Remark 60.24.8 (Base change map). In the situation of Remark 60.24.1 assume S = \mathop{\mathrm{Spec}}(A) and S' = \mathop{\mathrm{Spec}}(A') are affine. Let \mathcal{F}' be an \mathcal{O}_{X'/S'}-module. Let \mathcal{F} be the pullback of \mathcal{F}'. Then there is a canonical base change map
L(S' \to S)^*R\tau _{X'/S', *}\mathcal{F}' \longrightarrow R\tau _{X/S, *}\mathcal{F}
where \tau _{X/S} and \tau _{X'/S'} are the structure morphisms, see Remark 60.9.6. On global sections this gives a base change map
60.24.8.1
\begin{equation} \label{crystalline-equation-base-change-map} R\Gamma (\text{Cris}(X'/S'), \mathcal{F}') \otimes ^\mathbf {L}_{A'} A \longrightarrow R\Gamma (\text{Cris}(X/S), \mathcal{F}) \end{equation}
in D(A).
Hint: Compose the very general base change map of Cohomology on Sites, Remark 21.19.3 with the canonical map Lf_{\text{cris}}^*\mathcal{F}' \to f_{\text{cris}}^*\mathcal{F}' = \mathcal{F}.
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