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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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Modules of higher order invariants
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by Frank D. Grosshans and Sebastian Walcher PDF
Proc. Amer. Math. Soc. 143 (2015), 531-542 Request permission

Abstract:

Let $k$ be an algebraically closed field of characteristic $p\geq 0$. Let $A$ be a commutative $k$-algebra with multiplicative identity and let $M$ be an $A$-module. Let $G$ be a linear algebraic group acting rationally on both $A$ and $M$. In this paper we study $A^{G}$-modules of $n$th order invariants, $I_{n}(M,G)$. The $I_{n}(M,G)$ are defined inductively by $I_{0}(M,G)=\{0\}$ and $I_{n}(M,G)$ = $\{m\in M:g\cdot m-m\in I_{n-1}(M,G)$ for all $g\in G\}$. We show that some fundamental problems concerning these modules can be reduced to the case $I_{n}(k[G],G)$ where $G$ acts on itself by right translation. We study the questions as to when $I_{n}(M,G)$ is a finitely generated $A^{G}$-module and how the $I_{n}(M,G)$ are related to equivariant mappings. For the classical case of $\mathbb {G}_{a}$ acting on binary forms, we describe the $I_{n}(M,G)$ and determine when they are Cohen-Macaulay.
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Additional Information
  • Frank D. Grosshans
  • Affiliation: Department of Mathematics, West Chester University, West Chester, Pennsylvania 19383
  • Email: fgrosshans@wcupa.edu
  • Sebastian Walcher
  • Affiliation: Lehrstuhl A für Mathematik, RWTH Aachen, 52056 Aachen, Germany
  • Email: walcher@mathA.rwth-aachen.de
  • Received by editor(s): September 26, 2012
  • Received by editor(s) in revised form: May 22, 2013
  • Published electronically: October 10, 2014
  • Additional Notes: The authors thank the referee for a very careful reading of the manuscript and many helpful suggestions.
  • Communicated by: Harm Derksen
  • © Copyright 2014 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 143 (2015), 531-542
  • MSC (2010): Primary 13A50; Secondary 37C80
  • DOI: https://doi.org/10.1090/S0002-9939-2014-12268-4
  • MathSciNet review: 3283642