2019/12/02 by Bin Zhu, Zhu, Bin, Xiao Zhuang +1
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1912.00621
openalex publication_date 2019/12/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 K0(\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 K0(\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 K0(\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 \citet or an exact category \C \citem by subgroups of K0(\C).