The Grothendieck groups of n-cluster categories of type A_∞

2025/09/25 by Huimin Chang, Chang, Huimin
Mathematics · #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2509.20661

openalex publication_date 2025/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article, we investigate the Grothendieck groups K₀(\C_A_∞ⁿ) of n-cluster categories \C_A_∞ⁿ of type A_∞ introduced by T.~Holm and P.~Jørgensen. We prove that K₀(\C_A_∞ⁿ)≅ℤ for an arbitrary n≥ 1. As an application, this generalizes a result of Murphy for n=1.

Citations

• The Grothendieck group of an extriangulated category
• The Grothendieck Groups of Discrete Cluster Categories of Dynkin Type A_∞
• Grothendieck groups of triangulated categories via cluster tilting\n subcategories

Related

• The Grothendieck Groups of Discrete Cluster Categories of Dynkin Type A_∞
• Grothendieck ∞-groupoids, and still another definition of ∞-categories
• Grothendieck groups of repetitive cluster categories
• Cluster structures for the A_∞ singularity
• Relations for Grothendieck groups of n-cluster tilting subcategories