Institute of Mathematics

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.