On the category of modules of second kind for Galois coverings

Abstract. Let F : R→R/G be a Galois covering and mod 1 (R/G) (resp. mod 2 (R/G)) be a full subcategory of the module category mod(R/G), consisting of all R/G-modules of first (resp. second) kind with respect to F . The structure of the categories (mod(R/G))/ [mod 1 (R/G)] and mod 2 (R/G) is given in terms of representation categories of stabilizers of weakly-G-periodic modules for some class of coverings. Introduction. The covering technique in representation theory was introduced and developed for the investigation of representation-finite algebras and computing their representations (see [G], [Gr], [BG], [R]). It has been generalized and applied to the study of representation-infinite algebras (see [DS1], [DLS], [DS2], [P]). The covering methods in representation theory of algebras over a field k are based on interpretation of modules over the algebra as representations of some quiver with relations, or more generally modules over a locally bounded category. Following [BG] a k-category R is called locally bounded if all objects of R have local endomorphism rings, the different objects are nonisomorphic, and both sums y∈R dim k R(x, y) and y∈R dim k R(y, x) are finite for each x ∈ R. R-modules are then contravariant k-linear functors from R to the category of k-vector spaces. An R-module M is locally finitedimensional (resp. finite-dimensional ) if dim k M (x) is finite for each x ∈ R (resp. x∈R dim k M (x) is finite). We denote by MOD R the category of all R-modules, by Mod R (resp. mod R) the full subcategory of all locally finitedimensional (resp. finite-dimensional) R-modules and by Ind R (resp. ind R) the full subcategory of indecomposable objects of Mod R (resp. mod R)