Geodesic
Gorenstein dimension of abelian categories arising from cluster tilting subcategories
Czechoslovak Mathematical Journal, Tome 71 (2021) no. 2, pp. 435-453

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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.
DOI : 10.21136/CMJ.2021.0417-19
Classification : 18E10, 18G80
Keywords: extriangulated category; abelian category; cluster tilting subcategory; Gorenstein dimension 
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Liu, Yu; Zhou, Panyue. Gorenstein dimension of abelian categories arising from cluster tilting subcategories. Czechoslovak Mathematical Journal, Tome 71 (2021) no. 2, pp. 435-453. doi: 10.21136/CMJ.2021.0417-19

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