A characterization of cycle-finite generalized double tilted algebras

Skowyrski, Adam

doi:10.1016/j.jalgebra.2014.06.006

Cited by 7 publications

(2 citation statements)

References 33 publications

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“…Motivated by this fact, Skowroński studied in [26] cycles of maps of finite depth, which are originally called finite cycles. Indeed, Auslander-Reiten components whose non-directing modules lie only on cycles of maps of finite depth are described in [17], and algebras whose module category contains only cycles of maps of finite depth are extensively studied by many authors; see, for example, [16,17,23,26]. More generally, the connected components Γ of Γ A such that rad(Γ ) contains only short cycles of maps of finite depth are studied in [13].…”

Section: Introductionmentioning

confidence: 99%

Auslander–Reiten components with bounded short cycles

Liu

2017

Communications in Algebra

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We study Auslander-Reiten components of an artin algebra with bounded short cycles, namely, there exists a bound for the depths of maps appearing on short cycles of non-zero non-invertible maps between modules in the given component. First, we give a number of combinatorial characterizations of almost acyclic Auslander-Reiten components. Then, we show that an Auslander-Reiten component with bounded short cycles is closely related to the connecting component of a tilted quotient algebra. In particular, the number of such components is finite and each of them is almost acyclic with only finitely many DTr-orbits. As an application, we show that an artin algebra is representation-finite if and only if its module category has bounded short cycles. This includes a well known result of Ringel's, saying that a representation-directed algebra is representation-finite.

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Section: Introductionmentioning

confidence: 99%

Auslander–Reiten components with bounded short cycles

Liu

2017

Communications in Algebra

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“…The study of cycle-finite algebras have attached much attention. We refer to [10], [18], [19], [20], [33], [38], [39] for some general results on cycle-finite algebras and their module categories.…”

Section: Introductionmentioning

confidence: 99%