Non-orbicular modules for Galois coverings (Q2759167)

Let \(G\) be a group of \(k\)-linear automorphisms of a locally bounded \(k\)-category \(R\) acting freely on the objects of \(R\). Here \(k\) denotes a field. The action of \(G\) induces a quotient category \(R/G\) and a covering functor \(F\colon R\to R/G\) (see \textit{P. Gabriel} [Lect. Notes Math. 903, 68-105 (1981; Zbl 0481.16008)]).NEWLINENEWLINENEWLINEOne of the central problems in the theory of Galois coverings is to determine whether they preserve tame representation type (under some natural assumptions). An analysis of this question leads to the investigation of the category of modules of the ``second kind'' (see \textit{P. Dowbor} [Fundam. Math. 149, No. 1, 31-54 (1996; Zbl 0855.16018); Bull. Pol. Acad. Sci., Math. 44, No. 3, 341-352 (1996; Zbl 0889.16003)]) and several more detailed problems, in particular the question whether all indecomposable \(R/G\)-modules are orbicular in the tame situation (see also \textit{P. Dowbor} [J. Algebra 239, No. 1, 112-149 (2001; see the following review Zbl 0997.16008)]).NEWLINENEWLINENEWLINEAn indecomposable locally finite dimensional \(R\)-module \(B\) is called \(G\)-atom if its support is contained in the union of finitely many orbits of the induced action of the stabilizer of \(B\). An indecomposable locally finite dimensional \(R/G\)-module \(X\) is orbicular if the pull-up \(F_\bullet X=X\circ F\) is a direct sum of \(G\)-atoms belonging to one \(G\)-orbit.NEWLINENEWLINENEWLINEThe article contains a construction of (usually large families of) non-orbicular modules in the situation when there exists a sequence of \(G\)-atoms with special properties. Specialization of this result to the case when the sequence consists of an atom \(B\) and its left Kan extension \(\widetilde B\) yields (when \(B\) is not isomorphic to \(\widetilde B\)) a wild subcategory of non-orbicular \(R/G\)-modules. This suggests that it is important to look at the influence of the properties of \(G\)-atoms and their left Kan extensions on the representation type of \(R\); it is proved that if \(R\) is tame and \(G\) acts freely on \(R\) then \(B\cong\widetilde B\) for any infinite \(G\)-atom \(B\) with the top of the endomorphism ring isomorphic to \(k\). In order to do this, the author develops the technique of neighbourhoods of full subcategories of \(R\) and so called extension embeddings of matrix rings.