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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The $D$-module structure of $R[F]$-modules


Author: Manuel Blickle
Journal: Trans. Amer. Math. Soc. 355 (2003), 1647-1668
MSC (2000): Primary 13A35, 16S99, 16S32
Published electronically: November 22, 2002
MathSciNet review: 1946409
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Abstract: Let $R$ be a regular ring, essentially of finite type over a perfect field $k$. An $R$-module $\mathcal{M}$ is called a unit $R[F]$-module if it comes equipped with an isomorphism $F^{e*} \mathcal{M} \xrightarrow{\ \ }\mathcal{M}$, where $F$ denotes the Frobenius map on $\operatorname{Spec}R$, and $F^{e*}$ is the associated pullback functor. It is well known that $\mathcal{M}$ then carries a natural $D_R$-module structure. In this paper we investigate the relation between the unit $R[F]$-structure and the induced $D_R$-structure on $\mathcal{M}$. In particular, it is shown that if $k$ is algebraically closed and $\mathcal{M}$ is a simple finitely generated unit $R[F]$-module, then it is also simple as a $D_R$-module. An example showing the necessity of $k$ being algebraically closed is also given.


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Additional Information

Manuel Blickle
Affiliation: Universität Essen, FB6 Mathematik, 45117 Essen, Germany
Email: manuel.blickle@uni-essen.de

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03197-5
PII: S 0002-9947(02)03197-5
Keywords: Modules with Frobenius action, $D$-modules, $F$-modules
Received by editor(s): May 10, 2002
Received by editor(s) in revised form: July 10, 2002
Published electronically: November 22, 2002
Article copyright: © Copyright 2002 American Mathematical Society