A construction of non-regularly orbicular modules for Galois coverings. (English) Zbl 1121.16013

Dowbor, Piotr

J. Math. Soc. Japan 57, No. 4, 1077-1127 (2005).

Summary: 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 \(\Phi^{B(f,s)}\colon I_n\text{-spr}_{l(s)}(kG_B)\to\text{mod}(R/G)\), 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 group 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.

MSC:

16G10	Representations of associative Artinian rings
16G60	Representation type (finite, tame, wild, etc.) of associative algebras
16G50	Cohen-Macaulay modules in associative algebras

Keywords:

Galois coverings; Cohen-Macaulay modules; generalized tensor products; finite dimensional algebras; locally bounded \(k\)-categories; push-down functors; orbicular indecomposable modules; representation embeddings; skew group algebras