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arXiv · 2306.09832

On the generalizations of global dimensions and singularity categories

Xiaolei Zhang·Tiwei Zhao·Dingguo Wang
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Abstract

For each $n\in\mathbb{N}\cup\{\infty\}$, we introduce the notion of $n$-singularity category $\mathbf{D}_{n{\rm-}sg}(R)$ of a given ring $R$, which can be seen as a generalization of the classical singularity category. Moreover, the $n$-global dimension $n$-gldim$(R)$ of $R$ is investigated. We show that $\mathbf{D}_{n{\rm-}sg}(R)=0$ if and only if $n$-gldim$(R)$ is finite. Furthermore, we characterize the vanishing property of $n$-singularity categories in terms of recollements.

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Xiaolei Zhang, Tiwei Zhao, Dingguo Wang. 2023-06-16. On the generalizations of global dimensions and singularity categories. https://arxiv.org/abs/2306.09832

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