Notes on generalizations of local Ogus-Vologodsky correspondence

TL;DR

This paper generalizes the Ogus-Vologodsky correspondence by constructing functors between Higgs modules and modules with integrable connections over schemes with Frobenius lifts, extending equivalences to broader categories.

Contribution

It introduces a functorial framework connecting Higgs modules and integrable connection modules via level raising inverse image functors, generalizing previous local results to a broader setting.

Findings

The functor is an equivalence for quasi-nilpotent objects when m=1.

The functor induces an equivalence of categories for nilpotent objects under certain conditions.

Similar equivalences are established for modules with integrable p^m-Witt-connections.

Abstract

Given a smooth scheme over $\Z/p^n\Z$ with a lift of relative Frobenius to $\Z/p^{n+1}\Z$, we construct a functor from the category of Higgs modules to that of modules with integrable connections as the composite of the level raising inverse image functors from the category of modules with integrable $p^{m}$-connections to that of modules with integrable $p^{m-1}$-connections for $1 \leq m \leq n$. In the case $m=1$, we prove that the level raising inverse image functor is an equivalence when restricted to quasi-nilpotent objects, which generalizes a local result of Ogus-Vologodsky. We also prove that the above level raising inverse image functor for a smooth $p$-adic formal scheme induces an equivalence of $\Q$-linearized categories for general $m$ when restricted to nilpotent objects (in strong sense), under a strong condition on Frobenius lift. We also prove a similar result for the category of modules with integrable $p^{m}$-Witt-connections.