Cleft extensions of rings and singularity categories

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.

On the relative homology of cleft extensions of rings and abelian categories