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Cycles in module categories. (English) Zbl 0819.16013

Dlab, V. (ed.) et al., Finite dimensional algebras and related topics. Proceedings of the NATO Advanced Research Workshop on Representations of algebras and related topics. Ottawa, Canada, August 10-18, 1992. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 424, 309-345 (1994).
Let \(A\) be an Artin algebra, and \(\text{mod }A\) be the category of finitely generated right \(A\)-modules. A cycle in \(\text{mod }A\) is a sequence \(M_ 0 \to M_ 1 \to \cdots \to M_ n = M_ 0\) of non-zero non-isomorphisms between indecomposable modules. It is known that \(A\) is representation-finite if and only if the infinite radical of \(\text{mod }A\) is zero. Otherwise, \(\text{mod }A\) always admits cycles. In fact, all but at most finitely many DTr-orbits in the Auslander-Reiten quiver \(\Gamma_ A\) of \(A\) consist entirely of modules lying on cycles. Moreover, those indecomposable modules not lying on a cycle have a simpler structure because their supports are tilted algebras. This shows that in order to obtain information on the remaining indecomposables, one needs to study the properties of cycles in \(\text{mod }A\). The aim of this paper is to present a survey of the results obtained in this direction. While this approach to the representation theory of Artin algebras is relatively recent, it has been dealt with surprising success (the bibliography of this survey paper contains 74 items!). Also, the author indicates new research directions and poses a number of open problems. He shows, for instance, that the indecomposable modules which lie only on large cycles or finite cycles are better behaved than the others. He also shows how the shape of the components of \(\Gamma_ A\) depends on the properties of cycles passing through the modules lying in these components. He studies the relation between the representation type of \(A\), the cycles in \(\text{mod }A\), the properties of the infinite radical, and the shape of the components of \(\Gamma_ A\).
For the entire collection see [Zbl 0797.00008].

MSC:

16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
16G10 Representations of associative Artinian rings
16G60 Representation type (finite, tame, wild, etc.) of associative algebras
16G20 Representations of quivers and partially ordered sets
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