Lemma 60.18.2. In Situation 60.7.5. Assume moreover $X$ and $S$ are affine schemes. Consider the full subcategory $\mathcal{C} \subset \text{Cris}(X/S)$ consisting of divided power thickenings $(X, T, \delta )$ endowed with the chaotic topology (see Sites, Example 7.6.6). For any locally quasi-coherent $\mathcal{O}_{X/S}$-module $\mathcal{F}$ we have

$R\Gamma (\mathcal{C}, \mathcal{F}|_\mathcal {C}) = R\Gamma (\text{Cris}(X/S), \mathcal{F})$

Proof. Denote $\text{AffineCris}(X/S)$ the fully subcategory of $\text{Cris}(X/S)$ consisting of those objects $(U, T, \delta )$ with $U$ and $T$ affine. We turn this into a site by saying a family of morphisms $\{ (U_ i, T_ i, \delta _ i) \to (U, T, \delta )\} _{i \in I}$ of $\text{AffineCris}(X/S)$ is a covering if and only if it is a covering of $\text{Cris}(X/S)$. With this definition the inclusion functor

$\text{AffineCris}(X/S) \longrightarrow \text{Cris}(X/S)$

is a special cocontinuous functor as defined in Sites, Definition 7.29.2. The proof of this is exactly the same as the proof of Topologies, Lemma 34.3.10. Thus we see that the topos of sheaves on $\text{Cris}(X/S)$ is the same as the topos of sheaves on $\text{AffineCris}(X/S)$ via restriction by the displayed inclusion functor. Therefore we have to prove the corresponding statement for the inclusion $\mathcal{C} \subset \text{AffineCris}(X/S)$.

We will use without further mention that $\mathcal{C}$ and $\text{AffineCris}(X/S)$ have products and fibre products (details omitted, see Lemma 60.8.2). The inclusion functor $u : \mathcal{C} \to \text{AffineCris}(X/S)$ is fully faithful, continuous, and commutes with products and fibre products. We claim it defines a morphism of ringed sites

$f : (\text{AffineCris}(X/S), \mathcal{O}_{X/S}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_{X/S}|_\mathcal {C})$

To see this we will use Sites, Lemma 7.14.6. Note that $\mathcal{C}$ has fibre products and $u$ commutes with them so the categories $\mathcal{I}^ u_{(U, T, \delta )}$ are disjoint unions of directed categories (by Sites, Lemma 7.5.1 and Categories, Lemma 4.19.8). Hence it suffices to show that $\mathcal{I}^ u_{(U, T, \delta )}$ is connected. Nonempty follows from Lemma 60.5.6: since $U$ and $T$ are affine that lemma says there is at least one object $(X, T', \delta ')$ of $\mathcal{C}$ and a morphism $(U, T, \delta ) \to (X, T', \delta ')$ of divided power thickenings. Connectedness follows from the fact that $\mathcal{C}$ has products and that $u$ commutes with them (compare with the proof of Sites, Lemma 7.5.2).

Note that $f_*\mathcal{F} = \mathcal{F}|_\mathcal {C}$. Hence the lemma follows if $R^ pf_*\mathcal{F} = 0$ for $p > 0$, see Cohomology on Sites, Lemma 21.14.6. By Cohomology on Sites, Lemma 21.7.4 it suffices to show that $H^ p(\text{AffineCris}(X/S)/(X, T, \delta ), \mathcal{F}) = 0$ for all $(X, T, \delta )$. This follows from Lemma 60.18.1 because the topos of the site $\text{AffineCris}(X/S)/(X, T, \delta )$ is equivalent to the topos of the site $\text{Cris}(X/S)/(X, T, \delta )$ used in the lemma. $\square$

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