From Submodule Categories to the Stable Auslander Algebra

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.

22 pages. Diagrams changed to tikz-cd. Several citations added as well as small corrections since V2

References in corpus (2)

• Objective triangle functors
• Monomorphism categories, cotilting theory, and Gorenstein-projective modules

Cited by in corpus (7)

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• A new characterization of Auslander algebras
• Abelian quotients arising from extriangulated categories via morphism categories
• On strongly quasi-hereditary algebras