Gorenstein Subcategories and Relative Singularity Categories

Abstract:

      Let $\mathscr{A}$ be an abelian category, $\mathscr{T}$ a self-orthogonal subcategory of $\mathscr{A}$ and each object in $\mathscr{T}$ admit finite projective and injective dimensions. If the left Gorenstein subcategory $l\mathcal{G}(\mathscr{T})$ equals to the right orthogonal class of $\mathscr{T}$ and the right Gorenstein subcategory $r\mathcal{G}(\mathscr{T})$ equals to the left orthogonal class of $\mathscr{T}$, we prove that the Gorenstein subcategory $\mathcal{G}(\mathscr{T})$ equals to the intersection of the left orthogonal class of $\mathscr{T}$ and the right orthogonal class of $\mathscr{T}$, and prove that their stable categories are triangle equivalent to the relative singularity category of $\mathscr{A}$ with respect to $\mathscr{T}$. As applications, let $R$ be a left Noetherian ring with finite left self-injective dimension and ${_{R}}C_{S}$ a semidualizing bimodule, and let the supremum of the flat dimensions of all injective left $R$-modules be finite. We prove that if $_{R}C$ has finite injective (or flat) dimension and the right orthogonal class of $C$ contains $R$, then there exists a triangle-equivalence between the intersection of $C$-Gorenstein projective modules and Bass class with respect to $C$, and the relative singularity category with respect to $C$-projective modules. Some classical results are generalized.