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

TL;DR

This paper explores the structure of group actions on commutative algebras over fields of positive characteristic, characterizing trace-surjective actions, and establishing connections to Galois extensions and polynomial invariants for various finite groups.

Contribution

It introduces a categorical framework for trace-surjective group actions on algebras, extending previous work on p-groups to broader classes like abelian and p-elementary groups.

Findings

Characterization of trace-surjective actions via geometric stabilizer conditions

Existence of faithful polynomial ring actions with polynomial invariants for certain groups

Structure theorem for trace-surjective algebras in p-solvable groups

Abstract

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 A is a Galois extension of the ring of P invariants for every Sylow p-group P of G. We investigate categorical properties, 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 [8], [7] and [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. For p-solvable groups we obtain a structure theorem on trace-surjective algebras, generalizing the corresponding result for p-groups in [8].