z}$-graded local rings and Graded Isolated Singularities Li, Haonan Wu, Quanshui Commutative Algebra Algebraic Geometry Rings and Algebras 13A02, 13H05, 14A22, 16W50, 16S38 In this note we first study regular $\mathbb{Z}$-graded local rings. We characterize commutative noetherian regular $\mathbb{Z}$-graded local rings in similar ways as in the usual local case. Then, we characterize graded isolated singularity for a commutative $\mathbb{Z}$-graded semilocal algebra in terms of the global dimension of its associated noncommutative projective scheme. As a corollary, we obtain that a commutative affine $\mathbb{N}$-graded algebra generated in degree $1$ is a graded isolated singularity if and only if its associated noncommutative projective scheme is smooth; if and only if the category of coherent sheaves on its projective scheme has finite global dimension, which are known in literature. |
| title | Regular $\mathbb{Z}$-graded local rings and Graded Isolated Singularities |
| topic | Commutative Algebra Algebraic Geometry Rings and Algebras 13A02, 13H05, 14A22, 16W50, 16S38 |
| url | https://arxiv.org/abs/2410.05667 |