Czechoslovak Mathematical Journal, first online, pp. 1-19

Gorenstein dimension of abelian categories arising from cluster tilting subcategories

Yu Liu, Panyue Zhou

Received September 19, 2019.   Published online February 3, 2021.

Abstract:  Let $\mathscr{C}$ be a triangulated category and $\mathscr{X}$ be a cluster tilting subcategory of $\mathscr{C}$. Koenig and Zhu showed that the quotient category $\mathscr{C}/\mathscr{X}$ is Gorenstein of Gorenstein dimension at most one. But this is not always true when $\mathscr{C}$ becomes an exact category. The notion of an extriangulated category was introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. Now let $\mathscr{C}$ be an extriangulated category with enough projectives and enough injectives, and $\mathscr{X}$ a cluster tilting subcategory of $\mathscr{C}$. We show that under certain conditions, the quotient category $\mathscr{C}/\mathscr{X}$ is Gorenstein of Gorenstein dimension at most one. As an application, this result generalizes the work by Koenig and Zhu.
Keywords:  extriangulated category; abelian category; cluster tilting subcategory; Gorenstein dimension
Classification MSC:  18G80, 18E10
DOI:  10.21136/CMJ.2021.0417-19

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Affiliations:   Yu Liu, School of Mathematics, Southwest Jiaotong University, 610031, 111 N 1st Section, 2nd Ring Rd, Sha Xi Mei Shi Yi Tiao Jie, Jinniu District, Chengdu, Sichuan, P. R. China, e-mail:; Panyue Zhou (corresponding author), College of Mathematics, Hunan Institute of Science and Technology, Xueyuan Rd, 414006, Yueyang, Hunan, P. R. China, e-mail:

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