The mod $p$ Riemann-Hilbert correspondence and the perfect site

TL;DR

This paper explores the mod p Riemann-Hilbert correspondence, connecting étale sheaves and Frobenius modules over F_p-algebras, utilizing Breen's vanishing results on the perfect site.

Contribution

It provides an exposition of the mod p Riemann-Hilbert correspondence using Breen's vanishing results, clarifying the relationship between sheaves and Frobenius modules.

Findings

Establishes covariant and contravariant forms of the correspondence

Utilizes Breen's vanishing results on the perfect site

Clarifies the relationship between étale sheaves and Frobenius modules

Abstract

The mod $p$ Riemann-Hilbert correspondence (in covariant and contravariant forms) relates $\mathbb{F}_p$-\'etale sheaves on the spectrum of an $\mathbb{F}_p$-algebra $R$ and Frobenius modules over $R$. We give an exposition of these correspondences using Breen's vanishing results on the perfect site.