2020/03/15 by Hafezi, Rasool
#FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2003.06897
Let R be a right notherian ring. We introduce the concept of relative singularity category ΔX(R) of R with respect to a contravariantly finite subcategory X of \rmmod-R. Along with some finiteness conditions on X, we prove that ΔX(R) is triangle equivalent to a subcategory of the homotopy category \mathbbK_\rmac(X) of exact complexes over X. As an application, a new description of the classical singularity category \mathbbD_\rmsg(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.