From submodule categories to the stable Auslander algebra

Abstract

We construct two functors from the submodule category of a selfinjective representation-finite algebra Λ to the module category of the stable Auslander algebra of Λ. These functors factor through the module category of the Auslander algebra of Λ. 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 Λ 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.

A new characterization of Auslander algebras

“…Theorem B also implies that pd Λ M = 2 if and only if pd Λ soc M = 2. Before we complete this paper, Eiríksson gives a characterization of P 1 (Λ) in [7], which states that P 1 (Λ) consists of all Λ-modules cogenerated by projective Λ-modules. We investigate P 1 (Λ) from a different point of view and stress that the projective dimension of a Λmodule M is completely determined by the projective dimension of its socle.…”

Abelian quotients of categories of short exact sequences

“…If (C, E) is Frobenius, we can show that the three categories E(C), Mono(C) and Epi(C) are equivalent. Therefore, we have the following result, which generalizes [13,Theorem 1]. Corollary 1.…”

“…They showed that the quotient categories S(A)/[U 1 ] and S(A)/[U 2 ] are equivalent to mod-Π n−1 where Π n−1 is the preprojective algebra of type A n−1 . Recently due to Eiríksson [13,Theorem1], the above result was generalized for any self-injective algebra of finite representation type by replacing Π n−1 with B, the stable Auslander algebra of A.…”

Strongly Quasi-Hereditary Algebras and Rejective Subcategories

“…Also certain important algebras associated with preprojective algebras and elements in Coxeter groups are known to be right-strongly quasi-hereditary, e.g., [15, 19]. We refer to [6, 7, 13] for recent results on right-strongly quasi-hereditary algebras.…”