Gorenstein dimension of abelian categories

TL;DR

This paper extends the understanding of Gorenstein properties of quotient categories from triangulated to extriangulated categories, broadening the scope of algebraic structures with controlled Gorenstein dimension.

Contribution

It generalizes previous results by Koenig and Zhu to extriangulated categories with enough projectives and injectives, establishing Gorenstein dimension bounds for their quotients.

Findings

Quotient categories are Gorenstein of Gorenstein dimension at most one.

Generalization of Koenig and Zhu's work to extriangulated categories.

Conditions identified under which the Gorenstein property holds.

Abstract

Let C be triangulated category and X a cluster tilting subcategory of C. Koenig and Zhu showed that the quotient category C/X is Gorenstein of Gorenstein dimension at most one. The notion of an extriangulated category was introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. Now let C be extriangulated category with enough projectives and enough injectives, and X a cluster tilting subcategory of C. In this article, we show that under certain conditions the quotient category C/X is Gorenstein of Gorenstein dimension at most one. As an application, this result generalizes work by Koenig and Zhu.