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

TL;DR

This paper investigates the Krull-Gabriel dimension of various categories associated with tilted algebras, establishing equalities and bounds, and classifying the dimensions for certain classes of algebras.

Contribution

It proves that the Krull-Gabriel dimensions of cluster-tilted and cluster repetitive categories are equal and classifies their possible values for tame categories.

Findings

KG( ilde{C})=KG(reve{C}) extless=KG(\hat{C})

KG( ilde{C})=KG(reve{C})=KG(\hat{C}) extless extgreater ext{{0,2, ext{infinity}}}

Appendix offers an alternative method for determining the dimension

Abstract

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 $\widehat{C},\check{C},\widetilde{C}$ are the associated repetitive category, cluster repetitive category and cluster-tilted algebra, respectively. Our first result states that $KG(\widetilde{C})=KG(\check{C})\leq KG(\widehat{C})$. Since the Krull-Gabriel dimensions of tame locally support-finite repetitive categories are known, we further conclude that $KG(\widetilde{C})=KG(\check{C})=KG(\widehat{C})\in\{0,2,\infty\}$. Finally, in the Appendix Grzegorz Bobi\'nski presents a different way of determining the Krull-Gabriel dimension of the cluster-tilted algebras, by applying results of Geigle.