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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles 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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A variant of the Reynolds operator
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by Huah Chu, Shou-Jen Hu and Ming-chang Kang PDF
Proc. Amer. Math. Soc. 133 (2005), 2865-2871 Request permission

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
  • 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
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 133 (2005), 2865-2871
  • MSC (2000): Primary 13A50, 16D40, 16W22
  • DOI: https://doi.org/10.1090/S0002-9939-05-07845-7
  • MathSciNet review: 2159763