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Chapter 51: Crystalline Cohomology > Section 51.24: Some further results

Remark 51.24.2 (Mayer-Vietoris). In the situation of Remark 51.24.1 suppose we have an open covering $X = X' \cup X''$. Denote $X''' = X' \cap X''$. Let $f'$, $f''$, and $f''$ be the restriction of $f$ to $X'$, $X''$, and $X'''$. Moreover, Let $\mathcal{F}'$, $\mathcal{F}''$, and $\mathcal{F}'''$ be the restriction of $\mathcal{F}$ to the crystalline sites of $X'$, $X''$, and $X'''$. Then there exists a distinguished triangle $$ Rf_{\text{cris}, *}\mathcal{F} \longrightarrow Rf'_{\text{cris}, *}\mathcal{F}' \oplus Rf''_{\text{cris}, *}\mathcal{F}'' \longrightarrow Rf'''_{\text{cris}, *}\mathcal{F}''' \longrightarrow Rf_{\text{cris}, *}\mathcal{F}[1] $$ in $D(\mathcal{O}_{X'/S'})$.

Hints: This is a formal consequence of the fact that the subcategories $\text{Cris}(X'/S)$, $\text{Cris}(X''/S)$, $\text{Cris}(X'''/S)$ correspond to open subobjects of the final sheaf on $\text{Cris}(X/S)$ and that the last is the intersection of the first two.

    The code snippet corresponding to this tag is a part of the file crystalline.tex and is located in lines 4446–4470 (see updates for more information).

    \begin{remark}[Mayer-Vietoris]
    \label{remark-mayer-vietoris}
    In the situation of Remark \ref{remark-compute-direct-image}
    suppose we have an open covering $X = X' \cup X''$. Denote
    $X''' = X' \cap X''$. Let $f'$, $f''$, and $f''$ be the restriction of $f$
    to $X'$, $X''$, and $X'''$. Moreover, Let $\mathcal{F}'$, $\mathcal{F}''$,
    and $\mathcal{F}'''$ be the restriction of $\mathcal{F}$ to the crystalline
    sites of $X'$, $X''$, and $X'''$. Then there exists a distinguished triangle
    $$
    Rf_{\text{cris}, *}\mathcal{F}
    \longrightarrow
    Rf'_{\text{cris}, *}\mathcal{F}' \oplus Rf''_{\text{cris}, *}\mathcal{F}''
    \longrightarrow
    Rf'''_{\text{cris}, *}\mathcal{F}'''
    \longrightarrow
    Rf_{\text{cris}, *}\mathcal{F}[1]
    $$
    in $D(\mathcal{O}_{X'/S'})$.
    
    \medskip\noindent
    Hints: This is a formal consequence of the fact that the subcategories
    $\text{Cris}(X'/S)$, $\text{Cris}(X''/S)$, $\text{Cris}(X'''/S)$ correspond
    to open subobjects of the final sheaf on $\text{Cris}(X/S)$ and that the
    last is the intersection of the first two.
    \end{remark}

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