Pure semisimple n-cluster tilting subcategories

From the viewpoint of higher homological algebra, we introduce pure semisimple n-abelian categories, which are analogs of pure semisimple abelian categories. Let A be an Artin algebra and M be an n-cluster tilting subcategory of Mod-A. We show that M is pure semisimple if and only if each module in M is a direct sum of finitely generated modules. Let m be an n-cluster tilting subcategory of mod-A. We show that Add(m) is an n-cluster tilting subcategory of Mod-A if and only if m has an additive generator if and only if Mod(m) is locally finite. This generalizes Auslander's classical results on pure semisimplicity of Artin algebras. (C) 2019 Elsevier Inc. All rights reserved.