Gorenstein Dimension of Abelian Categories Arising from Cluster Tilting Subcategories

Abstract

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