  Let $W$ be the ring of the Witt vectors of a perfect field of characteristic
$p$, $\mathfrak{X}$ a smooth formal scheme over $W$, $\mathfrak{X}'$ the base
change of $\mathfrak{X}$ by the Frobenius morphism of $W$, $\mathfrak{X}_{2}'$
the reduction modulo $p^{2}$ of $\mathfrak{X}'$ and $X$ the special fiber of
$\mathfrak{X}$. We lift the Cartier transform of Ogus-Vologodsky defined by
$\mathfrak{X}_{2}'$ modulo $p^{n}$. More precisely, we construct a functor from
the category of $p^{n}$-torsion $\mathscr{O}_{\mathfrak{X}'}$-modules with
integrable $p$-connection to the category of $p^{n}$-torsion
$\mathscr{O}_{\mathfrak{X}}$-modules with integrable connection, each subject
to suitable nilpotence conditions. Our construction is based on Oyama's
reformulation of the Cartier transform of Ogus-Vologodsky in characteristic
$p$. If there exists a lifting $F:\mathfrak{X}\to \mathfrak{X}'$ of the
relative Frobenius morphism of $X$, our functor is compatible with a functor
constructed by Shiho from $F$. As an application, we give a new interpretation
of Faltings' relative Fontaine modules and of the computation of their
cohomology.
