Czechoslovak Mathematical Journal, Vol. 70, No. 2, pp. 483-504, 2020


One-sided Gorenstein subcategories

Weiling Song, Tiwei Zhao, Zhaoyong Huang

Received August 20, 2018.   Published online December 10, 2019.

Abstract:  We introduce the right (left) Gorenstein subcategory relative to an additive subcategory $\mathscr{C}$ of an abelian category $\mathscr{A}$, and prove that the right Gorenstein subcategory $r\mathcal{G}(\mathscr{C})$ is closed under extensions, kernels of epimorphisms, direct summands and finite direct sums. When $\mathscr{C}$ is self-orthogonal, we give a characterization for objects in $r\mathcal{G}(\mathscr{C})$, and prove that any object in $\mathscr{A}$ with finite $r\mathcal{G}(\mathscr{C})$-projective dimension is isomorphic to a kernel (or a cokernel) of a morphism from an object in $\mathscr{A}$ with finite $\mathscr{C}$-projective dimension to an object in $r\mathcal{G}(\mathscr{C})$. As an application, we obtain a weak Auslander-Buchweitz context related to the kernel of a hereditary cotorsion pair in $\mathscr{A}$ having enough injectives.
Keywords:  right Gorenstein subcategory; self-orthogonal subcategory; relative projective dimension; cotorsion pair; kernel; (weak) Auslander-Buchweitz context
Classification MSC:  18G25, 16E10, 18G10
DOI:  10.21136/CMJ.2019.0385-18


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Affiliations:   Weiling Song, Department of Applied Mathematics, College of Science, Nanjing Forestry University, 159 Longpan Road, Nanjing 210037, Jiangsu Province, P. R. China, e-mail: songwl@njfu.edu.cn; Tiwei Zhao (corresponding author), School of Mathematical Sciences, Qufu Normal University, 57 Jingxuan West Road, Qufu 273165, Shandong Province, P. R. China, e-mail: tiweizhao@qfnu.edu.cn; Zhaoyong Huang, Department of Mathematics, Nanjing University, Gulou Campus, No. 22, Hankou Road, Gulou District, Nanjing 210093, Jiangsu Province, P. R. China, e-mail: huangzy@nju.edu.cn


 
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