Gorenstein Dimension of Abelian Categories Arising from Cluster Tilting Subcategories

Abstract

Let $$\\mathscr{C}$$ C be a triangulated category and $$\\mathscr{K}$$ K be a cluster tilting subcategory of $$\\mathscr{C}$$ C . Koenig and Zhu showed that the quotient category $$\\mathscr{C}$$ C / $$\\mathscr{K}$$ K is Gorenstein of Gorenstein dimension at most one. But this is not always true when $$\\mathscr{C}$$ 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 $$\\mathscr{C}$$ C be an extriangulated category with enough projectives and enough injectives, and $$\\mathscr{K}$$ K a cluster tilting subcategory of $$\\mathscr{C}$$ C . We show that under certain conditions, the quotient category $$\\mathscr{C}$$ C / $$\\mathscr{K}$$ K is Gorenstein of Gorenstein dimension at most one. As an application, this result generalizes the work by Koenig and Zhu.