From submodule categories to the stable Auslander algebra

Eiriksson, Ögmundur

doi:10.1016/j.jalgebra.2017.05.012

Cited by 20 publications

(15 citation statements)

References 26 publications

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“…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.…”

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confidence: 99%

A new characterization of Auslander algebras

Shen

Zhang

2017

J. Algebra Appl.

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Let Λ be a finite dimensional Auslander algebra. For a Λ-module M , we prove that the projective dimension of M is at most one if and only if the projective dimension of its socle soc M is at most one. As an application, we give a new characterization of Auslander algebra Λ, and prove that a finite dimensional algebra Λ is an Auslander algebra provided its global dimension gl.d Λ ≤ 2 and an injective Λ-module is projective if and only if the projective dimension of its socle is at most one.

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confidence: 99%

A new characterization of Auslander algebras

Shen

Zhang

2017

J. Algebra Appl.

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“…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.…”

Section: Introductionmentioning

confidence: 57%

“…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.…”

Section: Introductionmentioning

confidence: 99%

Abelian quotients of categories of short exact sequences

Lin

2020

Journal of Algebra

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We mainly investigate abelian quotients of the categories of short exact sequences. The natural framework to consider the question is via identifying quotients of morphism categories as modules categories. These ideas not only can be used to recover the abelian quotients produced by clustertilting subcategories of both exact categories and triangulated categories, but also can be used to reach our goal. Let (C, E) be an exact category. We denote by E(C) the category of bounded complexes whose objects are given by short exact sequences in E and by SE(C) the full subcategory formed by split short exact sequences. In general, E(C) is just an exact category, but the quotient E(C)/[SE(C)] turns out to be abelian. In particular, if (C, E) is Frobenius, we present three equivalent abelian quotients of E(C) and point out that the equivalences are actually given by left and right rotations. The abelian quotient E(C)/[SE(C)] admits some nice properties. We explicitly describe the abelian structure, projective objects, injective objects and simple objects, which provide a new viewpoint to understanding Hilton-Rees Theorem and Auslander-Reiten theory. Furthermore, we present some analogous results both for n-exact versions and for triangulated versions.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Strongly Quasi-Hereditary Algebras and Rejective Subcategories

Tsukamoto¹

2018

Nagoya Math. J.

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Ringel's right-strongly quasi-hereditary algebras are a distinguished class of quasi-hereditary algebras of Cline-Parshall-Scott. We give characterizations of these algebras in terms of heredity chains and right rejective subcategories. We prove that any artin algebra of global dimension at most two is right-strongly quasi-hereditary. Moreover we show that the Auslander algebra of a representation-finite algebra A is strongly quasi-hereditary if and only if A is a Nakayama algebra.2010 Mathematics Subject Classification. 16G10, 18A40.

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From submodule categories to the stable Auslander algebra

Cited by 20 publications

References 26 publications

A new characterization of Auslander algebras

A new characterization of Auslander algebras

Abelian quotients of categories of short exact sequences

Strongly Quasi-Hereditary Algebras and Rejective Subcategories

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