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Proceedings of the American Mathematical Society

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

 
 

 

Modules of higher order invariants


Authors: Frank D. Grosshans and Sebastian Walcher
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
Published electronically: October 10, 2014
MathSciNet review: 3283642
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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

Keywords: Invariants, modules
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
Article copyright: © Copyright 2014 American Mathematical Society