Abelian Categories Arising from Cluster Tilting Subcategories

摘要

For a triangulated category T, if C is a cluster-tilting subcategory of T, then the factor category T/C is an abelian category. Under certain conditions, the converse also holds. This is a very important result of cluster-tilting theory, due to Koenig-Zhu and Beligiannis. Now let B be a suitable extriangulated category, which is a simultaneous generalization of triangulated categories and exact categories. We introduce the notion of pre-cluster tilting subcategory C of B, which is a generalization of cluster tilting subcategory. We show that C is cluster tilting if and only if the factor category B/C is abelian. Our result generalizes the related results on a triangulated category and is new for an exact category case.