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Tag 07IN

51.10. Sheaves on the crystalline site

Notation and assumptions as in Situation 51.7.5. In order to discuss the small and big crystalline sites of $X/S$ simultaneously in this section we let $$ \mathcal{C} = \text{CRIS}(X/S) \quad\text{or}\quad \mathcal{C} = \text{Cris}(X/S). $$ A sheaf $\mathcal{F}$ on $\mathcal{C}$ gives rise to a restriction $\mathcal{F}_T$ for every object $(U, T, \delta)$ of $\mathcal{C}$. Namely, $\mathcal{F}_T$ is the Zariski sheaf on the scheme $T$ defined by the rule $$ \mathcal{F}_T(W) = \mathcal{F}(U \cap W, W, \delta|_W) $$ for $W \subset T$ is open. Moreover, if $f : T \to T'$ is a morphism between objects $(U, T, \delta)$ and $(U', T', \delta')$ of $\mathcal{C}$, then there is a canonical comparison map \begin{equation} \tag{51.10.0.1} c_f : f^{-1}\mathcal{F}_{T'} \longrightarrow \mathcal{F}_T. \end{equation} Namely, if $W' \subset T'$ is open then $f$ induces a morphism $$ f|_{f^{-1}W'} : (U \cap f^{-1}(W'), f^{-1}W', \delta|_{f^{-1}W'}) \longrightarrow (U' \cap W', W', \delta|_{W'}) $$ of $\mathcal{C}$, hence we can use the restriction mapping $(f|_{f^{-1}W'})^*$ of $\mathcal{F}$ to define a map $\mathcal{F}_{T'}(W') \to \mathcal{F}_T(f^{-1}W')$. These maps are clearly compatible with further restriction, hence define an $f$-map from $\mathcal{F}_{T'}$ to $\mathcal{F}_T$ (see Sheaves, Section 6.21 and especially Sheaves, Definition 6.21.7). Thus a map $c_f$ as in (51.10.0.1). Note that if $f$ is an open immersion, then $c_f$ is an isomorphism, because in that case $\mathcal{F}_T$ is just the restriction of $\mathcal{F}_{T'}$ to $T$.

Conversely, given Zariski sheaves $\mathcal{F}_T$ for every object $(U, T, \delta)$ of $\mathcal{C}$ and comparison maps $c_f$ as above which (a) are isomorphisms for open immersions, and (b) satisfy a suitable cocycle condition, we obtain a sheaf on $\mathcal{C}$. This is proved exactly as in Topologies, Lemma 33.3.19.

The structure sheaf on $\mathcal{C}$ is the sheaf $\mathcal{O}_{X/S}$ defined by the rule $$ \mathcal{O}_{X/S} : (U, T, \delta) \longmapsto \Gamma(T, \mathcal{O}_T) $$ This is a sheaf by the definition of coverings in $\mathcal{C}$. Suppose that $\mathcal{F}$ is a sheaf of $\mathcal{O}_{X/S}$-modules. In this case the comparison mappings (51.10.0.1) define a comparison map \begin{equation} \tag{51.10.0.2} c_f : f^*\mathcal{F}_T \longrightarrow \mathcal{F}_{T'} \end{equation} of $\mathcal{O}_T$-modules.

Another type of example comes by starting with a sheaf $\mathcal{G}$ on $(\textit{Sch}/X)_{Zar}$ or $X_{Zar}$ (depending on whether $\mathcal{C} = \text{CRIS}(X/S)$ or $\mathcal{C} = \text{Cris}(X/S)$). Then $\underline{\mathcal{G}}$ defined by the rule $$ \underline{\mathcal{G}} : (U, T, \delta) \longmapsto \mathcal{G}(U) $$ is a sheaf on $\mathcal{C}$. In particular, if we take $\mathcal{G} = \mathbf{G}_a = \mathcal{O}_X$, then we obtain $$ \underline{\mathbf{G}_a} : (U, T, \delta) \longmapsto \Gamma(U, \mathcal{O}_U) $$ There is a surjective map of sheaves $\mathcal{O}_{X/S} \to \underline{\mathbf{G}_a}$ defined by the canonical maps $\Gamma(T, \mathcal{O}_T) \to \Gamma(U, \mathcal{O}_U)$ for objects $(U, T, \delta)$. The kernel of this map is denoted $\mathcal{J}_{X/S}$, hence a short exact sequence $$ 0 \to \mathcal{J}_{X/S} \to \mathcal{O}_{X/S} \to \underline{\mathbf{G}_a} \to 0 $$ Note that $\mathcal{J}_{X/S}$ comes equipped with a canonical divided power structure. After all, for each object $(U, T, \delta)$ the third component $\delta$ is a divided power structure on the kernel of $\mathcal{O}_T \to \mathcal{O}_U$. Hence the (big) crystalline topos is a divided power topos.

    The code snippet corresponding to this tag is a part of the file crystalline.tex and is located in lines 2028–2141 (see updates for more information).

