  Given a morphism $X \to S$ of log schemes of characteristic $p > 0$ and a
lifting of $X'$ over $S$ modulo $p^2$, we use Lorenzon's indexed algebras
$A_X^{gp}$ and $B_{X/S}$ to construct an equivalence between $O_X$-modules with
nilpotent integrable connection and indexed $B_{X/S}$-modules with nilpotent
$B_{X/S}$-linear Higgs field. If either satisfies a stricter nilpotence
condition, we find an isomorphism between the de Rham cohomology of the
connection and the Higgs cohomology of the Higgs field.
