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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
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

Sebastian Walcher
Affiliation: Lehrstuhl A für Mathematik, RWTH Aachen, 52056 Aachen, Germany

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