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.
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Comment #2478 by Daniel Litt on
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