Fleischmann, Peter, Woodcock, Chris F. (2018) Free actions of pgroups on affine varieties in characteristic p. Mathematical Proceedings of the Cambridge Philosophical Society, 165 (1). pp. 109135. ISSN 03050041. EISSN 14698064. (doi:10.1017/S0305004117000317) (KAR id:61213)
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Official URL: https://doi.org/10.1017/S0305004117000317 
Abstract
Let K be an algebraically closed field and n ? K n affine nspace. It is known that a finite group can only act freely on n if K has characteristic p > 0 and is a pgroup. In that case the group action is “nonlinear” and the ring of regular functions K[ n ] must be a tracesurjective K ? algebra.
Now let k be an arbitrary field of characteristic p > 0 and let G be a finite pgroup. In this paper we study the category of all finitely generated tracesurjective k ? G algebras. It has been shown in [13] that the objects in are precisely those finitely generated k ? G algebras A such that A G ? A is a Galoisextension in the sense of [7], with faithful action of G on A. Although is not an abelian category it has “sprojective objects”, which are analogues of projective modules, and it has (sprojective) categorical generators, which we will describe explicitly. We will show that sprojective objects and their rings of invariants are retracts of polynomial rings and therefore regular UFDs. The category also has “weakly initial objects”, which are closely related to the essential dimension of G over k. Our results yield a geometric structure theorem for free actions of finite pgroups on affine kvarieties. There are also close connections to open questions on retracts of polynomial rings, to embedding problems in standard modular Galoistheory of pgroups and, potentially, to a new constructive approach to homogeneous invariant theory
Item Type:  Article 

DOI/Identification number:  10.1017/S0305004117000317 
Subjects:  Q Science > QA Mathematics (inc Computing science) 
Divisions:  Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science 
Depositing User:  Christopher Woodcock 
Date Deposited:  06 Apr 2017 08:10 UTC 
Last Modified:  09 Dec 2022 06:26 UTC 
Resource URI:  https://kar.kent.ac.uk/id/eprint/61213 (The current URI for this page, for reference purposes) 
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