2019/03/27 by Ebrahimi, Ramin, Nasr-Isfahani, Alireza · 1 citation
#FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1903.11307
From the viewpoint of higher homological algebra, we introduce pure semisimple n-abelian category, which is analogs of pure semisimple abelian category. Let Λ be an Artin algebra and M be an n-cluster tilting subcategory of Mod-Λ. We show that M is pure semisimple if and only if each module in M is a direct sum of finitely generated modules. Let \mathfrakm be an n-cluster tilting subcategory of mod-Λ. We show that Add(\mathfrakm) is an n-cluster tilting subcategory of Mod-Λ if and only if \mathfrakm has an additive generator if and only if Mod(\mathfrakm) is locally finite. This generalizes Auslander's classical results on pure semisimplicity of Artin algebras.