Relative Singularity categories and singular equivalences

Abstract

Let $R$ be a right notherian ring. We introduce the concept of relative singularity category $\Delta_{\mathcal{X}}(R)$ of $R$ with respect to a contravariantly finite subcategory $\mathcal{X}$ of $\rm{mod}\mbox{-}R.$ Along with some finiteness conditions on $\mathcal{X}$, we prove that $\Delta_{\mathcal{X}}(R)$ is triangle equivalent to a subcategory of the homotopy category $\mathbb{K}_{\rm{ac}}(\mathcal{X})$ of exact complexes over $\mathcal{X}$. As an application, a new description of the classical singularity category $\mathbb{D}_{\rm{sg}}(R)$ is given. The relative singularity categories are applied to lift a stable equivalence between two suitable subcategories of the module categories of two given right notherian ring to get a singular equivalence between the rings. In different types of rings, including path rings, triangular matrix rings, trivial extension rings and tensor rings, we provide some consequences for their singularity categories.