Among the nice features this category may possess is the existence of separating tubular families, introduced by Ringel in [12]. A well-known example of a class of algebras having a separating tubular family is the class of tame concealed algebras: in this case, the family consists of stable tubes. Further, Ringel introduced a notion of extension or coextension by branches using modules from a separating tubular family, then he showed that this process does not affect the existence of separating tubular families, so that the tilted algebras of euclidean type and the tubular algebras also possess such families [12]. Separating tubular families also occur in the module categories of wild algebras: this is the case, for instance, for all wild canonical algebras.
Abstract. We describe the structure of all indecomposable modules in standard coils of the Auslander-Reiten quivers of finite-dimensional algebras over an algebraically closed field. We prove that the supports of such modules are obtained from algebras with sincere standard stable tubes by adding braids of two linear quivers. As an application we obtain a complete classification of non-directing indecomposable modules over all strongly simply connected algebras of polynomial growth.Introduction. Let K be an algebraically closed field, and A a basic, connected, finite-dimensional K-algebra. We denote by mod A the category of finite-dimensional right A-modules, by ind A the full subcategory of mod A consisting of a complete set of non-isomorphic indecomposable A-modules, by Γ A the Auslander-Reiten quiver of A and by τ = τ A the AuslanderReiten translation in Γ A . We identify the vertices of Γ A with the corresponding modules in ind A, and the components of Γ A with the corresponding full subcategories of ind A. A component Γ of Γ A is called standard if Γ is equivalent to the mesh-category K(Γ ) of Γ (see [10], [24]). It was shown in [31] that every standard component of Γ A with infinitely many τ -orbits and without projective and injective modules is a stable tube, that is, a translation quiver of the form ZA ∞ /(τ r ) for some r ≥ 1. A module X in ind A is called directing if it does not lie on a cycle X = X 0 → X 1 → · · · → X r = X, r ≥ 1, of non-zero non-isomorphisms in ind A. The structure of directing modules is fairly well understood (see [8], [11], [12], [20], [21], [25]) because as shown in [24] their supports are tilted algebras. On the other hand, the Auslander-Reiten quiver Γ A of an algebra A admits at most finitely many τ A -orbits containing directing modules (see [22], [29]).
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