On the generalizations of global dimensions and singularity categories

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.