# A prime-characteristic analogue of a theorem of Hartshorne-Polini

## TL;DR

This paper establishes a positive-characteristic analogue of a theorem by Hartshorne and Polini, relating the dimensions of certain module morphism spaces over regular rings in characteristic p.

## Contribution

It introduces a new characteristic p analogue of a known characteristic zero theorem, connecting Frobenius-stable parts and module morphism dimensions.

## Key findings

- Dimensions of module morphism spaces are equal to Frobenius-stable parts.
- Calculated the F-module length of local cohomology modules.
- Extended the understanding of F-module structures in positive characteristic.

## Abstract

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.