On Krull-Gabriel dimension of cluster repetitive categories and cluster-tilted algebras

Assume that K is an algebraically closed field and denote by KG ( R ) the Krull-Gabriel dimension of R , where R is a locally bounded K -category (or a bound quiver K -algebra). Assume that C is a tilted K -algebra and C ˆ , C ˇ , C ˜ are the associated repetitive category, cluster repetitive category and cluster-tilted algebra, respectively. Our first result states that KG ( C ˜ ) = KG ( C ˇ ) ≤ KG ( C ˆ ) . Since the Krull-Gabriel dimensions of tame locally support-finite repetitive categories are known, we further conclude that KG ( C ˜ ) = KG ( C ˇ ) = KG ( C ˆ ) ∈ { 0 , 2 , ∞ } . Finally, in the Appendix Grzegorz Bobiński presents a different way of determining the Krull-Gabriel dimension of the cluster-tilted algebras, by applying results of Geigle.