Modular group actions on algebras and p-local Galois extensions for finite groups

Let k be a field of positive characteristic p and let G be a finite group. In this paper we study the category TsG of finitely generated commutative k-algebras A on which G acts by algebra automorphisms with surjective trace. If A=k[X], the ring of regular functions of a variety X, then trace-surjective group actions on A are characterized geometrically by the fact that all point stabilizers on X are p′-subgroups or, equivalently, that AP≤A is a Galois extension for every Sylow p-group of G. We investigate categorical properties of TsG, using a version of Frobenius-reciprocity for group actions on k-algebras, which is based on tensor induction for modules. We also describe projective generators in TsG, extending and generalizing the investigations started in [7-9] in the case of p-groups. As an application we show that for an abelian or p-elementary group G and k large enough, there is always a faithful (possibly nonlinear) action on a polynomial ring such that the ring of invariants is also a polynomial ring. This would be false for linear group actions by a result of Serre. If A is a normal domain and G≤Autk(A) an arbitrary finite group, we show that AOp(G) is the integral closure of k[Soc(A)], the subalgebra of A generated by the simple kG-submodules in A. For p-solvable groups this leads to a structure theorem on trace-surjective algebras, generalizing the corresponding result for p-groups in [8].