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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.

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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.

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  • $\begingroup$ Thank you very much. It seems this is precisely what I was looking for. $\endgroup$
    – Lars
    Commented Jan 28, 2010 at 7:40
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    $\begingroup$ The reference in FGA is Bourbaki no. 190, p. 23 (Example 2). $\endgroup$
    – Thomas Riepe
    Commented Jan 28, 2010 at 11:21
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    $\begingroup$ This is a pretty old question, so you all might not be around anymore, but it's not clear at all to me, in any of these references, how such a connection gives a descent datum, is that made explicit anywhere? $\endgroup$
    – Jonathan Beardsley
    Commented Feb 21, 2015 at 15:27
  • $\begingroup$ 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]. $\endgroup$
    – equin
    Commented Jun 4, 2020 at 9:47

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