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$${\mathfrak U}$$-Fasersummen in darstellungsendlichen Algebren. ($${\mathfrak U}$$- fibresums in representation finite algebras). (German) Zbl 0659.16020
Let A be a finite-dimensional algebra over a field, and let $${\mathfrak U}$$ be an additive subcategory of the category A-mod of all finitely generated left A-modules. It is supposed additionally that $${\mathfrak U}$$ satisfies the conditions: $$Ext_ A({\mathfrak U},Sub {\mathfrak U})=0=Ext_ A({\mathfrak U},Fac {\mathfrak U})$$ (Sub $${\mathfrak U}$$ (Fac $${\mathfrak U})$$ means the additive subcategory of $${\mathfrak U}$$ consisting of all submodules (factormodules) of modules from $${\mathfrak U})$$. If $$M\in A$$-mod then $$Soc_{{\mathfrak U}}M=\{\sum$$ Im f: $$U\in {\mathfrak U}$$, $$f\in Hom_ A(U,M)\}$$ (“the $${\mathfrak U}$$-socle of M”). By $$\Gamma_{{\mathfrak U}}^ a$$subquiver of the Auslander-Reiten quiver of A is denoted; $$\Gamma_{{\mathfrak U}}$$ is generated by A-modules M with $$Soc_{{\mathfrak U}}M\neq 0$$. For any such $${\mathfrak U}^ a$$sort of a generalized amalgamated sum is determined, a so called $${\mathfrak U}$$-Fasersumme. It is shown that any indecomposable A-module M with $$Soc_{{\mathfrak U}}M\neq 0$$ is a $${\mathfrak U}$$-Fasersumme (Theorem 3.2). Theorem 3.3 and Corollary 3.5 characterize $$\Gamma_{{\mathfrak U}}$$ in terms of some idempotents of the basic algebra of A.
Reviewer: J.S.Ponizovskij

##### MSC:
 16Gxx Representation theory of associative rings and algebras 16P10 Finite rings and finite-dimensional associative algebras
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