The Stacks project

Remark 60.24.1 (Higher direct images). 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

be a commutative diagram of morphisms of schemes and assume $p$ is locally nilpotent on $X$ and $X'$. Let $\mathcal{F}$ be an $\mathcal{O}_{X/S}$-module on $\text{Cris}(X/S)$. Then $Rf_{\text{cris}, *}\mathcal{F}$ can be computed as follows.

Given an object $(U', T', \delta ')$ of $\text{Cris}(X'/S')$ set $U = X \times _{X'} U' = f^{-1}(U')$ (an open subscheme of $X$). Denote $(T_0, T, \delta )$ the divided power scheme over $S$ such that

is cartesian in the category of divided power schemes, see Lemma 60.7.4. There is an induced morphism $U \to T_0$ and we obtain a morphism $(U/T)_{\text{cris}} \to (X/S)_{\text{cris}}$, see Remark 60.9.3. Let $\mathcal{F}_ U$ be the pullback of $\mathcal{F}$. Let $\tau _{U/T} : (U/T)_{\text{cris}} \to T_{Zar}$ be the structure morphism. Then we have

where the left hand side is the restriction (see Section 60.10).

Hints: First, show that $\text{Cris}(U/T)$ is the localization (in the sense of Sites, Lemma 7.30.3) of $\text{Cris}(X/S)$ at the sheaf of sets $f_{\text{cris}}^{-1}h_{(U', T', \delta ')}$. Next, reduce the statement to the case where $\mathcal{F}$ is an injective module and pushforward of modules using that the pullback of an injective $\mathcal{O}_{X/S}$-module is an injective $\mathcal{O}_{U/T}$-module on $\text{Cris}(U/T)$. Finally, check the result holds for plain pushforward.