2020/07/13 by Joseph Reid, Reid, Joseph · 1 citation
Mathematics · Physics and Astronomy · #Abelian category #Abelian group #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Combinatorics #Composition (language) #Derived category #Discrete mathematics #FOS: Mathematics #Finitely-generated abelian group #Functor #Grothendieck group #Indecomposable module #Isomorphism (crystallography) #Linguistics #Mathematics #Nonlinear Waves and Solitons #Pure mathematics #Representation Theory (math.RT) #Subcategory #Translation (biology) #math.RT
paper · pdf · doi:10.48550/arxiv.2007.06350
arxiv created 2020/07/13 · openalex publication_date 2020/07/13 · arxiv updated 2020/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
ABSTRACT. Let Φ be a finite dimensional K-algebra and let \mathscrC = \textrmmod Φ be the abelian category of finitely generated right Φ-modules. In their 1985 paper ``Modules determined by their composition factors'', Auslander and Reiten showed that under certain conditions modules in \textrmmod Φ are determined by their composition factors, and show an important formula related to the Auslander-Reiten translation. Let \mathscrT be a d-cluster tilting subcategory of \mathscrC, which by definition is also d-abelian. In this paper we will define the Grothendieck group for a d-abelian category, and show that the Grothendieck groups of \mathscrC and \mathscrT are isomorphic. We show also that under certain conditions, the indecomposable objects of \mathscrT are determined up to isomorphism by their composition factors in \mathscrC. Finally, we generalise the formula from Auslander and Reiten involving the higher dimensional Auslander-Reiten translation.