The Krull-Gabriel Dimension of Cycle-Finite Artin Algebras

Abstract

We determine the Krull-Gabriel dimension of the cycle-finite categories of finitely generated modules over artin algebras and derive some consequences.

On Krull-Gabriel dimension and Galois coverings

“…Moreover, J. Schröer conjectures in [45] that KG(A) = n ≥ 2 if and only if rad ω(n−1) A = 0 and rad ωn A = 0. This conjecture is confirmed for several important classes of algebras, see for example Section 1 in [52] for the list.…”

“…H. Krause shows in [25, 11.4] that KG(A) = 1 for any algebra A. W. Geigle proves in [15, 4.3] that if A is a tame hereditary algebra, then KG(A) = 2. A. Skowroński shows in [52,Theorem 1.2] that if A is a cycle-finite algebra [2], [3] of domestic representation type, then KG(A) = 2, see also [29]. M. Wenderlich proves in [54] that if A is a strongly simply connected algebra [50], then A is of domestic type if and only if KG(A) is finite.…”

“…Indeed, we prove in Theorem 7.3 that if A is such an algebra, then KG(A) = 2 if A is domestic and KG(A) = ∞ otherwise. A crucial ingredient in the proof of this result is [52,Theorem 1.2] on the Krull-Gabriel dimension of cycle-finite algebras. Note that Theorem 7.3 supports Conjecture 1.1.…”

“…It follows from [52] that if A is a cycle-finite algebra, then KG(A) = 2. This fact is crucial in proof of the following result.…”

Cycles of modules and finite representation type

“…In fact, in the course of the proof of Theorem 1.1, we establish the following deeper facts. The main results of the paper have been recently applied in [41] to determine the Krull-Gabriel dimension of cycle-finite algebras.…”