On the generalizations of global dimensions and singularity categories

For each $n\in \mathbb{N}\cup \{\mathrm{\infty }\}$, we introduce the notion of $n$-singularity category ${\mathbf{D}}_{n-sg}(R)$ 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 ${\mathbf{D}}_{n-sg}(R)=0$ if and only if $n$-gldim$(R)$ is finite. Furthermore, we characterize the vanishing property of $n$-singularity categories in terms of recollements.