Galois coverings of representation-infinite algebras
If R is a locally bounded K-category, K some algebraically closed field, G a group of K-linear automorphisms of R acting freely on the objects of R, then it was proved by \textit{P. Gabriel} [in: Representations of algebras, Lect. Notes Math. 903, 68-105 (1981; Zbl 0481.16008), \textit{K. Bongartz}, \textit{P. Gabriel}, Invent. Math. 65, 331-378 (1982; Zbl 0482.16026), \textit{R. Martínez-Villa}, \textit{J. A. de la Peña}, Invent. Math. 72, 359-362 (1983; Zbl 0491.16028)] for example that R is locally representation finite if and only if so is R/G. Moreover the push down functor \(F_{\lambda}: mod R\to mod (R/G)\) induces a bijection between G-orbits of indecomposable R-modules and indecomposable R/G- modules. If R is not locally representation finite in general the functor \(F_{\lambda}\) is not surjective on object classes. The authors successfully investigate the full subcateogry \(mod_ 2(R/G)\) of mod (R/G) defined by those modules having no direct summand isomorphic to \(F_{\lambda}(M)\) for some \(M\in mod R.\) As a first result they state in (2.3): \(X\in mod_ 2(R/G)\) iff there exists a decomposition \(F.(X)=\oplus Y_ i\) (F. the pull up functor) where all \(Y_ i\) are weakly-G-periodic. (Y is called weakly periodic if the support supp Y of Y is infinite and \((supp\;Y)/G_ Y\) is finite, \(G_ Y\) the stabilizer of Y.) Main result of the paper is the following Theorem: Let R be a locally bounded K-category and G a group of automorphisms of R which acts freely on \((ind\;R)/\cong\). Let \({\mathfrak S}\) be a separating family of subcategories of R with respect to G and \({\mathfrak S}_ 0\) a fixed set of representatives of G-orbits of \({\mathfrak S}\). Then there exists an equivalence of categories \[ E:\coprod_{L\in {\mathfrak S}_ 0}(mod L/G_ L)/[mod_ 1L/G_ L]\to (mod R/G)/[mod_ 1R/G]. \] As a consequence, the Auslander-Reiten quiver \(\Gamma_{R/G}\) of R/G is isomorphic to the disjoint union of translation-quivers \(\Gamma_ R/G\coprod (\coprod_{L\in {\mathfrak S}_ 0}(\Gamma_{L/G_ L})_ 2)\), where \((\Gamma_{L/G_ L})_ 2\) is the union of connected components of \(\Gamma_{L/G_ l}\) whose points are \(L/G_ L\)-modules of second kind. (For the definition of a separating family of subcategories see (3.1) of the paper.) In {\S} 4 the authors show that each indecomposable locally finite dimensional R-module X is a limit of a sequence of finite dimensional (indecomposable) modules \((Y_ n)\), the fundamental R- sequence produced by S. {\S} 5 finally is devoted to carefully chosen examples and applications.
indecomposable locally finite dimensional R-module
scientific article; zbMATH DE number 4021386