Auslander-Gorenstein algebras and precluster tilting

Abstract

We generalize the notions of n-cluster tilting subcategories and τselfinjective algebras into n-precluster tilting subcategories and τn-selfinjective algebras, where we show that a subcategory naturally associated to n-precluster tilting subcategories has a higher Auslander–Reiten theory. Furthermore, we give a bijection between n-precluster tilting subcategories and n-minimal Auslander–Gorenstein algebras, which is a higher dimensional analog of Auslander– Solberg correspondence (Auslander–Solberg, 1993) as well as a Gorenstein analog of n-Auslander correspondence (Iyama, 2007). The Auslander–Reiten theory associated to an n-precluster tilting subcategory is used to classify the n-minimal Auslander–Gorenstein algebras into four disjoint classes. Our method is based on relative homological algebra due to Auslander–Solberg.