Let $R$ be an $F$-finite Noetherian regular ring containing an algebraically closed field $k$ of positive characteristic, and let $M$ be an $\F$-finite $\F$-module over $R$ in the sense of Lyubeznik (for example, any local cohomology module of $R$). We prove that the $\mathbb{F}_p$-dimension of the space of $\F$-module morphisms $M \rightarrow E(R/\fm)$ (where $\fm$ is any maximal ideal of $R$ and $E(R/\fm)$ is the $R$-injective hull of $R/\fm$) is equal to the $k$-dimension of the Frobenius stable part of $\Hom_R(M,E(R/\fm))$. This is a positive-characteristic analogue of a recent result of Hartshorne and Polini for holonomic $\D$-modules in characteristic zero. We use this result to calculate the $\F$-module length of certain local cohomology modules associated with projective schemes.

中文翻译：

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Hartshorne-Polini 定理的素特征模拟

令 $R$ 是一个 $F$-finite Noetherian 正则环，包含一个具有正特征的代数闭域 $k$，让 $M$ 是 $R$ 上的 $\F$-finite $\F$-module Lyubeznik 的意义（例如，$R$ 的任何局部上同调模块）。我们证明了 $\F$-模态射 $M\rightarrowE(R/\fm)$ 的空间的 $\mathbb{F}_p$-维度（其中 $\fm$ 是 $R $ 和 $E(R/\fm)$ 是 $R/\fm$ 的 $R$-内射外壳) 等于 $\Hom_R(M,E( R/\fm))$。这是 Hartshorne 和 Polini 对特征零中完整 $\D$-模的最近结果的正特征模拟。我们使用这个结果来计算与投影方案相关的某些局部上同调模的 $\F$-模长度。