Finitistic dimensions and piecewise hereditary property of skew group algebras

Abstract

Let $\Lambda$ be a finite dimensional algebra and $G$ be a finite group whose elements act on $\Lambda$ as algebra automorphisms. Under the assumption that $\Lambda$ 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 $\Lambda G$ and $\Lambda$ have the same finitistic dimension, and have the same strong global dimension if the fixed algebra $\Lambda^S$ is a direct summand of the $\Lambda^S$-bimodule $\Lambda$. Using a homological characterization of piecewise hereditary algebras proved by Happel and Zacharia, we deduce a criterion for $\Lambda G$ to be piecewise hereditary.