Gorenstein Subcategories and Relative Singularity Categories
Received:June 07, 2023  Revised:January 06, 2024
Key Words: abelian category   self-orthogonal   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).
Author NameAffiliation
Junfu WANG Changzhou College of Information Technology, Jiangsu 213164, P. R. China 
Tiwei ZHAO School of Artificial Intelligence, Jianghan University, Hubei 430056, P. R. China
School of Mathematical Sciences, Qufu Normal University, Shandong 273165, P. R. China 
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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.
Citation:
DOI:10.3770/j.issn:2095-2651.2024.03.004
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