A construction of non-regularly orbicular modules for Galois coverings

Abstract

For a given finite dimensional $k$-algebra $A$ which admits a presentation in the form $R/G$, where $G$ is an infinite group of $k$-linear automorphisms of a locally bounded $k$-category $R$, a class of modules lying out of the image of the "push-down" functor associated with the Galois covering $R\to R/G$, is studied. Namely, the problem of existence and construction of the so called non-regularly orbicular indecomposable $R/G$-modules is discussed. For a $G$-atom $B$ (with a stabilizer $G_B$), whose endomorphism algebra has a suitable structure,a representation embedding ${\mathrm{\Phi }}^{B(f,s)}{}_{|}:I_n-{\mathrm{s}\mathrm{p}\mathrm{r}}_{l\left(s\right)}\left(k{G}_B\right)\to \mathrm{m}\mathrm{o}\mathrm{d}(R/G)$, which yields large families of non-regularly orbicular indecomposable $R/G$-modules,is constructed (Theorem 2.2). An important role in consideration is played by a result interpreting some class of $R/G$-modules in terms of Cohen-Macaulay modules over certain skew grup algebra (Theorem 3.3). Also, Theorems 4.5 and 5.4, adapting the generalized tensor product construction and Galois covering scheme, respectively, for Cohen-Macaulay modules context, are proved and intensively used.