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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A variant of the Reynolds operator


Authors: Huah Chu, Shou-Jen Hu and Ming-chang Kang
Journal: Proc. Amer. Math. Soc. 133 (2005), 2865-2871
MSC (2000): Primary 13A50, 16D40, 16W22
Published electronically: March 31, 2005
MathSciNet review: 2159763
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Abstract: Let $G$ be a linearly reductive group over a field $k$, and let $R$ be a $k$-algebra with a rational action of $G$. Given rational $R$-$G$-modules $M$and $N$, we define for the induced $G$-action on Hom$_{R}(M,N)$ a generalized Reynolds operator, which exists even if the action on Hom $_{R}(M, N)$ is not rational. Given an $R$-module homomorphism $M \rightarrow N$, it produces, in a natural way, an $R$-module homomorphism which is $G$-equivariant. We use this generalized Reynolds operator to study properties of rational $R$-$G$ modules. In particular, we prove that if $M$ is invariantly generated (i.e. $M = R \cdot M^{G}$), then $M^{G}$ is a projective (resp. flat) $R^{G}$-module provided that $M$ is a projective (resp. flat) $R$-module. We also give a criterion whether an $R$-projective (or $R$-flat) rational $R$-$G$-module is extended from an $R^{G}$-module.


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Additional Information

Huah Chu
Affiliation: Department of Mathematics, National Taiwan University, Taipei, Taiwan

Shou-Jen Hu
Affiliation: Department of Mathematics, Tamkang University, Taipei, Taiwan

Ming-chang Kang
Affiliation: Department of Mathematics, National Taiwan University, Taipei, Taiwan
Email: kang@math.ntu.edu.tw

DOI: http://dx.doi.org/10.1090/S0002-9939-05-07845-7
PII: S 0002-9939(05)07845-7
Keywords: Rings of invariants, linearly reductive groups, Reynolds operator, rational $G$-spaces, projective modules, flat modules.
Received by editor(s): February 20, 2004
Received by editor(s) in revised form: May 30, 2004
Published electronically: March 31, 2005
Communicated by: Martin Lorenz
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.