## Tag `07J7`

## 51.17. Crystals in quasi-coherent modules

In Situation 51.5.1. Set $X = \mathop{\rm Spec}(C)$ and $S = \mathop{\rm Spec}(A)$. We are going to classify crystals in quasi-coherent modules on $\text{Cris}(X/S)$. Before we do so we fix some notation.

Choose a polynomial ring $P = A[x_i]$ over $A$ and a surjection $P \to C$ of $A$-algebras with kernel $J = \mathop{\rm Ker}(P \to C)$. Set \begin{equation} \tag{51.17.0.1} D = \mathop{\rm lim}\nolimits_e D_{P, \gamma}(J) / p^eD_{P, \gamma}(J) \end{equation} for the $p$-adically completed divided power envelope. This ring comes with a divided power ideal $\bar J$ and divided power structure $\bar \gamma$, see Lemma 51.5.5. Set $D_e = D/p^eD$ and denote $\bar J_e$ the image of $\bar J$ in $D_e$. We will use the short hand \begin{equation} \tag{51.17.0.2} \Omega_D = \mathop{\rm lim}\nolimits_e \Omega_{D_e/A, \bar\gamma} = \mathop{\rm lim}\nolimits_e \Omega_{D/A, \bar\gamma}/p^e\Omega_{D/A, \bar\gamma} \end{equation} for the $p$-adic completion of the module of divided power differentials, see Lemma 51.6.12. It is also the $p$-adic completion of $\Omega_{D_{P, \gamma}(J)/A, \bar\gamma}$ which is free on $\text{d}x_i$, see Lemma 51.6.6. Hence any element of $\Omega_D$ can be written uniquely as a sum $\sum f_i\text{d}x_i$ with for all $e$ only finitely many $f_i$ not in $p^eD$. Moreover, the maps $\text{d}_{D_e/A, \bar\gamma} : D_e \to \Omega_{D_e/A, \bar\gamma}$ fit together to define a divided power $A$-derivation \begin{equation} \tag{51.17.0.3} \text{d} : D \longrightarrow \Omega_D \end{equation} on $p$-adic completions.

We will also need the ''products $\mathop{\rm Spec}(D(n))$ of $\mathop{\rm Spec}(D)$'', see Proposition 51.21.1 and its proof for an explanation. Formally these are defined as follows. For $n \geq 0$ let $J(n) = \mathop{\rm Ker}(P \otimes_A \ldots \otimes_A P \to C)$ where the tensor product has $n + 1$ factors. We set \begin{equation} \tag{51.17.0.4} D(n) = \mathop{\rm lim}\nolimits_e D_{P \otimes_A \ldots \otimes_A P, \gamma}(J(n))/ p^eD_{P \otimes_A \ldots \otimes_A P, \gamma}(J(n)) \end{equation} equal to the $p$-adic completion of the divided power envelope. We denote $\bar J(n)$ its divided power ideal and $\bar \gamma(n)$ its divided powers. We also introduce $D(n)_e = D(n)/p^eD(n)$ as well as the $p$-adically completed module of differentials \begin{equation} \tag{51.17.0.5} \Omega_{D(n)} = \mathop{\rm lim}\nolimits_e \Omega_{D(n)_e/A, \bar\gamma} = \mathop{\rm lim}\nolimits_e \Omega_{D(n)/A, \bar\gamma}/p^e\Omega_{D(n)/A, \bar\gamma} \end{equation} and derivation \begin{equation} \tag{51.17.0.6} \text{d} : D(n) \longrightarrow \Omega_{D(n)} \end{equation} Of course we have $D = D(0)$. Note that the rings $D(0), D(1), D(2), \ldots$ form a cosimplicial object in the category of divided power rings.

Lemma 51.17.1. Let $D$ and $D(n)$ be as in (51.17.0.1) and (51.17.0.4). The coprojection $P \to P \otimes_A \ldots \otimes_A P$, $f \mapsto f \otimes 1 \otimes \ldots \otimes 1$ induces an isomorphism \begin{equation} \tag{51.17.1.1} D(n) = \mathop{\rm lim}\nolimits_e D\langle \xi_i(j) \rangle/p^eD\langle \xi_i(j) \rangle \end{equation} of algebras over $D$ with $$ \xi_i(j) = x_i \otimes 1 \otimes \ldots \otimes 1 - 1 \otimes \ldots \otimes 1 \otimes x_i \otimes 1 \otimes \ldots \otimes 1 $$ for $j = 1, \ldots, n$.

