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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The Noether map II


Authors: Mara D. Neusel and Müfit Sezer
Journal: Proc. Amer. Math. Soc. 135 (2007), 2347-2354
MSC (2000): Primary 13A50, 20J06
Published electronically: March 21, 2007
MathSciNet review: 2302555
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Abstract: Let $ \rho: G\hookrightarrow \mathrm{GL}(n, \mathbb{F})$ be a faithful representation of a finite group $ G$. In this paper we proceed with the study of the image of the associated Noether map

$\displaystyle \eta_G^G: \mathbb{F}[V(G)]^G \rightarrow\mathbb{F}[V]^G. $

In our 2005 paper it has been shown that the Noether map is surjective if $ V$ is a projective $ \mathbb{F} G$-module. This paper deals with the converse. The converse is in general not true: we illustrate this with an example. However, for $ p$-groups (where $ p$ is the characteristic of the ground field $ \mathbb{F}$) as well as for permutation representations of any group the surjectivity of the Noether map implies the projectivity of $ V$.


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Additional Information

Mara D. Neusel
Affiliation: Department of Mathematics, Texas Tech University, Lubbock, Texas 79409
Email: mara.d.neusel@ttu.edu

Müfit Sezer
Affiliation: Department of Mathematics and Statistics, Boğazici Üniversitesi, MS 1042, Bebek, Istanbul, Turkey
Email: mufit.sezer@boun.edu.tr

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08915-0
PII: S 0002-9939(07)08915-0
Keywords: Invariant theory of finite groups, Noether map, modular invariant theory, projective $\mathbb{F}G$-modules, $p$-groups, permutation representations
Received by editor(s): April 12, 2006
Published electronically: March 21, 2007
Additional Notes: The first author is partially supported by NSA Grant No. H98230-05-1-0026
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2007 American Mathematical Society