Auslander-Gorenstein algebras and precluster tilting

Abstract

We generalize the notions of $n$-cluster tilting subcategories and $\tau$-selfinjective algebras into $n$-precluster tilting subcategories and ${\tau }_n$-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.

32 pages, final published version. The terminology `n-minimal Auslander-Gorenstein algebra' is used

References in corpus (2)

• d-Representation-finite self-injective algebras
• Higher Auslander-Reiten sequences and $t$-structures

Cited by in corpus (4)

• Gorenstein modifications and $\mathbb{Q}$-Gorenstein rings
• Dominant dimension and tilting modules
• Auslander-Gorenstein algebras, standardly stratified algebras and dominant dimensions
• Nakayama-type phenomena in higher Auslander--Reiten theory