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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The Noether map II
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by Mara D. Neusel and Müfit Sezer PDF
Proc. Amer. Math. Soc. 135 (2007), 2347-2354 Request permission

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
  • Email: mara.d.neusel@ttu.edu
  • Müfit Sezer
  • Affiliation: Department of Mathematics and Statistics, Boğazici Üniversitesi, MS 1042, Bebek, Istanbul, Turkey
  • MR Author ID: 703561
  • Email: mufit.sezer@boun.edu.tr
  • 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
  • © Copyright 2007 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 135 (2007), 2347-2354
  • MSC (2000): Primary 13A50, 20J06
  • DOI: https://doi.org/10.1090/S0002-9939-07-08915-0
  • MathSciNet review: 2302555