Regular $\mathbb{Z}$-graded local rings and Graded Isolated Singularities

Haonan Li; Quanshui Wu

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arXiv · 2410.05667

Regular $\mathbb{Z}$-graded local rings and Graded Isolated Singularities

Haonan Li·Quanshui Wu

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Abstract

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.

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Haonan Li, Quanshui Wu. 2024-10-08. Regular $\mathbb{Z}$-graded local rings and Graded Isolated Singularities. https://arxiv.org/abs/2410.05667

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