From Submodule Categories to the Stable Auslander Algebra

TL;DR

This paper constructs functors linking submodule categories of self-injective, representation-finite algebras to their stable Auslander algebras, revealing structural equivalences and characterizing Nakayama algebras through tilting modules.

Contribution

It introduces functors from submodule categories to stable Auslander algebras, establishing equivalences and characterizing Nakayama algebras via tilting modules.

Findings

Functors factor through the module category of the Auslander algebra.

Kernels of these functors have finitely many indecomposables.

Self-injective Nakayama algebras uniquely admit a characteristic tilting module.

Abstract

We construct two functors from the submodule category of a self-injective representation-finite algebra $\Lambda$ to the module category of the stable Auslander algebra of $\Lambda$. These functors factor through the module category of the Auslander algebra of $\Lambda$. Moreover they induce equivalences from the quotient categories of the submodule category modulo their respective kernels and said kernels have finitely many indecomposable objects up to isomorphism. Their construction uses a recollement of the module category of the Auslander algebra induced by an idempotent and this recollement determines a characteristic tilting and cotilting module. If $\Lambda$ is taken to be a Nakayama algebra, then said tilting and cotilting module is a characteristic tilting module of a quasi-hereditary structure on the Auslander algebra. We prove that the self-injective Nakayama algebras are the only algebras with this property.