A higher dimensional Auslander-Iyama-Solberg correspondence

TL;DR

This paper extends the Auslander-Iyama-Solberg correspondence to higher dimensions, establishing a bijection between generalized relative Auslander-Gorenstein pairs and new classes of modules, with applications to modular representation theory.

Contribution

It introduces a higher dimensional version of the correspondence, replacing the module P with any self-orthogonal module Q of finite projective and injective dimension.

Findings

Established a bijection between relative Auslander-Gorenstein pairs and generalized precluster tilting modules.

Extended the correspondence to include modules Q with self-orthogonality and finite dimensions.

Connected the new theoretical framework to modular representation theory of general linear groups.

Abstract

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.