## 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.

1. 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.

2. 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
$$\label{crystalline-equation-forget-small} \text{Cris}(X/S) \longrightarrow X_{Zar},\quad (U, T, \delta ) \longmapsto U$$

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

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. $\square$

Remark 60.9.3 (Functoriality). Let $p$ be a prime number. Let $(S, \mathcal{I}, \gamma ) \to (S', \mathcal{I}', \gamma ')$ be a morphism of divided power schemes over $\mathbf{Z}_{(p)}$. Let

$\xymatrix{ X \ar[r]_ f \ar[d] & Y \ar[d] \\ S_0 \ar[r] & S'_0 }$

be a commutative diagram of morphisms of schemes and assume $p$ is locally nilpotent on $X$ and $Y$. By analogy with Topologies, Lemma 34.3.17 we define

$f_{\text{cris}} : (X/S)_{\text{cris}} \longrightarrow (Y/S')_{\text{cris}}$

by the formula $f_{\text{cris}} = \pi _ Y \circ f_{\text{CRIS}} \circ i_ X$ where $i_ X$ and $\pi _ Y$ are as in Lemma 60.9.2 for $X$ and $Y$ and where $f_{\text{CRIS}}$ is as in Remark 60.8.5.

Remark 60.9.4 (Comparison with Zariski site). In Situation 60.7.5. The functor (60.9.1.1) is continuous, cocontinuous, and commutes with products and fibred products. Hence we obtain a morphism of topoi

$u_{X/S} : (X/S)_{\text{cris}} \longrightarrow \mathop{\mathit{Sh}}\nolimits (X_{Zar})$

relating the small crystalline topos of $X/S$ with the small Zariski topos of $X$. See Sites, Section 7.21.

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. $\square$

Remark 60.9.6 (Structure morphism). In Situation 60.7.5. Consider the closed subscheme $S_0 = V(\mathcal{I}) \subset S$. If we assume that $p$ is locally nilpotent on $S_0$ (which is always the case in practice) then we obtain a situation as in Definition 60.8.1 with $S_0$ instead of $X$. Hence we get a site $\text{Cris}(S_0/S)$. If $f : X \to S_0$ is the structure morphism of $X$ over $S$, then we get a commutative diagram of ringed topoi

$\xymatrix{ (X/S)_{\text{cris}} \ar[r]_{f_{\text{cris}}} \ar[d]_{u_{X/S}} & (S_0/S)_{\text{cris}} \ar[d]^{u_{S_0/S}} \\ \mathop{\mathit{Sh}}\nolimits (X_{Zar}) \ar[r]^{f_{small}} & \mathop{\mathit{Sh}}\nolimits (S_{0, Zar}) \ar[rd] \\ & & \mathop{\mathit{Sh}}\nolimits (S_{Zar}) }$

see Remark 60.9.3. We think of the composition $(X/S)_{\text{cris}} \to \mathop{\mathit{Sh}}\nolimits (S_{Zar})$ as the structure morphism of the crystalline site. Even if $p$ is not locally nilpotent on $S_0$ the structure morphism

$\tau _{X/S} : (X/S)_{\text{cris}} \longrightarrow \mathop{\mathit{Sh}}\nolimits (S_{Zar})$

is defined as we can take the lower route through the diagram above.

Remark 60.9.7 (Compatibilities). The morphisms defined above satisfy numerous compatibilities. For example, in the situation of Remark 60.9.3 we obtain a commutative diagram of ringed topoi

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

where the vertical arrows are the structure morphisms.

[1] This clashes with our convention to denote the topos associated to a site $\mathcal{C}$ by $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$.

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