# Frobenius Descent

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"

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.

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.
• I am also late to the game, but a short answer to Jonathan's question is this. Let me work on $X = Spec R$. A connection on $M$ gives it a $D^{(0)}$-module structure (in the language of Berthelot). This $D^{(0)}$-module structure corresponds to a 0-PD-stratification $\{\epsilon_n: P^n_{(0)} \otimes_R M \simeq M \otimes_R P^n_{(0)}\}$ [D-modules arithmetiques I], and Berthelot shows that, if we denote $F: R' \to R$ the Frobenius on $R$, then $R \otimes_{R'} R$ is a quotient of $P^n_{(0)}$ for large enough $n$ [D-modules arithmetiques II]. – equin Jun 4 '20 at 9:47