  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.
