Algebras with cycle-finite Galois coverings

Abstract

Abstract. We prove that the finite dimensional algebras over an algebraically closed field which admit cycle-finite Galois coverings with torsion-free Galois groups are of tame representation type, and derive some consequences.

The Krull-Gabriel Dimension of Cycle-Finite Artin Algebras

“…The class of cycle-finite algebras is wide and contains the following distinguished classes of algebras: the algebras of finite representation type, the tame tilted algebras [24,49], the tame double tilted algebras [46], the tame generalized double tilted algebras [47], the tubular algebras [49,50], the tame quasi-tilted algebras [63], the tame generalized multicoil algebras [36], and the strongly simply connected algebras of polynomial growths [61]. Moreover, frequently interesting algebras admit Galois coverings by cycle-finite locally bounded categories, and applying covering techniques we may reduce their representation theory to that for the corresponding cycle-finite algebras (see [1,16,38,54,62,64,66]). The study of cycle-finite algebras has attracted much attention.…”

On the number of terms in the middle of almost split sequences over cycle-finite artin algebras

“…The class of cycle-finite algebras contains the following distinguished classes of algebras: the algebras of finite representation type, the hereditary algebras of Euclidean type [14], [15], the tame tilted algebras [17], [19], [35], the tame double tilted algebras [32], the tame generalized double tilted algebras [33], the tubular algebras [35], the iterated tubular algebras [30], the tame quasi-tilted algebras [22], [43], the tame generalized multicoil algebras [26], the algebras with cycle-finite derived categories [2], and the strongly simply connected algebras of polynomial growth [41]. On the other hand, frequently an algebra A admits a Galois covering R → R/G = A, where R is a cycle-finite locally bounded category and G is an admissible group of automorphisms of R, which allows to reduce the representation theory of A to the representation theory of cycle-finite algebras being finite convex subcategories of R (see [16], [28], [42] for some general results). For example, every finite dimensional selfinjective algebra of polynomial growth over an algebraically closed field admits a canonical standard form A (geometric socle deformation of A) such that A has a Galois covering R → R/G = A, where R is a cycle-finite selfinjective locally bounded category and G is an admissible infinite cyclic group of automorphisms of R, the Auslander-Reiten quiver Γ A of A is the orbit quiver Γ R /G of Γ R , and the stable Auslander-Reiten quivers of A and A are isomorphic (see [36] and [44]).…”

“…Then α(X) measures the complication of homomorphisms in mod A with domain τ A X and codomain X. Therefore, it is interesting to study the relation between an algebra A and the values α(X) for all modules X in ind A (we refer to [6], [8], [10], [21], [25], [28], [29], [31], [45], [46] for some results in this direction). In particular, it has been proved by R. Bautista and S. Brenner in [8] that, if A is of finite representation type and X a nonprojective module in ind A, then α(X) ≤ 4, and if α(X) = 4 then the middle term Y of an almost split sequence in mod A with the right term X admits an indecomposable projective-injective direct summand P , and hence X = P/soc(P ).…”

Cycle-Finite Module Categories

“…Moreover, frequently an algebra A admits a Galois covering R → R/G = A where R is a cycle-finite locally bounded category and G is an admissible group of automorphisms of R, which allows to reduce the representation theory of A to the representation theory of cycle-finite algebras being finite convex subcategories of R. For example, every selfinjective algebra A of polynomial growth admits a canonical standard form A (geometric socle deformation of A) such that A has a Galois covering R → R/G = A, where R is a cycle-finite selfinjective locally bounded category and G is an admissible infinite cyclic group of automorphisms of R, the Auslander-Reiten quiver Γ A of A is the orbit quiver Γ R /G of Γ R , and the stable Auslander-Reiten quivers of A and A are isomorphic (see [69], [80] for details). We also mention that, by the main result of [59], every algebra A which admits a cycle-finite Galois covering R → R/G = A with G torsion-free is tame.…”