Grothendieck groups in extriangulated categories

TL;DR

This paper explores the structure of Grothendieck groups in extriangulated categories, establishing conditions under which their relations are generated by Auslander-Reiten triangles and classifying subcategories via subgroups.

Contribution

It provides new insights into the relation subgroups of Grothendieck groups in extriangulated categories and links subgroups to dense (co)resolving subcategories, extending existing classification results.

Findings

Relations in $K_0( ext{C})$ are generated by Auslander-Reiten $ ext{E}$-triangles in locally finite categories.

In triangulated categories with a cluster tilting subcategory, relations are generated by Auslander-Reiten triangles iff the category is locally finite.

There is a one-to-one correspondence between subgroups of $K_0( ext{C})$ containing a generator's image and dense (co)resolving subcategories.

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 $K_{0}(\C)$.