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arXiv · 2210.11798

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

Alicja Jaworska-Pastuszak·Grzegorz Pastuszak·Grzegorz Bobiński
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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.

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Alicja Jaworska-Pastuszak, Grzegorz Pastuszak, Grzegorz Bobiński. 2022-10-21. On Krull-Gabriel dimension of cluster repetitive categories and cluster-tilted algebras. https://arxiv.org/abs/2210.11798

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