Fleischmann, P. and Woodcock, C.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 .
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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.
|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:||04 Jan 2012 09:13|
|Resource URI:||http://kar.kent.ac.uk/id/eprint/28564 (The current URI for this page, for reference purposes)|
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