## 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.## References

- Abraham Broer, Victor Reiner, Larry Smith, and Peter Webb,
*Extending the coinvariant theorems of Chevalley, Shephard-Todd, Mitchell, and Springer*, Proc. Lond. Math. Soc. (3)**103**(2011), no. 5, 747–785. MR**2852288**, DOI 10.1112/plms/pdq027 - H. E. A. Eddy Campbell and David L. Wehlau,
*Modular invariant theory*, Encyclopaedia of Mathematical Sciences, vol. 139, Springer-Verlag, Berlin, 2011. Invariant Theory and Algebraic Transformation Groups, 8. MR**2759466**, DOI 10.1007/978-3-642-17404-9 - Harm Derksen and Gregor Kemper,
*Computing invariants of algebraic groups in arbitrary characteristic*, Adv. Math.**217**(2008), no. 5, 2089–2129. MR**2388087**, DOI 10.1016/j.aim.2007.08.016 - Giuseppe Gaeta, Frank D. Grosshans, Jürgen Scheurle, and Sebastian Walcher,
*Reduction and reconstruction for symmetric ordinary differential equations*, J. Differential Equations**244**(2008), no. 7, 1810–1839. MR**2404440**, DOI 10.1016/j.jde.2008.01.009 - Frank D. Grosshans,
*Algebraic homogeneous spaces and invariant theory*, Lecture Notes in Mathematics, vol. 1673, Springer-Verlag, Berlin, 1997. MR**1489234**, DOI 10.1007/BFb0093525 - W. J. Haboush,
*Reductive groups are geometrically reductive*, Ann. of Math. (2)**102**(1975), no. 1, 67–83. MR**382294**, DOI 10.2307/1970974 - Lex Renner and Alvaro Rittatore,
*Observable actions of algebraic groups*, Transform. Groups**14**(2009), no. 4, 985–999. MR**2577204**, DOI 10.1007/s00031-009-9073-x - Michel Van den Bergh,
*A converse to Stanley’s conjecture for $\textrm {Sl}_2$*, Proc. Amer. Math. Soc.**121**(1994), no. 1, 47–51. MR**1181176**, DOI 10.1090/S0002-9939-1994-1181176-5

## 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