论文标题
在集群重复类别的Krull-Gabriel维度和集群倾斜代数
On Krull-Gabriel dimension of cluster repetitive categories and cluster-tilted algebras
论文作者
论文摘要
假设$ k $是一个代数封闭的字段，用$ kg（r）$表示$ r $的krull-gabriel尺寸，其中$ r $是本地限制的$ k $ -scategory（或bourcan Quiver $ k $ -k $ -algebra）。假设$ c $是一个倾斜的$ k $ -Algebra，$ \ wideHat {c}，\ check {c}，\ widetilde {c} $分别是关联的重复类别，群集重复类别和集群倾斜的代数。我们的第一个结果指出，$ kg（\ widetilde {c}）= kg（\ check {c}）\ leq kg（\ widehat {c}）$。由于已知驯服本地支持的重复类别的Krull-Gabriel维度，因此我们进一步得出结论，$ kg（\ widetilde {c}）= kg（\ check {c}）= kg（\ wideHat {c} {c} {c}）最后，在附录中，GrzegorzBobiński通过应用Geigle的结果来确定群集倾斜代数的Krull-Gabriel维度的不同方法。
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.
