# A note on abelian quotient categories

## TL;DR

This paper explores conditions under which a subcategory of a triangulated category yields an abelian quotient, linking Serre functors and cluster tilting, and generalizes previous results by Beligiannis.

## Contribution

It establishes a characterization of cluster tilting subcategories via abelian quotient categories and Serre functor conditions, extending prior work in the field.

## Key findings

- X is cluster tilting iff C/X is abelian and S(X)=X[2]
- Generalizes Beligiannis's results on abelian quotients
- Provides new criteria for cluster tilting in triangulated categories

## Abstract

Let C be a triangulated category with a Serre functor S and X a non-zero contravariantly finite rigid subcategory of C. Then X is cluster tilting if and only if the quotient category C/X is abelian and S(X)=X[2]. As an application, this result generalizes work by Beligiannis.