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 $\widehat{C},\check{C},\tilde{C}$ are the associated repetitive category, cluster repetitive category and cluster-tilted algebra, respectively. Our first result states that $KG(\tilde{C})=KG(\check{C})\le KG(\widehat{C})$. Since the Krull-Gabriel dimensions of tame locally support-finite repetitive categories are known, we further conclude that $KG(\tilde{C})=KG(\check{C})=KG(\widehat{C})\in \{0,2,\mathrm{\infty }\}$. 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.