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Paper   IPM / M / 16025
School of Mathematics
  Title:   Pure semisimple n-cluster tilting subcategories
  Author(s):  Alireza Nasr-Isfahani (Joint with R. Ebrahimi)
  Status:   Published
  Journal: J. Algebra
  Vol.:  549
  Year:  2020
  Pages:   177-194
  Supported by:  IPM
From the viewpoint of higher homological algebra, we intro-duce pure semisimple n-abelian categories, which are analogs of pure semisimple abelian categories. Let �?be an Artin al-gebra and Mbe an n-cluster tilting subcategory of Mod-�?. We show that Mis pure semisimple if and only if each mod-ule in Mis a direct sum of finitely generated modules. Let mbe an n-cluster tilting subcategory of mod-�?. We show that Add(m)is an n-cluster tilting subcategory of Mod-�?if and only if mhas an additive generator if and only if Mod(m)is locally finite. This generalizes Auslander�??s classical results on pure semisimplicity of Artin algebras

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