Proof.We have $$ P \otimes_A \ldots \otimes_A P = P[\xi_i(j)] $$ and $J(n)$ is generated by $J$ and the elements $\xi_i(j)$. Hence the lemma follows from Lemma 51.2.5. $\square$Lemma 51.17.2. Let $D$ and $D(n)$ be as in (51.17.0.1) and (51.17.0.4). Then $(D, \bar J, \bar\gamma)$ and $(D(n), \bar J(n), \bar\gamma(n))$ are objects of $\text{Cris}^\wedge(C/A)$, see Remark 51.5.4, and $$ D(n) = \coprod\nolimits_{j = 0, \ldots, n} D $$ in $\text{Cris}^\wedge(C/A)$.

Proof.The first assertion is clear. For the second, if $(B \to C, \delta)$ is an object of $\text{Cris}^\wedge(C/A)$, then we have $$ \mathop{\rm Mor}\nolimits_{\text{Cris}^\wedge(C/A)}(D, B) = \mathop{\rm Hom}\nolimits_A((P, J), (B, \mathop{\rm Ker}(B \to C))) $$ and similarly for $D(n)$ replacing $(P, J)$ by $(P \otimes_A \ldots \otimes_A P, J(n))$. The property on coproducts follows as $P \otimes_A \ldots \otimes_A P$ is a coproduct. $\square$In the lemma below we will consider pairs $(M, \nabla)$ satisfying the following conditions

- $M$ is a $p$-adically complete $D$-module,
- $\nabla : M \to M \otimes^\wedge_D \Omega_D$ is a connection, i.e., $\nabla(fm) = m \otimes \text{d}f + f\nabla(m)$,
- $\nabla$ is integrable (see Remark 51.6.10), and
- $\nabla$ is
topologically quasi-nilpotent: If we write $\nabla(m) = \sum \theta_i(m)\text{d}x_i$ for some operators $\theta_i : M \to M$, then for any $m \in M$ there are only finitely many pairs $(i, k)$ such that $\theta_i^k(m) \not \in pM$.The operators $\theta_i$ are sometimes denoted $\nabla_{\partial/\partial x_i}$ in the literature. In the following lemma we construct a functor from crystals in quasi-coherent modules on $\text{Cris}(X/S)$ to the category of such pairs. We will show this functor is an equivalence in Proposition 51.17.4.

Lemma 51.17.3. In the situation above there is a functor $$ \begin{matrix} \text{crystals in quasi-coherent} \\ \mathcal{O}_{X/S}\text{-modules on }\text{Cris}(X/S) \end{matrix} \longrightarrow \begin{matrix} \text{pairs }(M, \nabla)\text{ satisfying} \\ \text{(1), (2), (3), and (4)} \end{matrix} $$

Proof.Let $\mathcal{F}$ be a crystal in quasi-coherent modules on $X/S$. Set $T_e = \mathop{\rm Spec}(D_e)$ so that $(X, T_e, \bar\gamma)$ is an object of $\text{Cris}(X/S)$ for $e \gg 0$. We have morphisms $$ (X, T_e, \bar\gamma) \to (X, T_{e + 1}, \bar\gamma) \to \ldots $$ which are closed immersions. We set $$ M = \mathop{\rm lim}\nolimits_e \Gamma((X, T_e, \bar\gamma), \mathcal{F}) = \mathop{\rm lim}\nolimits_e \Gamma(T_e, \mathcal{F}_{T_e}) = \mathop{\rm lim}\nolimits_e M_e $$ 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.By Lemma 51.15.1 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 $$ \nabla : M \longrightarrow M \otimes^\wedge_D \Omega_D $$ 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 $$ M \otimes^\wedge_{D, p_0} D(1) \rightarrow M(1) \leftarrow M \otimes^\wedge_{D, p_1} D(1) $$ compare with the proof of Lemma 51.15.1. Denote $c$ the composition from left to right. Pick $m \in M$. Write $\xi_i = x_i \otimes 1 - 1 \otimes x_i$. Using (51.17.1.1) we can write uniquely $$ c(m \otimes 1) = \sum\nolimits_K \theta_K(m) \otimes \prod \xi_i^{[k_i]} $$ 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 $$ \nabla(m) = \sum \theta_i(m) \text{d}x_i. $$ 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 51.15.1 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 51.15.1 we see that $$ q_{02}^*c = q_{12}^*c \circ q_{01}^*c. $$ This means that $$ \sum\nolimits_{K''} \theta_{K''}(m) \otimes \prod {\zeta''_i}^{[k''_i]} = \sum\nolimits_{K', K} \theta_{K'}(\theta_K(m)) \otimes \prod {\zeta'_i}^{[k'_i]} \prod \zeta_i^{[k_i]} $$ in $M \otimes^\wedge_{D, q_2} D(2)$ where \begin{align*} \zeta_i & = x_i \otimes 1 \otimes 1 - 1 \otimes x_i \otimes 1,\\ \zeta'_i & = 1 \otimes x_i \otimes 1 - 1 \otimes 1 \otimes x_i,\\ \zeta''_i & = x_i \otimes 1 \otimes 1 - 1 \otimes 1 \otimes x_i. \end{align*} 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 51.17.1. 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 $$ \theta_K(m) = (\prod \theta_i^{k_i})(m). $$ 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 $p$. $\square$

