Algebras determined by their left and right parts

Assem, Ibrahim; Coelho, Flávio U.; Lanzilotta, Marcelo; Smith, David Stanley; Trepode, Sonia

doi:10.1090/conm/376/06949

Cited by 14 publications

(22 citation statements)

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“…The module category of a quasi-tilted algebra has been described in [11,15,16]. Several generalisations of this notion have been introduced over the years (see [4] for a survey). We recall the definition of laura algebras [3]: an algebra Λ is laura if L Λ ∪ R Λ is cofinite in indΛ.…”

Section: 4mentioning

confidence: 99%

From Trisections in Module Categories to Quasi-Directed Components

et al. 2011

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Section: 4mentioning

confidence: 99%

From Trisections in Module Categories to Quasi-Directed Components

et al. 2011

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“…In particular, we show that such components are convex and quasi-directed. Recall that a component Γ of Γ (mod A) is said to be generalized standard [28] [3], if it contains at most finitely many modules lying on cycles in ind A. We stress that the cycles considered in this definition are all cycles in ind A and not only cycles of irreducible morphisms or cycles in Γ .…”

Section: Hook-bounded Components Are Quasi-directedmentioning

confidence: 99%

“…We enlarge this definition as follows: an Auslander-Reiten component is called quasi-directed if it is generalized standard and contains only finitely many modules lying on cycles (of non-necessarily irreducible morphisms). This terminology was already used in [3]. We stress that these two definitions are equivalent for laura algebras since any "quasi-directed" component of a laura algebra is convex.…”

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confidence: 99%

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

Smith

2004

Journal of Algebra

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Let Γ be a connected component of the Auslander-Reiten quiver Γ (mod A) of an Artin algebra. We show that Γ is quasi-directed if and only if the paths in ind A, having end-points in Γ , satisfy certain conditions. Moreover, we describe the shapes of such components and identify the classes of algebras to which they are related.  2004 Elsevier Inc. All rights reserved.Let A be an Artin algebra. We are interested in studying the representation theory of A, thus in characterizing A by properties of the module category mod A of finitely generated right modules. One way to achieve this goal is to take a look at the paths in ind A, a full subcategory of mod A consisting of one representative from each isomorphism class of indecomposable modules. For instance, the quasi-tilted algebras [14], the shod algebras [9,21], the weakly shod algebras [10,11] and, recently, the laura algebras [2,22,33] are several important classes of algebras which have been totally characterized by properties of paths

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“…In the same work [14], they have also studied, for an algebra , two subcategories L and R of the module category mod . These subcategories played an important role in subsequent papers [1,3,4,7,14,18] and will be very useful in our considerations.…”

Section: Introductionmentioning

confidence: 98%

On the Derived Categories and Quasitilted Algebras

Coelho

Tosar²

2008

Algebr Represent Theor

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In this paper, we present some results on the bounded derived category of Artin algebras, and in particular on the indecomposable objects in these categories, using homological properties. Given a complex X * , we consider the set J X * = {i ∈ Z | H i (X * ) = 0} and we define the application l(X * ) = maxJ X * − minJ X * + 1. We give relationships between some homological properties of an algebra and the respective application l. On the other hand, using homological properties again, we determine two subcategories of the bounded derived category of an algebra, which turn out to be the bounded derived categories of quasi-tilted algebras. As a consequence of these results we obtain new characterizations of quasi-tilted and quasi-tilted gentle algebras.

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Algebras determined by their left and right parts

Cited by 14 publications

References 0 publications

From Trisections in Module Categories to Quasi-Directed Components

From Trisections in Module Categories to Quasi-Directed Components

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

On the Derived Categories and Quasitilted Algebras

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