Skew Group Algebras and their Yoneda Algebras

Skew group algebras appear in connection with the study of singularities [1], [2]. It was proved in [4], [6], [10] the preprojective algebra of an Euclidean diagram is Morita equivalent to a skew group algebra of a polynomial algebra. In [7] we investigated the Yoneda algebra of a selfinjective Koszul algebra and proved they have properties analogous to the commutative regular algebras, we call such algebras generalized Auslander regular. The aim of the paper is to prove that given a positively graded locally finite K-algebra Λ = ∑ί≥0 Λ_ί and a finite grading preserving group G of automorphisms of Λ, with characteristic K not dividing the order of G, then G acts naturally on the Yoneda algebra Γ =⊕κ≥0　Ext^k_Λ(Λ0,Λ0) and the skew group algebra Γ*G is isomorphic to the Yoneda algebra Λ*G = ⊕κ≥0 Ext^κ_Λ*_G(Λ0*G,Λ0*G). As an application we prove Λ is generalized Auslander regular if and only if Λ*G is generalized Auslander regular and Λ is Koszul if and only if Λ*G is so.