Proposition 51.17.4. The functor $$ \begin{matrix} \text{crystals in quasi-coherent} \\ \mathcal{O}_{X/S}\text{-modules on }\text{Cris}(X/S) \end{matrix} \longrightarrow \begin{matrix} \text{pairs }(M, \nabla)\text{ satisfying} \\ \text{(1), (2), (3), and (4)} \end{matrix} $$ of Lemma 51.17.3 is an equivalence of categories.

Proof.Let $(M, \nabla)$ be given. We are going to construct a crystal in quasi-coherent modules $\mathcal{F}$. Write $\nabla(m) = \sum \theta_i(m)\text{d}x_i$. Then $\theta_i \circ \theta_j = \theta_j \circ \theta_i$ and we can set $\theta_K(m) = (\prod \theta_i^{k_i})(m)$ for any multi-index $K = (k_i)$ with $k_i \geq 0$ and $\sum k_i < \infty$.Let $(U, T, \delta)$ be any object of $\text{Cris}(X/S)$ with $T$ affine. Say $T = \mathop{\rm Spec}(B)$ and the ideal of $U \to T$ is $J_B \subset B$. By Lemma 51.5.6 there exists an integer $e$ and a morphism $$ f : (U, T, \delta) \longrightarrow (X, T_e, \bar\gamma) $$ where $T_e = \mathop{\rm Spec}(D_e)$ as in the proof of Lemma 51.17.3. Choose such an $e$ and $f$; denote $f : D \to B$ also the corresponding divided power $A$-algebra map. We will set $\mathcal{F}_T$ equal to the quasi-coherent sheaf of $\mathcal{O}_T$-modules associated to the $B$-module $$ M \otimes_{D, f} B. $$ However, we have to show that this is independent of the choice of $f$. Suppose that $g : D \to B$ is a second such morphism. Since $f$ and $g$ are morphisms in $\text{Cris}(X/S)$ we see that the image of $f - g : D \to B$ is contained in the divided power ideal $J_B$. Write $\xi_i = f(x_i) - g(x_i) \in J_B$. By analogy with the proof of Lemma 51.17.3 we define an isomorphism $$ c_{f, g} : M \otimes_{D, f} B \longrightarrow M \otimes_{D, g} B $$ by the formula $$ m \otimes 1 \longmapsto \sum\nolimits_K \theta_K(m) \otimes \prod \xi_i^{[k_i]} $$ which makes sense by our remarks above and the fact that $\nabla$ is topologically quasi-nilpotent (so the sum is finite!). A computation shows that $$ c_{g, h} \circ c_{f, g} = c_{f, h} $$ if given a third morphism $h : (U, T, \delta) \longrightarrow (X, T_e, \bar\gamma)$. It is also true that $c_{f, f} = 1$. Hence these maps are all isomorphisms and we see that the module $\mathcal{F}_T$ is independent of the choice of $f$.

If $a : (U', T', \delta') \to (U, T, \delta)$ is a morphism of affine objects of $\text{Cris}(X/S)$, then choosing $f' = f \circ a$ it is clear that there exists a canonical isomorphism $a^*\mathcal{F}_T \to \mathcal{F}_{T'}$. We omit the verification that this map is independent of the choice of $f$. Using these maps as the restriction maps it is clear that we obtain a crystal in quasi-coherent modules on the full subcategory of $\text{Cris}(X/S)$ consisting of affine objects. We omit the proof that this extends to a crystal on all of $\text{Cris}(X/S)$. We also omit the proof that this procedure is a functor and that it is quasi-inverse to the functor constructed in Lemma 51.17.3. $\square$

Lemma 51.17.5. In Situation 51.5.1. Let $A \to P' \to C$ be ring maps with $A \to P'$ smooth and $P' \to C$ surjective with kernel $J'$. Let $D'$ be the $p$-adic completion of $D_{P', \gamma}(J')$. There are homomorphisms of divided power $A$-algebras $$ a : D \longrightarrow D',\quad b : D' \longrightarrow D $$ compatible with the maps $D \to C$ and $D' \to C$ such that $a \circ b = \text{id}_{D'}$. These maps induce an equivalence of categories of pairs $(M, \nabla)$ satisfying (1), (2), (3), and (4) over $D$ and pairs $(M', \nabla')$ satisfying (1), (2), (3), and (4) over $D'$. In particular, the equivalence of categories of Proposition 51.17.4 also holds for the corresponding functor towards pairs over $D'$.

