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I am trying to understand the proof of Cartier’s theorem on pages 370-371 (pages 17-18 of the PDF file) of Katz’s “Nilpotent connections and the monodromy theorem: applications of a result of Turrittin”, but I am stuck.

For simplicity, let us assume that we are given a smooth scheme $S$ over $\mathbb{F}_p$. Let $S^{(p)}$ denote the fibre product of the absolute Frobenius of $\mathbb{F}_p$ and the structure map of $S$ as an $\mathbb{F}_p$-scheme, and $F$ the relative Frobenius map. We want to prove that there is an equivalence of categories between the category of quasi-coherent sheaves on $S^{(p)}$ and the category of sheaves of $\mathcal O_S$-modules on $S$ with integrable ($\mathbb{F}_p$-)connection whose $p$-curvature is $0$. The functor in the forward direction simply maps a quasicoherent sheaf $\mathscr F$ on $S^{(p)}$ to its pullback $F^*\mathscr F$ under the relative Frobenius and the functor in the reverse direction maps a pair $(E, \nabla)$ to $E^{\nabla}$, the sheaf of germs of horizontal sections of $(E, \nabla)$. To prove the result, he shows that the canonical mapping of $\mathcal O_S$-modules from the pullback of $E^{\nabla}$ (under $F$) to $E$ is an isomorphism. I cannot understand how he does that; in particular, the construction he gives is completely opaque to me. Could you please explain the main idea of the proof to me? Any hints on why the claims he makes along the way are true would also be appreciated. If you could point me to alternative proofs of the theorem, that would be great!

Are there any standard books that contain the proof of this theorem? I have seen at least one version of it stated without proof in the language of $D_S$-modules. Are there any references that I could look at for complete proofs of these and other related results?

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I learned this from Victor Ginzburg. Let $S = \mathbb A^1_{k}$, with coordinate $x$ and corresponding vector field $\partial = \frac{d}{dx}$, where $k$ is a field of characteristic $p$. Let $(E,\nabla)$ be an $\mathcal O_X$-module with flat connection $\nabla$ with $p$-curvature 0. We need to show that $F^\ast E^\nabla \to E$ is an isomorphism. Given a general section $f$ of $E$, consider the operator $$ P = \sum_{ j=0}^{p-1} \frac{(-x)^j}{j!} \nabla_\partial^j.$$ Because $\nabla$ has $p$-curvature zero, $\nabla_\partial P = 0$, that is, the image of $P$ are flat sections. Further, $\nabla(Pe)(0) = e(0)$ for all sections $e$ of $E$. We can use this to write a "Taylor expansion" $$ Te = \sum_{k = 0}^{p-1} \frac{x^k}{k!} P(\nabla^k_\partial e).$$ You can check (again, using flatness and $p$-curvature zero) that $Te = e$. $T$ provides an inverse to the canonical map.

The proof for general smooth $S$ essentially reduces to this case via étale coordinates.

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  • $\begingroup$ Thank you so much for the quick reply! A couple of stupid questions: 1. Assuming that flat sections are the same as horizontal sections, did you mean $\nabla P = 0$ instead of ${\nabla}_{\partial} P = 0$? If something is in the kernel of $\nabla$, then it is definitely in the kernel of ${\nabla}_{\partial}$, but is the converse true? 2. Having zero $p$-curvature means having "enough" horizontal sections. So is the "main idea" behind the construction of $P$ simply to provide sufficiently many horizontal sections? 3. What exactly do you mean by evaluating the section $e$ at 0? $\endgroup$ – clarkkent Jul 25 '20 at 22:04
  • $\begingroup$ First, since $\partial$ generates the vector fields on $S$, $\nabla P = 0$ and $\nabla_\partial P = 0$ are equivalent. Yes, the main idea is that $P$ constructs sufficiently many horizontal sections. Evaluation means looking at the value of $e$ in the fiber. $\endgroup$ – Joshua Mundinger Jul 26 '20 at 20:23
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Cartier descent is historically important, since together with Galois descent, it was Grothendieck's source of inspiration for fppf descent.

As far as I remember, and with all due respect, Katz's proof is mainly of computational nature, there is not much to understand. It is still worth trying to follow it, since you will learn some techniques, and the conviction that the statement is true, and it has the big advantage of being elementary, and reasonably self-contained.

If you are looking for a modern treatment, my advice would be to look at Michael Groechning's very nice proof here:

Moduli of flat connections in positive characteristic

Mathematical Research Letters Volume 23 (2016) Number 4 Pages: 989 – 1047

https://dx.doi.org/10.4310/MRL.2016.v23.n4.a3

https://arxiv.org/abs/1201.0741

more specifically, Theorem 3.11. A word of warning: the proof is heavily conceptual, and you won't escape the language of $D$-modules, but why should you ? It is impossible to sum up the principle here, but to quote the author: "The proof given here relies on the interpretation of $D_X$ as Azumaya algebra over its centre". Another important ingredient is Morita theory.

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