# The Stacks Project

## Tag 07MS

Remark 54.24.8 (Base change map). In the situation of Remark 54.24.1 assume $S = \mathop{\rm Spec}(A)$ and $S' = \mathop{\rm Spec}(A')$ 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 54.9.6. On global sections this gives a base change map $$\tag{54.24.8.1} R\Gamma(\text{Cris}(X'/S'), \mathcal{F}') \otimes^\mathbf{L}_{A'} A \longrightarrow R\Gamma(\text{Cris}(X/S), \mathcal{F})$$ in $D(A)$.

Hint: Compose the very general base change map of Cohomology on Sites, Remark 21.20.3 with the canonical map $Lf_{\text{cris}}^*\mathcal{F}' \to f_{\text{cris}}^*\mathcal{F}' = \mathcal{F}$.

The code snippet corresponding to this tag is a part of the file crystalline.tex and is located in lines 4709–4738 (see updates for more information).

\begin{remark}[Base change map]
\label{remark-base-change}
In the situation of Remark \ref{remark-compute-direct-image}
assume $S = \Spec(A)$ and $S' = \Spec(A')$ 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 \ref{remark-structure-morphism}. On global sections this
gives a base change map

\label{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})

in $D(A)$.

\medskip\noindent
Hint: Compose the very general base change map of
Cohomology on Sites, Remark \ref{sites-cohomology-remark-base-change}
with the canonical map
$Lf_{\text{cris}}^*\mathcal{F}' \to f_{\text{cris}}^*\mathcal{F}' = \mathcal{F}$.
\end{remark}

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