斜群代数的分段维数与分段遗传性
Finitistic dimensions and piecewise hereditary property of skew group algebras
摘要：
设$Λ$是有限维代数，$G$是有限群，其元素作为代数自同构作用于$Λ$.假设$Λ$有一个本原正交幂等元的完备集$E$，在Sylow $p$ -子群$S \leqslant G$的作用下是闭的.如果$S$在$E$上的作用是自由的，我们证明了斜群代数$ΛG$和$Λ$具有相同的有限维数，并且当固定代数$Λ^S$是$Λ^S$ -双模$Λ$的直和项时，它们具有相同的强整体维数.利用Happel和Zacharia证明的分段遗传代数的同调特征，我们导出了$ΛG$是分段遗传的一个判别准则.
英文摘要：
Let $Λ$ be a finite dimensional algebra and $G$ be a finite group whose elements act on $Λ$ as algebra automorphisms. Under the assumption that $Λ$ has a complete set $E$ of primitive orthogonal idempotents, closed under the action of a Sylow $p$-subgroup $S \leqslant G$. If the action of $S$ on $E$ is free, we show that the skew group algebra $ΛG$ and $Λ$ have the same finitistic dimension, and have the same strong global dimension if the fixed algebra $Λ^S$ is a direct summand of the $Λ^S$-bimodule $Λ$. Using a homological characterization of piecewise hereditary algebras proved by Happel and Zacharia, we deduce a criterion for $ΛG$ to be piecewise hereditary.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
