The 𝐷–module structure of 𝑅[𝐹]–modules

Blickle, Manuel

doi:10.1090/S0002-9947-02-03197-5

The $D$–module structure of $R[F]$–modules
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Trans. Amer. Math. Soc. 355 (2003), 1647-1668 Request permission

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 $F^{e*} \mathcal {M} \to \mathcal {M}$, 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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