Non-linear group actions with polynomial invariant rings and a structure theorem for modular Galois extensions

Fleischmann, Peter and Woodcock, Chris F. (2011) Non-linear group actions with polynomial invariant rings and a structure theorem for modular Galois extensions. Proceedings of the London Mathematical Society, 103 (5). pp. 826-846. ISSN 0024-6115 . (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided)

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Let G be a finite p-group and k be a field of characteristic p>0. We show that G has a non-linear faithful action on a polynomial ring U of dimension n=log p(|G|) such that the invariant ring UG is also polynomial. This contrasts with the case of linear and graded group actions with polynomial rings of invariants, where the classical theorem of Chevalley–Shephard–Todd and Serre requires G to be generated by pseudo-reflections. Our result is part of a general theory of ‘trace surjective G-algebras’, which, in the case of p-groups, coincide with the Galois ring extensions in the sense of Chase, Harrison and Rosenberg [‘Galois theory and Galois cohomology of commutative rings’, Mem. Amer. Math. Soc. 52 (1965) 15–33]. We consider the dehomogenized symmetric algebra Dk, a polynomial ring with non-linear G-action, containing U as a retract and we show that DGk is a polynomial ring. Thus, U turns out to be universal in the sense that every trace surjective G-algebra can be constructed from U by ‘forming quotients and extending invariants’. As a consequence we obtain a general structure theorem for Galois extensions with given p-group as Galois group and any prescribed commutative k-algebra R as invariant ring. This is a generalization of the Artin–Schreier–Witt theory of modular Galois field extensions of degree ps.

Item Type: Article
Subjects: Q Science > QA Mathematics (inc Computing science) > QA150 Algebra
Divisions: Faculties > Science Technology and Medical Studies > School of Mathematics Statistics and Actuarial Science > Pure Mathematics
Depositing User: Chris F Woodcock
Date Deposited: 02 Jan 2012 00:35
Last Modified: 19 May 2014 13:26
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