    \section{Sheaves on the crystalline site}
    \label{section-sheaves}
    
    \noindent
    Notation and assumptions as in Situation \ref{situation-global}.
    In order to discuss the small and big crystalline sites of $X/S$
    simultaneously in this section we let
    $$
    \mathcal{C} = \text{CRIS}(X/S)
    \quad\text{or}\quad
    \mathcal{C} = \text{Cris}(X/S).
    $$
    A sheaf $\mathcal{F}$ on $\mathcal{C}$ gives rise to
    a {\it restriction} $\mathcal{F}_T$ for every object $(U, T, \delta)$
    of $\mathcal{C}$. Namely, $\mathcal{F}_T$ is the Zariski sheaf on
    the scheme $T$ defined by the rule
    $$
    \mathcal{F}_T(W) = \mathcal{F}(U \cap W, W, \delta|_W)
    $$
    for $W \subset T$ is open. Moreover, if $f : T \to T'$ is a morphism
    between objects
    $(U, T, \delta)$ and $(U', T', \delta')$ of $\mathcal{C}$, then there
    is a canonical {\it comparison} map
    \begin{equation}
    \label{equation-comparison}
    c_f : f^{-1}\mathcal{F}_{T'} \longrightarrow \mathcal{F}_T.
    \end{equation}
    Namely, if $W' \subset T'$ is open then $f$ induces a morphism
    $$
    f|_{f^{-1}W'} :
    (U \cap f^{-1}(W'), f^{-1}W', \delta|_{f^{-1}W'})
    \longrightarrow
    (U' \cap W', W', \delta|_{W'})
    $$
    of $\mathcal{C}$, hence we can use the restriction mapping
    $(f|_{f^{-1}W'})^*$ of $\mathcal{F}$ to define a map
    $\mathcal{F}_{T'}(W') \to \mathcal{F}_T(f^{-1}W')$.
    These maps are clearly compatible with further restriction, hence
    define an $f$-map from $\mathcal{F}_{T'}$ to $\mathcal{F}_T$ (see
    Sheaves, Section \ref{sheaves-section-presheaves-functorial}
    and especially
    Sheaves, Definition \ref{sheaves-definition-f-map}).
    Thus a map $c_f$ as in (\ref{equation-comparison}).
    Note that if $f$ is an open immersion, then $c_f$ is an
    isomorphism, because in that case $\mathcal{F}_T$ is just
    the restriction of $\mathcal{F}_{T'}$ to $T$.
    
    \medskip\noindent
    Conversely, given Zariski sheaves $\mathcal{F}_T$ for every object
    $(U, T, \delta)$ of $\mathcal{C}$ and comparison maps
    $c_f$ as above which (a) are isomorphisms for open immersions, and (b)
    satisfy a suitable cocycle condition, we obtain a sheaf on
    $\mathcal{C}$. This is proved exactly as in
    Topologies, Lemma \ref{topologies-lemma-characterize-sheaf-big}.
    
    \medskip\noindent
    The {\it structure sheaf} on $\mathcal{C}$ is the sheaf
    $\mathcal{O}_{X/S}$ defined by the rule
    $$
    \mathcal{O}_{X/S} :
    (U, T, \delta)
    \longmapsto
    \Gamma(T, \mathcal{O}_T)
    $$
    This is a sheaf by the definition of coverings in $\mathcal{C}$.
    Suppose that $\mathcal{F}$ is a sheaf of $\mathcal{O}_{X/S}$-modules.
    In this case the comparison mappings (\ref{equation-comparison})
    define a comparison map
    \begin{equation}
    \label{equation-comparison-modules}
    c_f : f^*\mathcal{F}_T \longrightarrow \mathcal{F}_{T'}
    \end{equation}
    of $\mathcal{O}_T$-modules.
    
    \medskip\noindent
    Another type of example comes by starting with a sheaf
    $\mathcal{G}$ on $(\Sch/X)_{Zar}$ or $X_{Zar}$ (depending on whether
    $\mathcal{C} = \text{CRIS}(X/S)$ or $\mathcal{C} = \text{Cris}(X/S)$).
    Then $\underline{\mathcal{G}}$ defined by the rule
    $$
    \underline{\mathcal{G}} :
    (U, T, \delta)
    \longmapsto
    \mathcal{G}(U)
    $$
    is a sheaf on $\mathcal{C}$. In particular, if we take
    $\mathcal{G} = \mathbf{G}_a = \mathcal{O}_X$, then we obtain
    $$
    \underline{\mathbf{G}_a} :
    (U, T, \delta)
    \longmapsto
    \Gamma(U, \mathcal{O}_U)
    $$
    There is a surjective map of sheaves
    $\mathcal{O}_{X/S} \to \underline{\mathbf{G}_a}$ defined by the
    canonical maps $\Gamma(T, \mathcal{O}_T) \to \Gamma(U, \mathcal{O}_U)$
    for objects $(U, T, \delta)$. The kernel of this map is denoted
    $\mathcal{J}_{X/S}$, hence a short exact sequence
    $$
    0 \to
    \mathcal{J}_{X/S} \to
    \mathcal{O}_{X/S} \to
    \underline{\mathbf{G}_a} \to 0
    $$
    Note that $\mathcal{J}_{X/S}$ comes equipped with a canonical
    divided power structure. After all, for each object $(U, T, \delta)$
    the third component $\delta$ {\it is} a divided power structure on the
    kernel of $\mathcal{O}_T \to \mathcal{O}_U$. Hence the (big)
    crystalline topos is a divided power topos.

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