Proof.We can pick the map $P = A[x_i] \to C$ such that it factors through a surjection of $A$-algebras $P \to P'$ (we may have to increase the number of variables in $P$ to do this). Hence we obtain a surjective map $a : D \to D'$ by functoriality of divided power envelopes and completion. Pick $e$ large enough so that $D_e$ is a divided power thickening of $C$ over $A$. Then $D_e \to C$ is a surjection whose kernel is locally nilpotent, see Divided Power Algebra, Lemma 23.2.6. Setting $D'_e = D'/p^eD'$ we see that the kernel of $D_e \to D'_e$ is locally nilpotent. Hence by Algebra, Lemma 10.136.17 we can find a lift $\beta_e : P' \to D_e$ of the map $P' \to D'_e$. Note that $D_{e + i + 1} \to D_{e + i} \times_{D'_{e + i}} D'_{e + i + 1}$ is surjective with square zero kernel for any $i \geq 0$ because $p^{e + i}D \to p^{e + i}D'$ is surjective. Applying the usual lifting property (Algebra, Proposition 10.136.13) successively to the diagrams $$ \xymatrix{ P' \ar[r] & D_{e + i} \times_{D'_{e + i}} D'_{e + i + 1} \\ A \ar[u] \ar[r] & D_{e + i + 1} \ar[u] } $$ we see that we can find an $A$-algebra map $\beta : P' \to D$ whose composition with $a$ is the given map $P' \to D'$. By the universal property of the divided power envelope we obtain a map $D_{P', \gamma}(J') \to D$. As $D$ is $p$-adically complete we obtain $b : D' \to D$ such that $a \circ b = \text{id}_{D'}$.Consider the base change functor $$ (M, \nabla) \longmapsto (M \otimes^\wedge_D D', \nabla') $$ from pairs for $D$ to pairs for $D'$, see Remark 51.6.11. Similarly, we have the base change functor corresponding to the divided power homomorphism $D' \to D$. To finish the proof of the lemma we have to show that the base change for the compositions $b \circ a : D \to D$ and $a \circ b : D' \to D'$ are isomorphic to the identity functor. This is clear for the second as $a \circ b = \text{id}_{D'}$. To prove it for the first, we use the functorial isomorphism $$ c_{\text{id}_D, b \circ a} : M \otimes_{D, \text{id}_D} D \longrightarrow M \otimes_{D, b \circ a} D $$ of the proof of Proposition 51.17.4. The only thing to prove is that these maps are horizontal, which we omit.

The last statement of the proof now follows. $\square$

Remark 51.17.6. The equivalence of Proposition 51.17.4 holds if we start with a surjection $P \to C$ where $P/A$ satisfies the strong lifting property of Algebra, Lemma 10.136.17. To prove this we can argue as in the proof of Lemma 51.17.5. (Details will be added here if we ever need this.) Presumably there is also a direct proof of this result, but the advantage of using polynomial rings is that the rings $D(n)$ are $p$-adic completions of divided power polynomial rings and the algebra is simplified.

