Galois coverings of representation-infinite algebras

Algebras with cycle-finite Galois coverings

“…The main results of the paper extend results proved in [Bo2] for algebras of finite type, in [DS3], [ES], [Po], [PoS] for biserial algebras, and in [Sk9] for algebras with strongly simply connected (in the sense of [Sk3]) Galois coverings.…”

“…We denote by ind 1 R/G (respectively, ind 2 R/G) the full subcategory of mod 1 R/G (respectively, mod 2 R/G) formed by the indecomposable modules. Following [DS3] the modules from ind 1 R/G (respectively, ind 2 R/G) are called indecomposable modules of the first kind (respectively, indecomposable modules of the second kind ). The category R is said to be G-exhaustive if mod R/G = mod 1 R/G [DS3].…”

“…Following [DS3] the modules from ind 1 R/G (respectively, ind 2 R/G) are called indecomposable modules of the first kind (respectively, indecomposable modules of the second kind ). The category R is said to be G-exhaustive if mod R/G = mod 1 R/G [DS3]. From [DS3, (2.5)] we know that R is G-exhaustive provided R is locally support-finite, that is, for each object x of R, the full subcategory of R consisting of the objects of all supports supp M , where M is a module in ind R with M (x) = 0, is a finite category.…”

“…The problem whether the G-exhaustivity of R forces R to be a locally support-finite category still remains open, although it was confirmed in some important cases (see [AS5], [Sk9]). We refer also to [Do1], [Do2], [Do3], [Do4], [DS3] for some results on indecomposable R/G-modules of the second kind.…”

“…In order to deal with arbitrary algebras, covering techniques were introduced and developed at the beginning of the 1980s (see [BoG], [DS1], [DS3], [Ga]). Frequently, an algebra A admits a Galois covering R → R/G = A, where R is a triangular locally bounded category and G is a torsion-free group acting freely on the objects of R, which allows us to reduce the representation theory of A to that for the corresponding algebras of finite global dimension inside R. Moreover, Geiss proved in [Ge] (see also [CB3]) that if an algebra A admits a tame degeneration (in the variety of algebras of a fixed dimension), then A is tame.…”