Relative singularity categories and singular equivalences

Let R be a right noetherian 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 $${\text {{mod{-}}}}R.$$ 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 $$\mathbb {K} _\mathrm{{ac}}(\mathcal {X} )$$ of exact complexes over $$\mathcal {X} $$ . As an application, a new description of the classical singularity category $$\mathbb {D} _\mathrm{{sg}}(R)$$ 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 noetherian rings 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.