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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.
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

[1] Demonet, L., Liu, Y.: Quotients of exact categories by cluster tilting subcategories as module categories. J. Pure Appl. Algebra 217 (2013), 2282-2297. | DOI | MR | JFM

[2] Koenig, S., Zhu, B.: From triangulated categories to abelian categories: Cluster tilting in a general framework. Math. Z. 258 (2008), 143-160. | DOI | MR | JFM

[3] Liu, Y.: Abelian quotients associated with fully rigid subcategories. Available at , 14 pages. | arXiv

[4] Liu, Y., Nakaoka, H.: Hearts of twin cotorsion pairs on extriangulated categories. J. Algebra 528 (2019), 96-149. | DOI | MR | JFM

[5] Nakaoka, H., Palu, Y.: Extriangulated categories, Hovey twin cotorsion pairs and model structures. Cah. Topol. Géom. Différ. Catég. 60 (2019), 117-193. | MR | JFM

[6] Zhou, P., Zhu, B.: Triangulated quotient categories revisited. J. Algebra 502 (2018), 196-232. | DOI | MR | JFM

[7] Zhou, P., Zhu, B.: Cluster-tilting subcategories in extriangulated categories. Theory Appl. Categ. 34 (2019), 221-242. | MR | JFM

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