The code snippet corresponding to this tag is a part of the file `crystalline.tex` and is located in lines 2926–3365 (see updates for more information).

```
\section{Crystals in quasi-coherent modules}
\label{section-quasi-coherent-crystals}
\noindent
In Situation \ref{situation-affine}.
Set $X = \Spec(C)$ and $S = \Spec(A)$. We are going to
classify crystals in quasi-coherent modules on $\text{Cris}(X/S)$.
Before we do so we fix some notation.
\medskip\noindent
Choose a polynomial ring $P = A[x_i]$ over $A$ and a surjection $P \to C$
of $A$-algebras with kernel $J = \Ker(P \to C)$. Set
\begin{equation}
\label{equation-D}
D = \lim_e D_{P, \gamma}(J) / p^eD_{P, \gamma}(J)
\end{equation}
for the $p$-adically completed divided power envelope.
This ring comes with a divided power ideal $\bar J$ and divided power
structure $\bar \gamma$, see Lemma \ref{lemma-list-properties}.
Set $D_e = D/p^eD$ and denote $\bar J_e$ the image of $\bar J$ in $D_e$.
We will use the short hand
\begin{equation}
\label{equation-omega-D}
\Omega_D = \lim_e \Omega_{D_e/A, \bar\gamma} =
\lim_e \Omega_{D/A, \bar\gamma}/p^e\Omega_{D/A, \bar\gamma}
\end{equation}
for the $p$-adic completion of the module of divided power differentials,
see Lemma \ref{lemma-differentials-completion}.
It is also the $p$-adic completion of
$\Omega_{D_{P, \gamma}(J)/A, \bar\gamma}$
which is free on $\text{d}x_i$, see
Lemma \ref{lemma-module-differentials-divided-power-envelope}.
Hence any element of $\Omega_D$ can be written uniquely as a sum
$\sum f_i\text{d}x_i$ with for all $e$ only finitely many $f_i$
not in $p^eD$. Moreover, the maps
$\text{d}_{D_e/A, \bar\gamma} : D_e \to \Omega_{D_e/A, \bar\gamma}$
fit together to define a divided power $A$-derivation
\begin{equation}
\label{equation-derivation-D}
\text{d} : D \longrightarrow \Omega_D
\end{equation}
on $p$-adic completions.
\medskip\noindent
We will also need the ``products $\Spec(D(n))$ of $\Spec(D)$'', see
Proposition \ref{proposition-compute-cohomology} and its proof for an
explanation. Formally these are defined as follows. For $n \geq 0$ let
$J(n) = \Ker(P \otimes_A \ldots \otimes_A P \to C)$ where
the tensor product has $n + 1$ factors. We set
\begin{equation}
\label{equation-Dn}
D(n) = \lim_e
D_{P \otimes_A \ldots \otimes_A P, \gamma}(J(n))/
p^eD_{P \otimes_A \ldots \otimes_A P, \gamma}(J(n))
\end{equation}
equal to the $p$-adic completion of the divided power envelope.
We denote $\bar J(n)$ its divided power ideal and $\bar \gamma(n)$
its divided powers. We also introduce $D(n)_e = D(n)/p^eD(n)$ as well
as the $p$-adically completed module of differentials
\begin{equation}
\label{equation-omega-Dn}
\Omega_{D(n)} = \lim_e \Omega_{D(n)_e/A, \bar\gamma} =
\lim_e \Omega_{D(n)/A, \bar\gamma}/p^e\Omega_{D(n)/A, \bar\gamma}
\end{equation}
and derivation
\begin{equation}
\label{equation-derivation-Dn}
\text{d} : D(n) \longrightarrow \Omega_{D(n)}
\end{equation}
Of course we have $D = D(0)$. Note that the rings $D(0), D(1), D(2), \ldots$
form a cosimplicial object in the category of divided power rings.
\begin{lemma}
\label{lemma-structure-Dn}
Let $D$ and $D(n)$ be as in (\ref{equation-D}) and (\ref{equation-Dn}).
The coprojection $P \to P \otimes_A \ldots \otimes_A P$,
$f \mapsto f \otimes 1 \otimes \ldots \otimes 1$
induces an isomorphism
\begin{equation}
\label{equation-structure-Dn}
D(n) = \lim_e D\langle \xi_i(j) \rangle/p^eD\langle \xi_i(j) \rangle
\end{equation}
of algebras over $D$ with
$$
\xi_i(j) = x_i \otimes 1 \otimes \ldots \otimes 1 -
1 \otimes \ldots \otimes 1 \otimes x_i \otimes 1 \otimes \ldots \otimes 1
$$
for $j = 1, \ldots, n$.
\end{lemma}
\begin{proof}
We have
$$
P \otimes_A \ldots \otimes_A P = P[\xi_i(j)]
$$
and $J(n)$ is generated by $J$ and the elements $\xi_i(j)$.
Hence the lemma follows from
Lemma \ref{lemma-divided-power-envelope-add-variables}.
\end{proof}
\begin{lemma}
\label{lemma-property-Dn}
Let $D$ and $D(n)$ be as in (\ref{equation-D}) and (\ref{equation-Dn}).
