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 K 0 ( 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 K 0 ( 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 K 0 ( 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 K 0 ( C ) .