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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

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