## 60.9 The crystalline site

Since (60.8.1.1) commutes with products and fibre products, we see that looking at those $(U, T, \delta )$ such that $U \to X$ is an open immersion defines a full subcategory preserved under fibre products (and more generally finite nonempty limits). Hence the following definition makes sense.

Definition 60.9.1. In Situation 60.7.5.

The (small) *crystalline site* of $X$ over $(S, \mathcal{I}, \gamma )$, denoted $\text{Cris}(X/S, \mathcal{I}, \gamma )$ or simply $\text{Cris}(X/S)$ is the full subcategory of $\text{CRIS}(X/S)$ consisting of those $(U, T, \delta )$ in $\text{CRIS}(X/S)$ such that $U \to X$ is an open immersion. It comes endowed with the Zariski topology.

The topos of sheaves on $\text{Cris}(X/S)$ is denoted $(X/S)_{\text{cris}}$ or sometimes $(X/S, \mathcal{I}, \gamma )_{\text{cris}}$^{1}.

For any $(U, T, \delta )$ in $\text{Cris}(X/S)$ the morphism $U \to X$ defines an object of the small Zariski site $X_{Zar}$ of $X$. Hence a canonical forgetful functor

60.9.1.1
\begin{equation} \label{crystalline-equation-forget-small} \text{Cris}(X/S) \longrightarrow X_{Zar},\quad (U, T, \delta ) \longmapsto U \end{equation}

and a left adjoint

60.9.1.2
\begin{equation} \label{crystalline-equation-endow-trivial-small} X_{Zar} \longrightarrow \text{Cris}(X/S),\quad U \longmapsto (U, U, \emptyset ) \end{equation}

which is sometimes useful.

We can compare the small and big crystalline sites, just like we can compare the small and big Zariski sites of a scheme, see Topologies, Lemma 34.3.14.

Lemma 60.9.2. Assumptions as in Definition 60.8.1. The inclusion functor

\[ \text{Cris}(X/S) \to \text{CRIS}(X/S) \]

commutes with finite nonempty limits, is fully faithful, continuous, and cocontinuous. There are morphisms of topoi

\[ (X/S)_{\text{cris}} \xrightarrow {i} (X/S)_{\text{CRIS}} \xrightarrow {\pi } (X/S)_{\text{cris}} \]

whose composition is the identity and of which the first is induced by the inclusion functor. Moreover, $\pi _* = i^{-1}$.

**Proof.**
For the first assertion see Lemma 60.8.2. This gives us a morphism of topoi $i : (X/S)_{\text{cris}} \to (X/S)_{\text{CRIS}}$ and a left adjoint $i_!$ such that $i^{-1}i_! = i^{-1}i_* = \text{id}$, see Sites, Lemmas 7.21.5, 7.21.6, and 7.21.7. We claim that $i_!$ is exact. If this is true, then we can define $\pi $ by the rules $\pi ^{-1} = i_!$ and $\pi _* = i^{-1}$ and everything is clear. To prove the claim, note that we already know that $i_!$ is right exact and preserves fibre products (see references given). Hence it suffices to show that $i_! * = *$ where $*$ indicates the final object in the category of sheaves of sets. To see this it suffices to produce a set of objects $(U_ i, T_ i, \delta _ i)$, $i \in I$ of $\text{Cris}(X/S)$ such that

\[ \coprod \nolimits _{i \in I} h_{(U_ i, T_ i, \delta _ i)} \to * \]

is surjective in $(X/S)_{\text{CRIS}}$ (details omitted; hint: use that $\text{Cris}(X/S)$ has products and that the functor $\text{Cris}(X/S) \to \text{CRIS}(X/S)$ commutes with them). In the affine case this follows from Lemma 60.5.6. We omit the proof in general.
$\square$

Lemma 60.9.5. In Situation 60.7.5. Let $X' \subset X$ and $S' \subset S$ be open subschemes such that $X'$ maps into $S'$. Then there is a fully faithful functor $\text{Cris}(X'/S') \to \text{Cris}(X/S)$ which gives rise to a morphism of topoi fitting into the commutative diagram

\[ \xymatrix{ (X'/S')_{\text{cris}} \ar[r] \ar[d]_{u_{X'/S'}} & (X/S)_{\text{cris}} \ar[d]^{u_{X/S}} \\ \mathop{\mathit{Sh}}\nolimits (X'_{Zar}) \ar[r] & \mathop{\mathit{Sh}}\nolimits (X_{Zar}) } \]

Moreover, this diagram is an example of localization of morphisms of topoi as in Sites, Lemma 7.31.1.

**Proof.**
The fully faithful functor comes from thinking of objects of $\text{Cris}(X'/S')$ as divided power thickenings $(U, T, \delta )$ of $X$ where $U \to X$ factors through $X' \subset X$ (since then automatically $T \to S$ will factor through $S'$). This functor is clearly cocontinuous hence we obtain a morphism of topoi as indicated. Let $h_{X'} \in \mathop{\mathit{Sh}}\nolimits (X_{Zar})$ be the representable sheaf associated to $X'$ viewed as an object of $X_{Zar}$. It is clear that $\mathop{\mathit{Sh}}\nolimits (X'_{Zar})$ is the localization $\mathop{\mathit{Sh}}\nolimits (X_{Zar})/h_{X'}$. On the other hand, the category $\text{Cris}(X/S)/u_{X/S}^{-1}h_{X'}$ (see Sites, Lemma 7.30.3) is canonically identified with $\text{Cris}(X'/S')$ by the functor above. This finishes the proof.
$\square$

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