On generalized standard Auslander–Reiten components having only finitely many non-directing modules

Smith, David Stanley

doi:10.1016/j.jalgebra.2004.03.022

Cited by 7 publications

(17 citation statements)

References 28 publications

(51 reference statements)

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“…We recall that a component Γ of Γ (mod A) is quasi-directed [1,2,30] if it is generalized standard [26], that is, rad ∞ (X, Y ) = 0 for all X, Y in Γ , and almost directed , that is, it contains only finitely many non-directing modules. Moreover, Γ is convex if any path from X to Y , with X, Y in Γ , contains only modules from Γ .…”

Section: 4mentioning

confidence: 99%

“…Observe that the equivalence of (c) and (d) has been obtained in [30] under the assumption that the given modules X and Y belong to the same component of Γ (mod A); this assumption turns out to be unnecessary. As we see, we easily deduce from this corollary the equivalence of the statements (b), (c), (d) and (e) of our main theorem.…”

Section: 2mentioning

confidence: 99%

“…Moreover, we can assume that P 2 / ∈ Γ M . Consequently, f 1 ∈ rad ∞ (P 1 , P 2 ) and it follows from [21,27,30], for instance, that for each t ≥ 0 there exists a path…”

Section: The Infinite Radicalmentioning

confidence: 99%

“…In particular, it is shown in [29] that an algebra A is laura if and only if it is quasitilted or its Auslander-Reiten quiver contains a faithful quasi-directed component, which is, in this case, convex. In [30], …”

mentioning

confidence: 99%

“…In this paper, we introduce a notion of distance between indecomposable modules which allows us to unify the approaches used in [1] and [30]. We deduce from this new characterizations of laura algebras (see Sec.…”

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confidence: 99%

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Laura algebras and quasi-directed components

Lanzilotta

Smith

2006

Colloq. Math.

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Abstract. Using a notion of distance between indecomposable modules we deduce new characterizations of laura algebras and quasi-directed Auslander-Reiten components. Afterwards, we investigate the infinite radical of Artin algebras and show that there exist infinitely many non-directing modules between two indecomposable modules X and Y if rad ∞ A (X, Y ) = 0. We draw as inference that a convex component is quasi-directed if and only if it is almost directed.1. Introduction. The aim of the representation theory of Artin algebras is to study an algebra A by means of the category mod A of finitely generated right A-modules, which often turns out to be easier to handle. A key example arises in [12] where Happel, Reiten and Smalø show that an algebra A is quasitilted, that is, A is the endomorphism ring of a tilting object in a locally finite hereditary abelian category, if and only if A has global dimension at most two and any indecomposable A-module lies either in L A or in R A . Recall that L A is the full subcategory of mod A consisting of all indecomposable A-modules whose predecessors have projective dimension at most one and R A is defined dually.Following this example, the module category, and particularly L A and R A , gained in importance, and new classes of algebras, defined by the homological properties of their indecomposable modules, appeared: the shod algebras [7], the weakly shod algebras [9] and the laura algebras [1,20]; each of them generalizing the previous ones. We refer the reader to [2] for a complete review on these classes of algebras.Laura algebras have been introduced independently by Assem and Coelho [1] and Reiten and Skowroński [20] as a generalization of the representation-finite algebras and the weakly shod algebras. Their nice properties have made them rather interesting and hugely investigated; see, for instance, [1,20,29,2,3,11]. In particular, it is shown in [29] that an algebra A is laura if and only if it is quasitilted or its Auslander-Reiten quiver contains a faithful quasi-directed component, which is, in this case, convex. In [30],

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Section: 4mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

“…Moreover, we can assume that P 2 / ∈ Γ M . Consequently, f 1 ∈ rad ∞ (P 1 , P 2 ) and it follows from [21,27,30], for instance, that for each t ≥ 0 there exists a path…”

Section: The Infinite Radicalmentioning

confidence: 99%

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confidence: 99%

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Laura algebras and quasi-directed components

Lanzilotta

Smith

2006

Colloq. Math.

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Almost laura algebras

Smith

2008

Journal of Algebra

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In this paper, we propose a generalization for the class of laura algebras, called almost laura. We show that this new class of algebras retains most of the essential features of laura algebras, especially concerning the important role played by the non-semiregular components in their Auslander-Reiten quivers. Also, we study more intensively the left supported almost laura algebras, showing that these are characterized by the presence of a generalized standard, convex and faithful component. Finally, we prove that almost laura algebras behave well with respect to full subcategories, split-by-nilpotent extensions and skew group algebras.In the representation theory of algebras, a prevalent technique consists of modifying certain features of a well-known family of algebras in order to obtain one whose representation theory is, to a large extent, predictable. For instance, in [22], Happel, Reiten and Smalø defined the quasitilted algebras (that is the endomorphism algebras of tilting objects over a hereditary abelian category), thus obtaining a common treatment of both the classes of tilted and canonical algebras. To overcome some difficulties caused by the categorical language, they introduced the left and the right parts of the module category of an algebra A, respectively denoted L A and R A . They showed that an algebra A is quasitilted if and only if its global dimension is at most two and any indecomposable A-module lies in L A ∪ R A .

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