The Stacks project

Lemma 60.18.3. In Situation 60.5.1. Set $\mathcal{C} = (\text{Cris}(C/A))^{opp}$ and $\mathcal{C}^\wedge = (\text{Cris}^\wedge (C/A))^{opp}$ endowed with the chaotic topology, see Remark 60.5.4 for notation. There is a morphism of topoi

\[ g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{C}^\wedge ) \]

such that if $\mathcal{F}$ is a sheaf of abelian groups on $\mathcal{C}$, then

\[ R^ pg_*\mathcal{F}(B \to C, \delta ) = \left\{ \begin{matrix} \mathop{\mathrm{lim}}\nolimits _ e \mathcal{F}(B_ e \to C, \delta ) & \text{if }p = 0 \\ R^1\mathop{\mathrm{lim}}\nolimits _ e \mathcal{F}(B_ e \to C, \delta ) & \text{if }p = 1 \\ 0 & \text{else} \end{matrix} \right. \]

where $B_ e = B/p^ eB$ for $e \gg 0$.

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

\[ \mathcal{I}(B_ e \to C, \delta ) \leftarrow \mathcal{I}(B_{e + 1} \to C, \delta ) \]

are surjective as the morphisms

\[ (B_ e \to C, \delta ) \longrightarrow (B_{e + 1} \to C, \delta ) \]

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). $\square$

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