2024/05/04 by Tiago Cruz, Cruz, Tiago, Chrysostomos Psaroudakis +1 · 1 citation
Mathematics · #16E10 #16E65 (Primary) 16G50 #16G10 #16G20 #16S50 #18G25 (Secondary) #20G43 #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2405.02736
openalex publication_date 2024/05/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.