## 59.17 Crystals in quasi-coherent modules

In Situation 59.5.1. Set $X = \mathop{\mathrm{Spec}}(C)$ and $S = \mathop{\mathrm{Spec}}(A)$. 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 $J = \mathop{\mathrm{Ker}}(P \to C)$. Set

59.17.0.1
\begin{equation} \label{crystalline-equation-D} D = \mathop{\mathrm{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 59.5.5. 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

59.17.0.2
\begin{equation} \label{crystalline-equation-omega-D} \Omega _ D = \mathop{\mathrm{lim}}\nolimits _ e \Omega _{D_ e/A, \bar\gamma } = \mathop{\mathrm{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 59.6.10. 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 59.6.6. 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

59.17.0.3
\begin{equation} \label{crystalline-equation-derivation-D} \text{d} : D \longrightarrow \Omega _ D \end{equation}

on $p$-adic completions.

We will also need the “products $\mathop{\mathrm{Spec}}(D(n))$ of $\mathop{\mathrm{Spec}}(D)$”, see Proposition 59.21.1 and its proof for an explanation. Formally these are defined as follows. For $n \geq 0$ let $J(n) = \mathop{\mathrm{Ker}}(P \otimes _ A \ldots \otimes _ A P \to C)$ where the tensor product has $n + 1$ factors. We set

59.17.0.4
\begin{equation} \label{crystalline-equation-Dn} D(n) = \mathop{\mathrm{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

59.17.0.5
\begin{equation} \label{crystalline-equation-omega-Dn} \Omega _{D(n)} = \mathop{\mathrm{lim}}\nolimits _ e \Omega _{D(n)_ e/A, \bar\gamma } = \mathop{\mathrm{lim}}\nolimits _ e \Omega _{D(n)/A, \bar\gamma }/p^ e\Omega _{D(n)/A, \bar\gamma } \end{equation}

and derivation

59.17.0.6
\begin{equation} \label{crystalline-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.

Lemma 59.17.1. Let $D$ and $D(n)$ be as in (59.17.0.1) and (59.17.0.4). The coprojection $P \to P \otimes _ A \ldots \otimes _ A P$, $f \mapsto f \otimes 1 \otimes \ldots \otimes 1$ induces an isomorphism

59.17.1.1
\begin{equation} \label{crystalline-equation-structure-Dn} D(n) = \mathop{\mathrm{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 $j = 1, \ldots , n$ where the second $x_ i$ is placed in the $j + 1$st slot; recall that $D(n)$ is constructed starting with the $n + 1$-fold tensor product of $P$ over $A$.

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 59.2.5. $\square$

Lemma 59.17.2. Let $D$ and $D(n)$ be as in (59.17.0.1) and (59.17.0.4). 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 59.5.4, 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

$\mathop{Mor}\nolimits _{\text{Cris}^\wedge (C/A)}(D, B) = \mathop{\mathrm{Hom}}\nolimits _ A((P, J), (B, \mathop{\mathrm{Ker}}(B \to C)))$

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

1. $M$ is a $p$-adically complete $D$-module,

2. $\nabla : M \to M \otimes ^\wedge _ D \Omega _ D$ is a connection, i.e., $\nabla (fm) = m \otimes \text{d}f + f\nabla (m)$,

3. $\nabla$ is integrable (see Remark 59.6.8), and

4. $\nabla$ 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 59.17.4.

Lemma 59.17.3. In the situation above there is a functor

$\begin{matrix} \text{crystals in quasi-coherent} \\ \mathcal{O}_{X/S}\text{-modules on }\text{Cris}(X/S) \end{matrix} \longrightarrow \begin{matrix} \text{pairs }(M, \nabla )\text{ satisfying} \\ \text{(07JB), (07JC), (07JD), and (07JE)} \end{matrix}$

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

$(X, T_ e, \bar\gamma ) \to (X, T_{e + 1}, \bar\gamma ) \to \ldots$

which are closed immersions. We set

$M = \mathop{\mathrm{lim}}\nolimits _ e \Gamma ((X, T_ e, \bar\gamma ), \mathcal{F}) = \mathop{\mathrm{lim}}\nolimits _ e \Gamma (T_ e, \mathcal{F}_{T_ e}) = \mathop{\mathrm{lim}}\nolimits _ e M_ e$

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, 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

$\nabla : M \longrightarrow M \otimes ^\wedge _ D \Omega _ D$

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 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

$c(m \otimes 1) = \sum \nolimits _ K \theta _ K(m) \otimes \prod \xi _ i^{[k_ i]}$

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 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

$q_{02}^*c = q_{12}^*c \circ q_{01}^*c.$

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 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

$\theta _ K(m) = (\prod \theta _ i^{k_ i})(m).$

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 59.17.4. The functor

$\begin{matrix} \text{crystals in quasi-coherent} \\ \mathcal{O}_{X/S}\text{-modules on }\text{Cris}(X/S) \end{matrix} \longrightarrow \begin{matrix} \text{pairs }(M, \nabla )\text{ satisfying} \\ \text{(07JB), (07JC), (07JD), and (07JE)} \end{matrix}$

of Lemma 59.17.3 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 $T = \mathop{\mathrm{Spec}}(B)$ and the ideal of $U \to T$ is $J_ B \subset B$. By Lemma 59.5.6 there exists an integer $e$ and a morphism

$f : (U, T, \delta ) \longrightarrow (X, T_ e, \bar\gamma )$

where $T_ e = \mathop{\mathrm{Spec}}(D_ e)$ as in the proof of Lemma 59.17.3. 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 59.17.3 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 59.17.3. $\square$

Lemma 59.17.5. In Situation 59.5.1. 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 59.17.4 also holds for the corresponding functor towards pairs over $D'$.

Proof. Before we embark on the proof we briefly explain how to formulate condition (4) for the case of pairs $(M', \nabla ')$ over $D'$. Since $A \to P'$ is smooth the $P'$-module $\Omega _{P'/A}$ is finite projective. Thus we can choose a map $\Omega _{P'/A} \to \bigoplus _{i = 1, \ldots , n} P'$ wich identifies $\Omega _{P'/A}$ with a direct summand of the target. This determines a map $\Omega _{D'} \to \bigoplus _{i = 1, \ldots , n} D'$ identifying the source with a direct summand of the target. Thus for $m' \in M'$ we can write $\nabla '(m') = \sum \theta '_ i(m')$ with $\theta '_ i(m') \in D'$. The topogical quasi-nilpotence of $\nabla '$ means: for any $m' \in M'$ there are only finitely many pairs $(i, k)$ such that $(\theta '_ i)^ k(m') \not\in pM'$. (Since in this case there are only finitely many $i$ this just means that $\theta _ i^ k(m') \in pM'$ for all $k \gg 0$.)

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 10.138.17 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 10.138.13) 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 59.6.9. 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 59.17.4. The only thing to prove is that these maps are horizontal, which we omit.

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

Remark 59.17.6. The equivalence of Proposition 59.17.4 holds if we start with a surjection $P \to C$ where $P/A$ satisfies the strong lifting property of Algebra, Lemma 10.138.17. To prove this we can argue as in the proof of Lemma 59.17.5. (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.

## Comments (2)

Comment #1446 by Matthieu Romagny on

Typo: just before the statement of 46.17.3, replace "equivalent" by "equivalence"

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