60.17 Crystals in quasi-coherent modules
In Situation 60.5.1. 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
60.17.0.1
\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 60.5.5. 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
60.17.0.2
\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 60.6.10. 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 60.6.6. 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
60.17.0.3
\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 60.21.1 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
60.17.0.4
\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
60.17.0.5
\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
60.17.0.6
\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 60.17.1. Let $D$ and $D(n)$ be as in (60.17.0.1) and (60.17.0.4). The coprojection $P \to P \otimes _ A \ldots \otimes _ A P$, $f \mapsto f \otimes 1 \otimes \ldots \otimes 1$ induces an isomorphism
60.17.1.1
\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 60.2.5.
$\square$
Lemma 60.17.2. Let $D$ and $D(n)$ be as in (60.17.0.1) and (60.17.0.4). 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 60.5.4, 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
\[ \mathop{\mathrm{Mor}}\nolimits _{\text{Cris}^\wedge (C/A)}(D, B) = \mathop{\mathrm{Hom}}\nolimits _ A((P, J), (B, \mathop{\mathrm{Ker}}(B \to C))) \]
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
$M$ 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 60.6.8), 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 60.17.4.
Lemma 60.17.3. 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 60.15.1 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 60.15.1. Denote $c$ the composition from left to right. Pick $m \in M$. Write $\xi _ i = x_ i \otimes 1 - 1 \otimes x_ i$. Using (60.17.1.1) 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 60.15.1 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 60.15.1 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 60.17.1. 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 60.17.4. 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 60.17.3 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 60.5.6 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 60.17.3. 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 60.17.3 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 60.17.3.
$\square$
Lemma 60.17.5. In Situation 60.5.1. 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)1 over $D'$. In particular, the equivalence of categories of Proposition 60.17.4 also holds for the corresponding functor towards pairs over $D'$.
Proof.
First, suppose that $P' = A[y_1, \ldots , y_ m]$ is a polynomial algebra over $A$. In this case, we can find ring maps $P \to P'$ and $P' \to P$ compatible with the maps to $C$ which induce maps $a : D \to D'$ and $b : D' \to D$ as in the lemma. Using completed base change along $a$ and $b$ we obtain functors between the categories of modules with connection satisfying properties (1), (2), (3), and (4) simply because these these categories are equivalent to the category of quasi-coherent crystals by Proposition 60.17.4 (and this equivalence is compatible with the base change operation as shown in the proof of the proposition).
Proof for general smooth $P'$. By the first paragraph of the proof we may assume $P = A[y_1, \ldots , y_ m]$ which gives us a surjection $P \to P'$ compatible with the map to $C$. 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 functors
\[ F : (M, \nabla ) \longmapsto (M \otimes ^\wedge _{D, a} D', \nabla ') \quad \text{and}\quad G : (M', \nabla ') \longmapsto (M' \otimes ^\wedge _{D', b} D, \nabla ) \]
on modules with connections satisfying (1), (2), and (3). See Remark 60.6.9. Since $a \circ b = \text{id}_{D'}$ we see that $F \circ G$ is the identity functor. Let us say that $(M', \nabla ')$ has property (4) if this is true for $G(M', \nabla ')$. A formal argument now shows that to finish the proof it suffices to show that $G(F(M, \nabla ))$ is isomorphic to $(M, \nabla )$ in the case that $(M, \nabla )$ satisfies all four conditions (1), (2), (3), and (4). For this 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 60.17.4 (which requires the topological quasi-nilpotency of $\nabla $ which we have assumed). It remains to prove that this map is horizontal, i.e., compatible with connections, which we omit.
The last statement of the proof now follows.
$\square$
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