Notes on generalizations of local Ogus-Vologodsky correspondence

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.