# The Stacks Project

## Tag 07IN

### 54.10. Sheaves on the crystalline site

Notation and assumptions as in Situation 54.7.5. 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 $$\tag{54.10.0.1} c_f : f^{-1}\mathcal{F}_{T'} \longrightarrow \mathcal{F}_T.$$ 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 (54.10.0.1). 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 (54.10.0.1) define a comparison map $$\tag{54.10.0.2} c_f : f^*\mathcal{F}_T \longrightarrow \mathcal{F}_{T'}$$ 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

\label{equation-comparison}
c_f : f^{-1}\mathcal{F}_{T'} \longrightarrow \mathcal{F}_T.

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

\label{equation-comparison-modules}
c_f : f^*\mathcal{F}_T \longrightarrow \mathcal{F}_{T'}

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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