The Stacks project

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
\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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