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.References
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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