Let $S$ be a scheme of positive characteristic $p$ and $X$ a smooth $S$-scheme. Let $F:X\rightarrow X^{(p)}$ denote the relative Frobenius. A result by Cartier (often called Cartier descent or Frobenius descent) then states that the category of quasi-coherent $\mathcal{O}\_{X^{(p)}}$-modules is equivalent to the category of quasi-coherent $\mathcal{O}\_X$-modules $(E,\nabla)$ with integrable connection of $p$-curvature $0$ (which means $\nabla(D)^p-\nabla(D^p)=0$ for all $S$-derivations $D:\mathcal{O}\_X\rightarrow \mathcal{O}\_X$).
The equivalence is given by


$$ (E,\nabla)\longmapsto E^\nabla$$


and


$$ E\mapsto (F^\*E,\nabla^{can})$$


where $\nabla^{can}$ is the canonical connection locally given by $f\otimes s\mapsto (1\otimes s)\otimes df$, for 


$$f\otimes s\in \mathcal{O}\_X(U)\otimes E(U).$$
(tensor over the sections of the structure sheaf of $X^{(p)}$, somehow jtex can't handle that)


The proof of this theorem can be found in 5.1. in Katz' paper ["Nilpotent connections and the monodromy theorem"](http://www.numdam.org/numdam-bin/item?id=PMIHES_1970__39__175_0)



> My question is: As $X/S$ is smooth, the relative Frobenius is faithfully flat (at least it is if $S$ is the spectrum of a perfect field), can the above theorem be interpreted as an instance of faithfully flat descent along $F$? In other words, does the connection $\nabla$ give rise to a descent datum for $E$ with respect to $F$? 


I know that connections are "first-order descent data", i.e. modules with connection descend along first order thickenings, but I don't see how this applies here.



Frobenius Descent
I believe that the answer is yes, and that this may have been one of Grothendieck's motivations for developing the general theory of flat descent. (If I remember correctly,
in the first (?) expose of FGA, in which he explains flat descent, Grothendieck has a reference to work of Cartier involving descent in the context of inseparable extensions,
and I would guess that it is a reference to this Cartier, or Frobenius, descent. Can
anyone cofirm this?)


To put a connection on $E$ is to extend the action of $\mathcal O\_X$ to an action
of $\mathcal D\_X$, the ring of differential operators generated (in local coordinates) by
$\partial /\partial x\_1,\ldots,\partial/\partial x\_n.$ (This is not the same as the full ring
of differential operators in charateristic $p$.) The $p$-curvatures generate an ideal
in this ring; I think it is just the ideal generated by $(\partial/\partial x\_i)^p$.
So if $E$ has vanishing $p$-curvature, the action of $\mathcal D\_X$ factors through
the quotient by this ideal. One can now interpret this information in terms
of descent data. 


A precise description is given in Prop. 2.6.2 of Berthelot's book
[D-modules arithmetiques II; Descente par Frobenius](http://perso.univ-rennes1.fr/pierre.berthelot/).


