Quotients of exact categories by cluster tilting subcategories as module categories

TL;DR

This paper demonstrates that certain subquotient categories derived from exact categories are abelian, extending known results from triangulated categories to a broader exact category context, especially involving cluster tilting subcategories.

Contribution

It generalizes the result that quotients by cluster tilting subcategories are abelian from triangulated to exact categories, establishing an equivalence with module categories.

Findings

Subquotient categories of exact categories are abelian.

B/M is equivalent to finitely presented modules over the stable category of M.

Extension of known triangulated category results to exact categories.

Abstract

We prove that some subquotient categories of exact categories are abelian. This generalizes a result by Koenig-Zhu in the case of (algebraic) triangulated categories. As a particular case, if an exact category B with enough projectives and injectives has a cluster tilting subcategory M, then B/M is abelian. More precisely, it is equivalent to the category of finitely presented modules over the stable category of M.