The Stacks Project


Tag 07MS

Chapter 54: Crystalline Cohomology > Section 54.24: Some further results

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 \begin{equation} \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}) \end{equation} 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
    \begin{equation}
    \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})
    \end{equation}
    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}

    Comments (0)

    There are no comments yet for this tag.

    There are also 2 comments on Section 54.24: Crystalline Cohomology.

    Add a comment on tag 07MS

    Your email address will not be published. Required fields are marked.

    In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).

    All contributions are licensed under the GNU Free Documentation License.




    In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

    This captcha seems more appropriate than the usual illegible gibberish, right?