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On tame algebras and bocses. (English) Zbl 0661.16026
Let \({\mathcal A}\) be a finite dimensional algebra over an algebraically closed field. The author proves a stronger version of Yu. Drozd’s “Wild and Tame Theorem” [Lect. Notes Math. 832, 242-258 (1980; Zbl 0457.16018)] which says that for \({\mathcal A}\) either in each dimension d there exist only finitely many 1-parameter families of indecomposable \({\mathcal A}\)-modules containing all, up to isomorphism, d-dimensional indecomposables, or the module theory of \({\mathcal A}\) is as bad as that of \(k<x,y>\), the free associative algebra on two indeterminates. His new proof, as the original one, relies also on proving of the analogous theorem in the theory of representations of BOCSes (i.e. bimodules over the category equipped with a coalgebra structure) and then applying the procedure attaching to \({\mathcal A}\) some BOCS \(\underline {\mathcal A}\) in such a way that the categories of representations of \({\mathcal A}\) and \(\underline {\mathcal A}\) are nicely related. Finally the author concludes that if \({\mathcal A}\) is tame then in each dimension almost all indecomposable modules are isomorphic to their Auslander-Reiten translates and in consequence lie in homogeneous tubes.
Reviewer: P.Dowbor

16G10 Representations of associative Artinian rings
16P10 Finite rings and finite-dimensional associative algebras
16B50 Category-theoretic methods and results in associative algebras (except as in 16D90)
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