  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.
