Lemma 59.17.3. In the situation above there is a functor

**Proof.**
Let $\mathcal{F}$ be a crystal in quasi-coherent modules on $X/S$. Set $T_ e = \mathop{\mathrm{Spec}}(D_ e)$ so that $(X, T_ e, \bar\gamma )$ is an object of $\text{Cris}(X/S)$ for $e \gg 0$. We have morphisms

which are closed immersions. We set

Note that since $\mathcal{F}$ is locally quasi-coherent we have $\mathcal{F}_{T_ e} = \widetilde{M_ e}$. Since $\mathcal{F}$ is a crystal we have $M_ e = M_{e + 1}/p^ eM_{e + 1}$. Hence we see that $M_ e = M/p^ eM$ and that $M$ is $p$-adically complete, see Algebra, Lemma 10.98.2.

By Lemma 59.15.1 we know that $\mathcal{F}$ comes endowed with a canonical integrable connection $\nabla : \mathcal{F} \to \mathcal{F} \otimes \Omega _{X/S}$. If we evaluate this connection on the objects $T_ e$ constructed above we obtain a canonical integrable connection

To see that this is topologically nilpotent we work out what this means.

Now we can do the same procedure for the rings $D(n)$. This produces a $p$-adically complete $D(n)$-module $M(n)$. Again using the crystal property of $\mathcal{F}$ we obtain isomorphisms

compare with the proof of Lemma 59.15.1. Denote $c$ the composition from left to right. Pick $m \in M$. Write $\xi _ i = x_ i \otimes 1 - 1 \otimes x_ i$. Using (59.17.1.1) we can write uniquely

for some $\theta _ K(m) \in M$ where the sum is over multi-indices $K = (k_ i)$ with $k_ i \geq 0$ and $\sum k_ i < \infty $. Set $\theta _ i = \theta _ K$ where $K$ has a $1$ in the $i$th spot and zeros elsewhere. We have

as can be seen by comparing with the definition of $\nabla $. Namely, the defining equation is $p_1^*m = \nabla (m) - c(p_0^*m)$ in Lemma 59.15.1 but the sign works out because in the Stacks project we consistently use $\text{d}f = p_1(f) - p_0(f)$ modulo the ideal of the diagonal squared, and hence $\xi _ i = x_ i \otimes 1 - 1 \otimes x_ i$ maps to $-\text{d}x_ i$ modulo the ideal of the diagonal squared.

Denote $q_ i : D \to D(2)$ and $q_{ij} : D(1) \to D(2)$ the coprojections corresponding to the indices $i, j$. As in the last paragraph of the proof of Lemma 59.15.1 we see that

This means that

in $M \otimes ^\wedge _{D, q_2} D(2)$ where

In particular $\zeta ''_ i = \zeta _ i + \zeta '_ i$ and we have that $D(2)$ is the $p$-adic completion of the divided power polynomial ring in $\zeta _ i, \zeta '_ i$ over $q_2(D)$, see Lemma 59.17.1. Comparing coefficients in the expression above it follows immediately that $\theta _ i \circ \theta _ j = \theta _ j \circ \theta _ i$ (this provides an alternative proof of the integrability of $\nabla $) and that

In particular, as the sum expressing $c(m \otimes 1)$ above has to converge $p$-adically we conclude that for each $i$ and each $m \in M$ only a finite number of $\theta _ i^ k(m)$ are allowed to be nonzero modulo $p$. $\square$

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