Finitistic dimensions and piecewise hereditary property of skew group algebras

TL;DR

This paper investigates the homological properties of skew group algebras, establishing conditions under which they share finitistic and global dimensions with the original algebra, and providing a criterion for being piecewise hereditary.

Contribution

It introduces new conditions relating the action of a Sylow p-subgroup on an algebra to the preservation of homological dimensions and piecewise hereditary property in skew group algebras.

Findings

Finitistic dimension of $\Lambda G$ equals that of $\Lambda$ under free Sylow p-subgroup action.

Same strong global dimension for $\Lambda G$ and $\Lambda$ if fixed algebra is a direct summand.

Provides a homological criterion for $\Lambda G$ to be piecewise hereditary.

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.