On Krull-Gabriel dimension of cluster repetitive categories and cluster-tilted algebras
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910644642512896 |
|---|---|
| author | Jaworska-Pastuszak, Alicja Pastuszak, Grzegorz Bobiński, Grzegorz |
| author_facet | Jaworska-Pastuszak, Alicja Pastuszak, Grzegorz Bobiński, Grzegorz |
| contents | 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 Bobiński presents a different way of determining the Krull-Gabriel dimension of the cluster-tilted algebras, by applying results of Geigle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_11798 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On Krull-Gabriel dimension of cluster repetitive categories and cluster-tilted algebras Jaworska-Pastuszak, Alicja Pastuszak, Grzegorz Bobiński, Grzegorz Representation Theory Rings and Algebras 16G20 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 Bobiński presents a different way of determining the Krull-Gabriel dimension of the cluster-tilted algebras, by applying results of Geigle. |
| title | On Krull-Gabriel dimension of cluster repetitive categories and cluster-tilted algebras |
| topic | Representation Theory Rings and Algebras 16G20 |
| url | https://arxiv.org/abs/2210.11798 |