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Tag 07JH

Chapter 51: Crystalline Cohomology > Section 51.17: Crystals in quasi-coherent modules

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. $\square$

    The code snippet corresponding to this tag is a part of the file crystalline.tex and is located in lines 3190–3207 (see updates for more information).

    \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}

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