# Article

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Keywords:
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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