The Stacks project

Proof. As $U \to T$ is a thickening we see that $U$ is affine if and only if $T$ is affine, see Limits, Lemma 32.11.1. Having said this, let us apply Cohomology on Sites, Lemma 21.10.9 to the collection $\mathcal{B}$ of affine objects $(U, T, \delta )$ and the collection $\text{Cov}$ of affine open coverings $\mathcal{U} = \{ (U_ i, T_ i, \delta _ i) \to (U, T, \delta )\} $. The Čech complex ${\check C}^*(\mathcal{U}, \mathcal{F})$ for such a covering is simply the Čech complex of the quasi-coherent $\mathcal{O}_ T$-module $\mathcal{F}_ T$ (here we are using the assumption that $\mathcal{F}$ is locally quasi-coherent) with respect to the affine open covering $\{ T_ i \to T\} $ of the affine scheme $T$. Hence the Čech cohomology is zero by Cohomology of Schemes, Lemma 30.2.6 and 30.2.2. Thus the hypothesis of Cohomology on Sites, Lemma 21.10.9 are satisfied and we win. $\square$

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

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

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. $\square$

Proof. Any functor between categories defines a morphism between chaotic topoi in the same direction, for example because such a functor can be considered as a cocontinuous functor between sites, see Sites, Section 7.21. Proof of the description of $g_*\mathcal{F}$ is omitted. Note that in the statement we take $(B_ e \to C, \delta )$ is an object of $\text{Cris}(C/A)$ only for $e$ large enough. Let $\mathcal{I}$ be an injective abelian sheaf on $\mathcal{C}$. Then the transition maps

are surjective as the morphisms

are monomorphisms in the category $\mathcal{C}$. Hence for an injective abelian sheaf both sides of the displayed formula of the lemma agree. Taking an injective resolution of $\mathcal{F}$ one easily obtains the result (sheaves are presheaves, so exactness is measured on the level of groups of sections over objects). $\square$

Proof. Note that $H^ q(X^ p, \mathcal{F}) = 0$ for all $q > 0$ as any presheaf is a sheaf on $\mathcal{C}$. The assumption on $X$ is that $h_ X \to *$ is surjective. Using that $H^ q(X, \mathcal{F}) = H^ q(h_ X, \mathcal{F})$ and $H^ q(\mathcal{C}, \mathcal{F}) = H^ q(*, \mathcal{F})$ we see that our statement is a special case of Cohomology on Sites, Lemma 21.13.2. $\square$