Gorenstein Subcategories and Relative Singularity Categories 
Received:June 07, 2023 Revised:January 06, 2024 
Key Words:
abelian category selforthogonal Gorenstein subcategories semidualizing bimodules

Fund Project:Supported by the Project of Natural Science Foundation of Changzhou College of Information Technology (Grant No.CXZK202204Y) and the Project of Youth Innovation Team of Universities of Shandong Province (Grant No.2022KJ314). 

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Abstract: 
Let $\mathscr{A}$ be an abelian category, $\mathscr{T}$ a selforthogonal 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 selfinjective 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 triangleequivalence 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. 
Citation: 
DOI:10.3770/j.issn:20952651.2024.03.004 
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