Krull Dimension of Tame Generalized Multicoil Algebras

Abstract

We determine the Krull dimension of the module category of finite dimensional tame generalized multicoil algebras over an algebraically closed field, which are domestic.

On Krull-Gabriel dimension and Galois coverings

“…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.…”

The Krull-Gabriel Dimension of Cycle-Finite Artin Algebras

“…In particular, for n = 2, we have the following special case of Schröer's conjecture: for an algebra A of infinite representation, KG(A) = 2 if and only if m≥1 (rad ∞ A ) m = 0. It has been confirmed for the following classes of algebras: the tilted algebras of Euclidean type [22,23], the algebras stably equivalent to tame hereditary algebras [23], the algebras with directing indecomposable projective modules [65], the enveloping algebras of restricted Lie algebras [19] (more generally, the infinitesimal group schemes [20]) in odd characteristic, the strongly simply connected algebras [61,69], the 1-domestic string algebras [44,45,51], and recently the tame generalized multicoil algebras [31].…”