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Tag 07J7

54.17. Crystals in quasi-coherent modules

In Situation 54.5.1. 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{54.17.0.1} 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 54.5.5. 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{54.17.0.2} \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 54.6.12. 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 54.6.6. 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{54.17.0.3} \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 54.21.1 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{54.17.0.4} 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{54.17.0.5} \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{54.17.0.6} \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 54.17.1. Let $D$ and $D(n)$ be as in (54.17.0.1) and (54.17.0.4). 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{54.17.1.1} 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 54.2.5. $\square$

Lemma 54.17.2. Let $D$ and $D(n)$ be as in (54.17.0.1) and (54.17.0.4). 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 54.5.4, 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

  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 54.6.10), 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 54.17.4.

Lemma 54.17.3. 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 54.15.1 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 54.15.1. Denote $c$ the composition from left to right. Pick $m \in M$. Write $\xi_i = x_i \otimes 1 - 1 \otimes x_i$. Using (54.17.1.1) 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 54.15.1 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 54.15.1 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 54.17.1. 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 54.17.4. 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 54.17.3 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 54.5.6 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 54.17.3. 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 54.17.3 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 54.17.3. $\square$

Lemma 54.17.5. In Situation 54.5.1. 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 54.17.4 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 54.6.11. 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 54.17.4. The only thing to prove is that these maps are horizontal, which we omit.

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

Remark 54.17.6. The equivalence of Proposition 54.17.4 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 54.17.5. (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}

    Comments (2)

    Comment #1446 by Matthieu Romagny on April 24, 2015 a 12:06 pm UTC

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

    Comment #1457 by Johan (site) on May 15, 2015 a 12:51 pm UTC

    Thanks! This and your other typo are fixed here.

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