Grothendieck groups in extriangulated categories (1912.00621v4)

Abstract: The aim of the paper is to discuss the relation subgroups of the Grothendieck groups of extriangulated categories and certain other subgroups. It is shown that a locally finite extriangulated category $\C$ has Auslander-Reiten $\E-$triangles and the relations of the Grothendieck group $K_{0}(\C)$ are generated by the Auslander-Rieten $\E-$triangles. A partial converse result is given when restricting to the triangulated categories with a cluster tilting subcategory: in the triangulated category $\C$ with a cluster tilting subcategory, the relations of the Grothendieck group $K_0(\C)$ are generated by Auslander-Reiten triangles if and only if the triangulated category $\C$ is locally finite. It is also shown that there is a one-to-one correspondence between subgroups of $K_{0}(\C)$ containing the image of $\mathcal G$ and dense $\mathcal G-$(co)resolving subcategories of $\C$ where $\mathcal G$ is a generator of $\C,$ which generalizes results about classifying subcategories of a triangulated \cite{t} or an exact category $\C$ \cite{m} by subgroups of $K_{0}(\C)$.