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Keywords:
extriangulated category; abelian category; cluster tilting subcategory; Gorenstein dimension
extriangulated category; abelian category; cluster tilting subcategory; Gorenstein dimension
Summary:
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.
References:
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