Then $(D, \bar J, \bar\gamma)$ and $(D(n), \bar J(n), \bar\gamma(n))$
are objects of $\text{Cris}^\wedge(C/A)$, see
Remark \ref{remark-completed-affine-site}, and
$$
D(n) = \coprod\nolimits_{j = 0, \ldots, n} D
$$
in $\text{Cris}^\wedge(C/A)$.
\end{lemma}
\begin{proof}
The first assertion is clear. For the second, if $(B \to C, \delta)$ is an
object of $\text{Cris}^\wedge(C/A)$, then we have
$$
\Mor_{\text{Cris}^\wedge(C/A)}(D, B) =
\Hom_A((P, J), (B, \Ker(B \to C)))
$$
and similarly for $D(n)$ replacing $(P, J)$ by
$(P \otimes_A \ldots \otimes_A P, J(n))$. The property on coproducts follows
as $P \otimes_A \ldots \otimes_A P$ is a coproduct.
\end{proof}
\noindent
In the lemma below we will consider pairs $(M, \nabla)$ satisfying the
following conditions
\begin{enumerate}
\item
\label{item-complete}
$M$ is a $p$-adically complete $D$-module,
\item
\label{item-connection}
$\nabla : M \to M \otimes^\wedge_D \Omega_D$ is a connection, i.e.,
$\nabla(fm) = m \otimes \text{d}f + f\nabla(m)$,
\item
\label{item-integrable}
$\nabla$ is integrable
(see Remark \ref{remark-connection}), and
\item
\label{item-topologically-quasi-nilpotent}
$\nabla$ is {\it topologically quasi-nilpotent}: If we write
$\nabla(m) = \sum \theta_i(m)\text{d}x_i$ for some operators
$\theta_i : M \to M$, then for any $m \in M$ there are only finitely
many pairs $(i, k)$ such that $\theta_i^k(m) \not \in pM$.
\end{enumerate}
The operators $\theta_i$ are sometimes denoted
$\nabla_{\partial/\partial x_i}$ in the literature.
In the following lemma we construct a functor from crystals in quasi-coherent
modules on $\text{Cris}(X/S)$ to the category of such pairs. We will show
this functor is an equivalence in
Proposition \ref{proposition-crystals-on-affine}.
\begin{lemma}
\label{lemma-crystals-on-affine}
In the situation above there is a functor
$$
\begin{matrix}
\text{crystals in quasi-coherent} \\
\mathcal{O}_{X/S}\text{-modules on }\text{Cris}(X/S)
\end{matrix}
\longrightarrow
\begin{matrix}
\text{pairs }(M, \nabla)\text{ satisfying} \\
\text{(\ref{item-complete}), (\ref{item-connection}),
(\ref{item-integrable}), and (\ref{item-topologically-quasi-nilpotent})}
\end{matrix}
$$
\end{lemma}
\begin{proof}
Let $\mathcal{F}$ be a crystal in quasi-coherent modules on $X/S$.
Set $T_e = \Spec(D_e)$ so that $(X, T_e, \bar\gamma)$ is an object
of $\text{Cris}(X/S)$ for $e \gg 0$. We have morphisms
$$
(X, T_e, \bar\gamma) \to (X, T_{e + 1}, \bar\gamma) \to \ldots
$$
which are closed immersions. We set
$$
M =
\lim_e \Gamma((X, T_e, \bar\gamma), \mathcal{F}) =
\lim_e \Gamma(T_e, \mathcal{F}_{T_e}) = \lim_e M_e
$$
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.
\medskip\noindent
By Lemma \ref{lemma-automatic-connection} 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
$$
\nabla : M \longrightarrow M \otimes^\wedge_D \Omega_D
$$
To see that this is topologically nilpotent we work out what this means.
\medskip\noindent
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
$$
M \otimes^\wedge_{D, p_0} D(1) \rightarrow M(1)
\leftarrow M \otimes^\wedge_{D, p_1} D(1)
$$
compare with the proof of Lemma \ref{lemma-automatic-connection}.
Denote $c$ the composition from left to right. Pick $m \in M$.
Write $\xi_i = x_i \otimes 1 - 1 \otimes x_i$.
Using (\ref{equation-structure-Dn}) we can write uniquely
$$
c(m \otimes 1) = \sum\nolimits_K \theta_K(m) \otimes \prod \xi_i^{[k_i]}
$$
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
$$
\nabla(m) = \sum \theta_i(m) \text{d}x_i.
$$
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 \ref{lemma-automatic-connection}
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.
\medskip\noindent
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 \ref{lemma-automatic-connection}
we see that
$$
q_{02}^*c = q_{12}^*c \circ q_{01}^*c.
$$
This means that
$$
\sum\nolimits_{K''} \theta_{K''}(m) \otimes \prod {\zeta''_i}^{[k''_i]}
=
\sum\nolimits_{K', K} \theta_{K'}(\theta_K(m))
