Singularity categories and singular equivalences for resolving subcategories

Abstract

Let $$\mathcal {X}$$ be a resolving subcategory of an abelian category. In this paper we investigate the singularity category $$\mathsf {D_{sg}}(\underline{\mathcal {X}})=\mathsf {D^b}({\mathsf {mod}}\,\underline{\mathcal {X}})/\mathsf {K^b}({\mathsf {proj}}({\mathsf {mod}}\,\underline{\mathcal {X}}))$$ of the stable category $$\underline{\mathcal {X}}$$ of $$\mathcal {X}$$ . We consider when the singularity category is triangle equivalent to the stable category of Gorenstein projective objects, and when the stable categories of two resolving subcategories have triangle equivalent singularity categories. Applying this to the module category of a Gorenstein ring, we prove that the complete intersections over which the stable categories of resolving subcategories have trivial singularity categories are the simple hypersurface singularities of type $$(\mathsf {A}_1)$$ . We also generalize several results of Yoshino on totally reflexive modules.