Grothendieck groups in extriangulated categories

Abstract

Extriangulated categories were introduced by Nakaoka and Palu to give a unification of properties in exact categories and triangulated categories. We consider in this article the Grothendieck group K0(C) of an extriangulated category C. We show that a locally finite extriangulated category C always has almost split extensions and in this case the relations of the Grothendieck group K0(C) are generated by the Auslander-Rieten E-triangles. We give a partial converse result when restricting to the triangulated categories with a cluster tilting subcategory: in the triangulated category with a cluster tilting subcategory, the relations of the Grothendieck group are generated by AR-triangles if and only if the triangulated category is locally finite. We also show that there is a one-to-one correspondence between subgroups of K0(C) containing the image of G and dense G−(co)resolving subcategories of C where G a generator of C, which generalizes results about classifying subcategories of a triangulated or exact category C by subgroups of K0(C).