Galois ring extensions and localized modular rings of invariants of p-groups

Fleischmann, Peter and Woodcock, Chris F. (2013) Galois ring extensions and localized modular rings of invariants of p-groups. Transformation Groups, 18 (1). pp. 131-147. ISSN 1083-4362. (doi:https://doi.org/10.1007/s00031-012-9205-6) (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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Official URL
http://dx.doi.org/10.1007/s00031-012-9205-6

Abstract

We apply recent results on Galois-ring extensions and trace surjective algebras to analyze dehomogenized modular invariant rings of finite p-groups, as well as related localizations. We describe criteria for the dehomogenized invariant ring to be polynomial or at least regular and we show that for regular affine algebras with possibly non-linear action by a p-group, the singular locus of the invariant ring is contained in the variety of the transfer ideal. If V is the regular module of an arbitrary finite p-group, or V is any faithful representation of a cyclic p-group, we show that there is a suitable invariant linear form, inverting which renders the ring of invariants into a "localized polynomial ring" with dehomogenization being a polynomial ring. This is in surprising contrast to the fact that for a faithful representation of a cyclic group of order larger than p, the ring of invariants itself cannot be a polynomial ring by a result of Serre. Our results here generalize observations made by Richman [R] and by Campbell and Chuai [CCH].

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: 27 Feb 2013 00:07 UTC
Last Modified: 19 May 2014 13:24 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/33275 (The current URI for this page, for reference purposes)
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