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60.10 Sheaves on the crystalline site

Notation and assumptions as in Situation 60.7.5. 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} \label{crystalline-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 6.21 and especially Sheaves, Definition 6.21.7). Thus a map $c_ f$ as in ( 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 34.3.20.

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 ( define a comparison map
\begin{equation} \label{crystalline-equation-comparison-modules} 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 $(\mathit{Sch}/X)_{Zar}$ 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.

Comments (4)

Comment #4066 by Dan Dore on

Maybe it's a good idea to remind the reader here of Definition 7.45.1, defining global sections for a site without a terminal object, then spelling out what this means for the cristalline site?

Comment #4143 by on

This whole chapter needs a thorough revision. For the moment I'll only fix mathematical errors. But yes, I do agree it is a good idea and hopefully your comment will help others who visit this page. Thank you.

Comment #5434 by Hao on

A small typo: In 07IQ we should interchange and .

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