# Homomorphisms, Localizations and a new Algorithm to construct Invariant Rings of Finite Groups

Fleischmann, Peter, Kemper, Gregor, Woodcock, Chris F. (2007) Homomorphisms, Localizations and a new Algorithm to construct Invariant Rings of Finite Groups. Journal of Algebra, 309 (2). pp. 497-517. ISSN 0021-8693. (doi:10.1016/j.jalgebra.2005.06.038) (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) (KAR id:732)

 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. Official URLhttp://dx.doi.org/10.1016/j.jalgebra.2005.06.038

## Abstract

Let G be a finite group acting on a polynomial ring A over the field K and let AG denote the corresponding ring of invariants. Let B be the subalgebra of AG generated by all homogeneous elements of degree less than or equal to the group order |G|. Then in general B is not equal to AG if the characteristic of K divides |G|. However we prove that the field of fractions Quot(B) coincides with the field of invariants Quot(AG)=Quot(A)G. We also study various localizations and homomorphisms of modular invariant rings as tools to construct generators for AG. We prove that there is always a nonzero transfer cAG of degree <|G|, such that the localization (AG)c can be generated by fractions of homogeneous invariants of degrees less than 2|G|?1. If with finite-dimensional -module V, then c can be chosen in degree one and 2|G|?1 can be replaced by |G|. Let denote the image of the classical Noether-homomorphism (see the definition in the paper). We prove that contains the transfer ideal and thus can be used to calculate generators for AG by standard elimination techniques using Gröbner-bases. This provides a new construction algorithm for AG.

Item Type: Article 10.1016/j.jalgebra.2005.06.038 Modular invariant theory; Computational algebra; Localization Q Science > QA Mathematics (inc Computing science) Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science Judith Broom 19 Dec 2007 18:27 UTC 16 Nov 2021 09:39 UTC https://kar.kent.ac.uk/id/eprint/732 (The current URI for this page, for reference purposes) https://orcid.org/0000-0003-4713-0040