\otimes \prod {\zeta'_i}^{[k'_i]} \prod \zeta_i^{[k_i]}
$$
in $M \otimes^\wedge_{D, q_2} D(2)$ where
\begin{align*}
\zeta_i & = x_i \otimes 1 \otimes 1 - 1 \otimes x_i \otimes 1,\\
\zeta'_i & = 1 \otimes x_i \otimes 1 - 1 \otimes 1 \otimes x_i,\\
\zeta''_i & = x_i \otimes 1 \otimes 1 - 1 \otimes 1 \otimes x_i.
\end{align*}
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 \ref{lemma-structure-Dn}.
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
$$
\theta_K(m) = (\prod \theta_i^{k_i})(m).
$$
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 $p$.
\end{proof}
\begin{proposition}
\label{proposition-crystals-on-affine}
The functor
$$
\begin{matrix}
\text{crystals in quasi-coherent} \\
\mathcal{O}_{X/S}\text{-modules on }\text{Cris}(X/S)
\end{matrix}
\longrightarrow
\begin{matrix}
\text{pairs }(M, \nabla)\text{ satisfying} \\
\text{(\ref{item-complete}), (\ref{item-connection}),
(\ref{item-integrable}), and (\ref{item-topologically-quasi-nilpotent})}
\end{matrix}
$$
of Lemma \ref{lemma-crystals-on-affine}
is an equivalence of categories.
\end{proposition}
\begin{proof}
Let $(M, \nabla)$ be given. We are going to construct
a crystal in quasi-coherent modules $\mathcal{F}$.
Write $\nabla(m) = \sum \theta_i(m)\text{d}x_i$.
Then $\theta_i \circ \theta_j = \theta_j \circ \theta_i$ and we
can set $\theta_K(m) = (\prod \theta_i^{k_i})(m)$ for any multi-index
$K = (k_i)$ with $k_i \geq 0$ and $\sum k_i < \infty$.
\medskip\noindent
Let $(U, T, \delta)$ be any object of $\text{Cris}(X/S)$ with $T$ affine.
Say $T = \Spec(B)$ and the ideal of $U \to T$ is $J_B \subset B$.
By Lemma \ref{lemma-set-generators} there exists an integer $e$ and a morphism
$$
f : (U, T, \delta) \longrightarrow (X, T_e, \bar\gamma)
$$
where $T_e = \Spec(D_e)$ as in the proof of
Lemma \ref{lemma-crystals-on-affine}.
Choose such an $e$ and $f$; denote $f : D \to B$ also the corresponding
divided power $A$-algebra map. We will set $\mathcal{F}_T$ equal to the
quasi-coherent sheaf of $\mathcal{O}_T$-modules associated to the $B$-module
$$
M \otimes_{D, f} B.
$$
However, we have to show that this is independent of the choice of $f$.
Suppose that $g : D \to B$ is a second such morphism. Since $f$ and $g$
are morphisms in $\text{Cris}(X/S)$ we see that the image of
$f - g : D \to B$ is contained in the divided power ideal $J_B$.
Write $\xi_i = f(x_i) - g(x_i) \in J_B$. By analogy with the proof
of Lemma \ref{lemma-crystals-on-affine} we define an isomorphism
$$
c_{f, g} : M \otimes_{D, f} B \longrightarrow M \otimes_{D, g} B
$$
by the formula
$$
m \otimes 1 \longmapsto
\sum\nolimits_K \theta_K(m) \otimes \prod \xi_i^{[k_i]}
$$
which makes sense by our remarks above and the fact that $\nabla$
is topologically quasi-nilpotent (so the sum is finite!).
A computation shows that
$$
c_{g, h} \circ c_{f, g} = c_{f, h}
$$
if given a third morphism
$h : (U, T, \delta) \longrightarrow (X, T_e, \bar\gamma)$.
It is also true that $c_{f, f} = 1$.
Hence these maps are all isomorphisms and we see that
the module $\mathcal{F}_T$ is independent of the choice of $f$.
\medskip\noindent
If $a : (U', T', \delta') \to (U, T, \delta)$ is a morphism of affine objects
of $\text{Cris}(X/S)$, then choosing $f' = f \circ a$ it is clear
that there exists a canonical isomorphism
$a^*\mathcal{F}_T \to \mathcal{F}_{T'}$. We omit the verification that this
map is independent of the choice of $f$. Using these maps as the restriction
maps it is clear that we obtain a crystal in quasi-coherent modules
on the full subcategory of $\text{Cris}(X/S)$ consisting of affine objects.
We omit the proof that this extends to a crystal on all of
$\text{Cris}(X/S)$. We also omit the proof that this procedure is a functor
and that it is quasi-inverse to the functor constructed in
Lemma \ref{lemma-crystals-on-affine}.
\end{proof}
\begin{lemma}
\label{lemma-crystals-on-affine-smooth}
In Situation \ref{situation-affine}.
Let $A \to P' \to C$ be ring maps with $A \to P'$ smooth and $P' \to C$
