Tilted algebras and short chains of modules

Jaworska, Alicja; Malicki, Piotr; Skowroński, Andrzej

doi:10.1007/s00209-012-0993-0

Cited by 14 publications

(16 citation statements)

References 34 publications

(58 reference statements)

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“…In connection with the final part of the above proof, we mention that, by a recent result proved by Jaworska, Malicki and Skowroński in [29], an algebra A is a tilted algebra if and only if there exists a sincere module M in mod A such that for any module X in ind A, we have Hom A (X, M ) = 0 or Hom A (M, τ A X) = 0. Moreover, all modules M in a module category mod A not being the middle of short chains have been described completely in [30].…”

Section: Cyclic Componentsmentioning

confidence: 77%

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Finite cycles of indecomposable modules

Malicki

Peña

Skowroński

2015

Journal of Pure and Applied Algebra

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Abstract. We solve a long standing open problem concerning the structure of finite cycles in the category mod A of finitely generated modules over an arbitrary artin algebra A, that is, the chains of homomorphisms M 0 f 1 −−→ M 1 → · · · → M r−1 fr −−→ Mr = M 0 between indecomposable modules in mod A which do not belong to the infinite radical of mod A. In particular, we describe completely the structure of an arbitrary module category mod A whose all cycles are finite. The main structural results of the paper allow to derive several interesting combinatorial and homological properties of indecomposable modules lying on finite cycles. For example, we prove that for all but finitely many isomorphism classes of indecomposable modules M lying on finite cycles of a module category mod A the Euler characteristic of M is well defined and nonnegative. As an another application of these results we obtain a characterization of all cycle-finite module categories mod A having only a finite number of functorially finite torsion classes. Moreover, new types of examples illustrating the main results of the paper are presented. IntroductionThroughout the paper, by an algebra is meant an artin algebra over a fixed commutative artin ring K, which we shall assume (without loss of generality) to be basic and indecomposable. For an algebra A, we denote by mod A the category of finitely generated right A-modules and by ind A the full subcategory of mod A formed by the indecomposable modules. The Jacobson radical rad A of mod A is the ideal generated by all nonisomorphisms between modules in ind A, and the infinite radical rad An important combinatorial and homological invariant of the module category mod A of an algebra A is its Auslander-Reiten quiver Γ A . Recall that Γ A is a valued translation quiver whose vertices are the isomorphism classes {X} of modules X in ind A, the arrows correspond to irreducible homomorphisms between modules in ind A, and the translation is the Auslander-Reiten translation τ A = DTr.We shall not distinguish between a module in X in ind A and the corresponding vertex {X} of Γ A . If A is an algebra of finite representation type, then every nonzero nonisomorphism in ind A is a finite sum of composition of irreducible homomorphisms between modules in ind A, and hence we may recover mod A from the translation quiver Γ A . In general, Γ A describes only the quotient category mod A/rad ∞ A . Let A be an algebra and M a module in ind A. An important information concerning the structure of M is coded in the structure and properties of its support algebra Supp(M ) defined as follows. Consider a decomposition A = P M ⊕ Q M of A in mod A such that the simple summands of the semisimple module P M /radP M are exactly the simple composition factors of M . Then Supp(M ) = A/t A (M ), where t A (M ) is the ideal in A generated by the images of all homomorphisms from Q M to A in mod A.We note that M is an indecomposable module over Supp(M ). Clearly, we may realistically hope to describe the structure of Supp(M ) only ...

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Section: Cyclic Componentsmentioning

confidence: 77%

“…We would like to mention that there exist generalized double tilted algebras of infinite representation type of arbitrary global dimension d ∈ N ∪ {∞}. We refer also to [29], [40], [67] for useful characterizations of tilted algebras.…”

Section: Main Results and Related Backgroundmentioning

confidence: 99%

Finite cycles of indecomposable modules

Malicki

Peña

Skowroński

2015

Journal of Pure and Applied Algebra

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“…We note that, by a result proved in [24] and [37], an algebra A is a tilted algebra if and only if Γ A admits a component C with a faithful section ∆ such that Hom A (X, τ A Y ) = 0 for all modules X and Y from ∆. We refer also to [18] for another characterization of tilted algebras involving short chains of modules.…”

Section: Preliminary Resultsmentioning

confidence: 99%

On the number of terms in the middle of almost split sequences over cycle-finite artin algebras

2014

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“…The well-known criteria of Liu and Skowroński states that an algebra C is tilted if and only if Γ C contains a component C with a faithful section Σ such that Hom C (X, τ C Y) = 0 for all modules X, Y from Σ. A recent characterization of tilted algebras was established in [4]. We recall that a C-module M is sincere if Hom C (C, M) 0 for every indecomposable summand of C. In particular, let T Σ be the direct sum of modules lying on the section Σ.…”

Section: Tilted Algebrasmentioning

confidence: 99%