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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 \[ \eta _G^G: \mathbb {F}[V(G)]^G \rightarrow \mathbb {F}[V]^G. \] 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

Müfit Sezer
Affiliation: Department of Mathematics and Statistics, Boğazici Üniversitesi, MS 1042, Bebek, Istanbul, Turkey
MR Author ID: 703561

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