Homological dimensions of skew group algebras

Abstract

Let [Formula: see text] be a finite dimensional algebra over a field [Formula: see text] and [Formula: see text] a subgroup of a finite group [Formula: see text]. In this paper, we consider the Gorenstein global dimensions and representation dimensions of the skew group algebras [Formula: see text] and [Formula: see text]. Under the assumption that [Formula: see text] is a separable extension over [Formula: see text], we show that [Formula: see text] and [Formula: see text] share the same Gorenstein global dimensions and representation dimensions. As an application, we give an affirmative answer for a conjecture raised in [Adjoint functors and representation dimensions, Acta Math. Sinica 22(2) (2006) 625–640]. Several known results are then obtained as corollaries.