A higher dimensional Auslander-Iyama-Solberg correspondence

In this paper, we prove a higher dimensional version of Auslander-Iyama-Solberg correspondence. Iyama and Solberg have shown a bijection between n-minimal Auslander-Gorenstein algebras and n-precluster tilting modules. If A is an n-minimal Auslander-Gorenstein algebra, then the pair $(A,P)$ is a relative $(n+1)$-Auslander-Gorenstein pair in the sense of the authors, where P is the minimal faithful projective-injective left A-module. We establish a higher dimensional Auslander-Iyama-Solberg, where P is replaced by any self-orthogonal module Q having finite projective and injective dimension. This new correspondence provides a bijection between relative Auslander--Gorenstein pairs and a new class of objects that generalise precluster tilting modules. This way, we obtain a new correspondence coming from the modular representation theory of general linear groups.