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

Fleischmann, Peter and Woodcock, Chris F. (2015) Modular group actions on algebras and p-local Galois extensions for finite groups. Journal of Algebra, 442 . pp. 316-353. ISSN 0021-8693. (doi:https://doi.org/10.1016/j.jalgebra.2015.06.003) (Full text available)

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http://dx.doi.org/10.1016/j.jalgebra.2015.06.003

Abstract

Let k be a field of positive characteristic p and let G be a finite group. In this paper we study the category TsGTsG of finitely generated commutative k-algebras A on which G acts by algebra automorphisms with surjective trace. If A=k[X]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′p′-subgroups or, equivalently, that AP≤AAP≤A is a Galois extension for every Sylow p-group of G . We investigate categorical properties of TsGTsG, 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 TsGTsG, extending and generalizing the investigations started in [7], [8] 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. If A is a normal domain and G≤Autk(A)G≤Autk(A) an arbitrary finite group, we show that AOp(G)AOp(G) is the integral closure of k[Soc(A)]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].

Item Type: Article
Subjects: Q Science > QA Mathematics (inc Computing science) > QA150 Algebra
Q Science > QA Mathematics (inc Computing science) > QA171 Representation theory
Divisions: Faculties > Sciences > School of Mathematics Statistics and Actuarial Science > Pure Mathematics
Depositing User: Chris F Woodcock
Date Deposited: 31 Oct 2015 15:43 UTC
Last Modified: 01 Dec 2017 00:00 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/51317 (The current URI for this page, for reference purposes)
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