Fleischmann, Peter, Woodcock, Chris F. (2015) Modular group actions on algebras and plocal Galois extensions for finite groups. Journal of Algebra, 442 . pp. 316353. ISSN 00218693. (doi:10.1016/j.jalgebra.2015.06.003) (KAR id:51317)
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Official URL: 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 kalgebras 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 tracesurjective 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 pgroup of G . We investigate categorical properties of TsGTsG, using a version of Frobeniusreciprocity 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 pgroups. As an application we show that for an abelian or pelementary 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 kGsubmodules in A. For psolvable groups this leads to a structure theorem on tracesurjective algebras, generalizing the corresponding result for pgroups in [8].
Item Type:  Article 

DOI/Identification number:  10.1016/j.jalgebra.2015.06.003 
Subjects: 
Q Science > QA Mathematics (inc Computing science) > QA150 Algebra Q Science > QA Mathematics (inc Computing science) > QA171 Representation theory 
Divisions:  Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science 
Depositing User:  Christopher Woodcock 
Date Deposited:  31 Oct 2015 15:43 UTC 
Last Modified:  16 Feb 2021 13:29 UTC 
Resource URI:  https://kar.kent.ac.uk/id/eprint/51317 (The current URI for this page, for reference purposes) 
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