surjective with kernel $J'$. Let $D'$ be the $p$-adic completion of
$D_{P', \gamma}(J')$. There are homomorphisms of divided power $A$-algebras
$$
a : D \longrightarrow D',\quad b : D' \longrightarrow D
$$
compatible with the maps $D \to C$ and $D' \to C$ such that
$a \circ b = \text{id}_{D'}$. These maps induce
an equivalence of categories of pairs $(M, \nabla)$ satisfying
(\ref{item-complete}), (\ref{item-connection}),
(\ref{item-integrable}), and (\ref{item-topologically-quasi-nilpotent})
over $D$ and pairs $(M', \nabla')$ satisfying
(\ref{item-complete}), (\ref{item-connection}),
(\ref{item-integrable}), and (\ref{item-topologically-quasi-nilpotent})
over $D'$. In particular, the equivalence of categories of
Proposition \ref{proposition-crystals-on-affine}
also holds for the corresponding functor towards pairs over $D'$.
\end{lemma}
\begin{proof}
We can pick the map $P = A[x_i] \to C$ such that it factors through
a surjection of $A$-algebras $P \to P'$ (we may have to increase the
number of variables in $P$ to do this). Hence we obtain a surjective
map $a : D \to D'$ by functoriality of divided power envelopes and
completion. Pick $e$ large enough so that $D_e$ is a divided power
thickening of $C$ over $A$. Then $D_e \to C$ is a surjection whose kernel
is locally nilpotent, see Divided Power Algebra, Lemma \ref{dpa-lemma-nil}.
Setting $D'_e = D'/p^eD'$
we see that the kernel of $D_e \to D'_e$ is locally nilpotent.
Hence by Algebra, Lemma \ref{algebra-lemma-smooth-strong-lift}
we can find a lift $\beta_e : P' \to D_e$ of the map $P' \to D'_e$.
Note that $D_{e + i + 1} \to D_{e + i} \times_{D'_{e + i}} D'_{e + i + 1}$
is surjective with square zero kernel for any $i \geq 0$ because
$p^{e + i}D \to p^{e + i}D'$ is surjective. Applying the usual lifting
property (Algebra, Proposition \ref{algebra-proposition-smooth-formally-smooth})
successively to the diagrams
$$
\xymatrix{
P' \ar[r] & D_{e + i} \times_{D'_{e + i}} D'_{e + i + 1} \\
A \ar[u] \ar[r] & D_{e + i + 1} \ar[u]
}
$$
we see that we can find an $A$-algebra map $\beta : P' \to D$ whose
composition with $a$ is the given map $P' \to D'$.
By the universal property of the divided power envelope we obtain a
map $D_{P', \gamma}(J') \to D$. As $D$ is $p$-adically complete we
obtain $b : D' \to D$ such that $a \circ b = \text{id}_{D'}$.
\medskip\noindent
Consider the base change functor
$$
(M, \nabla) \longmapsto
(M \otimes^\wedge_D D', \nabla')
$$
from pairs for $D$ to pairs for $D'$, see
Remark \ref{remark-base-change-connection}.
Similarly, we have the base change functor corresponding to the divided
power homomorphism $D' \to D$. To finish the proof of the lemma we have
to show that the base change for the compositions $b \circ a : D \to D$
and $a \circ b : D' \to D'$ are isomorphic to the identity functor.
This is clear for the second as $a \circ b = \text{id}_{D'}$.
To prove it for the first, we use the functorial isomorphism
$$
c_{\text{id}_D, b \circ a} :
M \otimes_{D, \text{id}_D} D
\longrightarrow
M \otimes_{D, b \circ a} D
$$
of the proof of Proposition \ref{proposition-crystals-on-affine}.
The only thing to prove is that these maps are horizontal, which we omit.
\medskip\noindent
The last statement of the proof now follows.
\end{proof}
\begin{remark}
\label{remark-equivalence-more-general}
The equivalence of Proposition \ref{proposition-crystals-on-affine}
holds if we start with a surjection $P \to C$ where $P/A$ satisfies the
strong lifting property of
Algebra, Lemma \ref{algebra-lemma-smooth-strong-lift}.
To prove this we can argue as in the proof of
Lemma \ref{lemma-crystals-on-affine-smooth}.
(Details will be added here if we ever need this.)
Presumably there is also a direct proof of this result, but the advantage
of using polynomial rings is that the rings $D(n)$ are $p$-adic completions
of divided power polynomial rings and the algebra is simplified.
\end{remark}
```

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