Gorenstein Dimension of Abelian Categories Arising from Cluster Tilting Subcategories.

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. [ABSTRACT FROM AUTHOR]