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

Assume that K is an algebraically closed field and denote by $\mathrm{KG}\left(R\right)$ 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 $\hat{C},\check{C},\tilde{C}$ are the associated repetitive category, cluster repetitive category and cluster-tilted algebra, respectively. Our first result states that $\mathrm{KG}\left(\tilde{C}\right)=\mathrm{KG}\left(\check{C}\right)\le \mathrm{KG}\left(\hat{C}\right)$. Since the Krull-Gabriel dimensions of tame locally support-finite repetitive categories are known, we further conclude that $\mathrm{KG}\left(\tilde{C}\right)=\mathrm{KG}\left(\check{C}\right)=\mathrm{KG}\left(\hat{C}\right)\in \{0,2,\infty \}$. 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.