## 59.8 The big crystalline site

We first define the big site. Given a divided power scheme $(S, \mathcal{I}, \gamma )$ we say $(T, \mathcal{J}, \delta )$ is a divided power scheme over $(S, \mathcal{I}, \gamma )$ if $T$ comes endowed with a morphism $T \to S$ of divided power schemes. Similarly, we say a divided power thickening $(U, T, \delta )$ is a divided power thickening over $(S, \mathcal{I}, \gamma )$ if $T$ comes endowed with a morphism $T \to S$ of divided power schemes.

Definition 59.8.1. In Situation 59.7.5.

1. A divided power thickening of $X$ relative to $(S, \mathcal{I}, \gamma )$ is given by a divided power thickening $(U, T, \delta )$ over $(S, \mathcal{I}, \gamma )$ and an $S$-morphism $U \to X$.

2. A morphism of divided power thickenings of $X$ relative to $(S, \mathcal{I}, \gamma )$ is defined in the obvious manner.

The category of divided power thickenings of $X$ relative to $(S, \mathcal{I}, \gamma )$ is denoted $\text{CRIS}(X/S, \mathcal{I}, \gamma )$ or simply $\text{CRIS}(X/S)$.

For any $(U, T, \delta )$ in $\text{CRIS}(X/S)$ we have that $p$ is locally nilpotent on $T$, see discussion after Definition 59.7.3. A good way to visualize all the data associated to $(U, T, \delta )$ is the commutative diagram

$\xymatrix{ T \ar[dd] & U \ar[l] \ar[d] \\ & X \ar[d] \\ S & S_0 \ar[l] }$

where $S_0 = V(\mathcal{I}) \subset S$. Morphisms of $\text{CRIS}(X/S)$ can be similarly visualized as huge commutative diagrams. In particular, there is a canonical forgetful functor

59.8.1.1
$$\label{crystalline-equation-forget} \text{CRIS}(X/S) \longrightarrow \mathit{Sch}/X,\quad (U, T, \delta ) \longmapsto U$$

as well as its one sided inverse (and left adjoint)

59.8.1.2
$$\label{crystalline-equation-endow-trivial} \mathit{Sch}/X \longrightarrow \text{CRIS}(X/S),\quad U \longmapsto (U, U, \emptyset )$$

which is sometimes useful.

Lemma 59.8.2. In Situation 59.7.5. The category $\text{CRIS}(X/S)$ has all finite nonempty limits, in particular products of pairs and fibre products. The functor (59.8.1.1) commutes with limits.

Proof. Omitted. Hint: See Lemma 59.5.3 for the affine case. See also Divided Power Algebra, Remark 23.3.5. $\square$

Lemma 59.8.3. In Situation 59.7.5. Let

$\xymatrix{ (U_3, T_3, \delta _3) \ar[d] \ar[r] & (U_2, T_2, \delta _2) \ar[d] \\ (U_1, T_1, \delta _1) \ar[r] & (U, T, \delta ) }$

be a fibre square in the category of divided power thickenings of $X$ relative to $(S, \mathcal{I}, \gamma )$. If $T_2 \to T$ is flat and $U_2 = T_2 \times _ T U$, then $T_3 = T_1 \times _ T T_2$ (as schemes).

Proof. This is true because a divided power structure extends uniquely along a flat ring map. See Divided Power Algebra, Lemma 23.4.2. $\square$

The lemma above means that the base change of a flat morphism of divided power thickenings is another flat morphism, and in fact is the “usual” base change of the morphism. This implies that the following definition makes sense.

Definition 59.8.4. In Situation 59.7.5.

1. A family of morphisms $\{ (U_ i, T_ i, \delta _ i) \to (U, T, \delta )\}$ of divided power thickenings of $X/S$ is a Zariski, étale, smooth, syntomic, or fppf covering if and only if

1. $U_ i = U \times _ T T_ i$ for all $i$ and

2. $\{ T_ i \to T\}$ is a Zariski, étale, smooth, syntomic, or fppf covering.

2. The big crystalline site of $X$ over $(S, \mathcal{I}, \gamma )$, is the category $\text{CRIS}(X/S)$ endowed with the Zariski topology.

3. 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.

There are some obvious functorialities concerning these topoi.

Remark 59.8.5 (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 $\mathbf{Z}_{(p)}$. Set $S_0 = V(\mathcal{I})$ and $S'_0 = V(\mathcal{I}')$. 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 $Y$. Then we get a continuous and cocontinuous functor

$\text{CRIS}(X/S) \longrightarrow \text{CRIS}(Y/S')$

by letting $(U, T, \delta )$ correspond to $(U, T, \delta )$ with $U \to X \to Y$ as the $S'$-morphism from $U$ to $Y$. Hence we get a morphism of topoi

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

see Sites, Section 7.21.

Remark 59.8.6 (Comparison with Zariski site). In Situation 59.7.5. The functor (59.8.1.1) is cocontinuous (details omitted) and commutes with products and fibred products (Lemma 59.8.2). Hence we obtain a morphism of topoi

$U_{X/S} : (X/S)_{\text{CRIS}} \longrightarrow \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/X)_{Zar})$

from the big crystalline topos of $X/S$ to the big Zariski topos of $X$. See Sites, Section 7.21.

Remark 59.8.7 (Structure morphism). In Situation 59.7.5. 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 59.8.1 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 morphisms of ringed topoi

$\xymatrix{ (X/S)_{\text{CRIS}} \ar[r]_{f_{\text{CRIS}}} \ar[d]_{U_{X/S}} & (S_0/S)_{\text{CRIS}} \ar[d]^{U_{S_0/S}} \\ \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/X)_{Zar}) \ar[r]^{f_{big}} & \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S_0)_{Zar}) \ar[rd] \\ & & \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{Zar}) }$

by Remark 59.8.5. We think of the composition $(X/S)_{\text{CRIS}} \to \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{Zar})$ as the structure morphism of the big crystalline site. Even if $p$ is not locally nilpotent on $S_0$ the structure morphism

$(X/S)_{\text{CRIS}} \longrightarrow \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{Zar})$

is defined as we can take the lower route through the diagram above. Thus it is the morphism of topoi corresponding to the cocontinuous functor $\text{CRIS}(X/S) \to (\mathit{Sch}/S)_{Zar}$ given by the rule $(U, T, \delta )/S \mapsto U/S$, see Sites, Section 7.21.

Remark 59.8.8 (Compatibilities). The morphisms defined above satisfy numerous compatibilities. For example, in the situation of Remark 59.8.5 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 $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$.

Comment #6265 by Mingchen on

On the second line after 07I6, I think you mean discussions after Lemma 07ME, instead of after Definition 07I4.

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